\(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx\) [65]

Optimal result

Integrand size = 31, antiderivative size = 1744 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\frac {2 b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{3 g^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2}}{g^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2}}{4 g^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 g \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2}}{9 g^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{g^5 (1-c x) (1+c x)}+\frac {b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{g^5}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{4 g^2}-\frac {2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 g^3}-\frac {c^2 d^2 x^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 g}+\frac {c d^2 f \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b g^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b g^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c g^6 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c g^4 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}-\frac {a d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2} \text {arctanh}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {-1+c^2 x^2}}\right )}{g^6 (1-c x) (1+c x)}+\frac {b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt {d-c^2 d x^2} \text {arccosh}(c x) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt {d-c^2 d x^2} \text {arccosh}(c x) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

-a*d^2*(c^2*f^2-g^2)^(5/2)*arctanh((c^2*f*x+g)/(c^2*f^2-g^2)^(1/2)/(c^2*x^2-1)^(1/2))*(c^2*x^2-1)^(1/2)*(-c^2*
d*x^2+d)^(1/2)/g^6/(-c*x+1)/(c*x+1)+a*d^2*(c^2*f^2-g^2)^2*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/g^5/(-c*x+1)/(c*x+
1)-b*c*d^2*(c^2*f^2-g^2)^2*x*(-c^2*d*x^2+d)^(1/2)/g^5/(c*x-1)^(1/2)/(c*x+1)^(1/2)+b*d^2*(c^2*f^2-g^2)^(5/2)*po
lylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))*(-c^2*d*x^2+d)^(1/2)/g^6/(c*x-1)^(1/2)
/(c*x+1)^(1/2)-b*d^2*(c^2*f^2-g^2)^(5/2)*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/
2)))*(-c^2*d*x^2+d)^(1/2)/g^6/(c*x-1)^(1/2)/(c*x+1)^(1/2)+b*d^2*(c^2*f^2-g^2)^(5/2)*arccosh(c*x)*ln(1+(c*x+(c*
x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))*(-c^2*d*x^2+d)^(1/2)/g^6/(c*x-1)^(1/2)/(c*x+1)^(1/2)-b*
d^2*(c^2*f^2-g^2)^(5/2)*arccosh(c*x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))*(-c^2
*d*x^2+d)^(1/2)/g^6/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/2*c^2*d^2*f*(c^2*f^2-2*g^2)*x*(a+b*arccosh(c*x))*(-c^2*d*x^2
+d)^(1/2)/g^4-1/5*c^2*d^2*x^2*(-c*x+1)*(c*x+1)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/g+2/15*b*c*d^2*x*(-c^2*
d*x^2+d)^(1/2)/g/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/45*b*c^3*d^2*x^3*(-c^2*d*x^2+d)^(1/2)/g/(c*x-1)^(1/2)/(c*x+1)^(
1/2)-1/25*b*c^5*d^2*x^5*(-c^2*d*x^2+d)^(1/2)/g/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/2*d^2*(c^2*f^2-g^2)^3*(a+b*arccos
h(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c/g^6/(g*x+f)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/4*b*c^3*d^2*f*(c^2*f^2-2*g^2)*x^2
*(-c^2*d*x^2+d)^(1/2)/g^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/4*c*d^2*f*(c^2*f^2-2*g^2)*(a+b*arccosh(c*x))^2*(-c^2*d
*x^2+d)^(1/2)/b/g^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/2*c*d^2*(c^2*f^2-g^2)^2*x*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d
)^(1/2)/b/g^5/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2/15*d^2*(-c*x+1)*(c*x+1)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/g+
1/8*c^2*d^2*f*x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/g^2-1/4*c^4*d^2*f*x^3*(a+b*arccosh(c*x))*(-c^2*d*x^2+d
)^(1/2)/g^2-1/3*d^2*(c^2*f^2-2*g^2)*(-c*x+1)*(c*x+1)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/g^3+b*d^2*(c^2*f^
2-g^2)^2*arccosh(c*x)*(-c^2*d*x^2+d)^(1/2)/g^5-1/2*d^2*(c^2*f^2-g^2)^2*(-c^2*x^2+1)*(a+b*arccosh(c*x))^2*(-c^2
*d*x^2+d)^(1/2)/b/c/g^4/(g*x+f)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/3*b*c*d^2*(c^2*f^2-2*g^2)*x*(-c^2*d*x^2+d)^(1/2)
/g^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/16*b*c^3*d^2*f*x^2*(-c^2*d*x^2+d)^(1/2)/g^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/9
*b*c^3*d^2*(c^2*f^2-2*g^2)*x^3*(-c^2*d*x^2+d)^(1/2)/g^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/16*b*c^5*d^2*f*x^4*(-c^2
*d*x^2+d)^(1/2)/g^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/16*c*d^2*f*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/g^2/(
c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 3.18 (sec) , antiderivative size = 1744, normalized size of antiderivative = 1.00, number of steps used = 39, number of rules used = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.032, Rules used = {5972, 5977, 5896, 5893, 30, 5915, 41, 5927, 5939, 102, 12, 75, 5923, 5976, 697, 5970, 6874, 95, 214, 1624, 1668, 739, 212, 5993, 5992, 8, 5980, 3401, 2296, 2221, 2317, 2438} \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=-\frac {b d^2 x^5 \sqrt {d-c^2 d x^2} c^5}{25 g \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 f x^4 \sqrt {d-c^2 d x^2} c^5}{16 g^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 f x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) c^4}{4 g^2}-\frac {b d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} c^3}{9 g^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 x^3 \sqrt {d-c^2 d x^2} c^3}{45 g \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} c^3}{4 g^4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b d^2 f x^2 \sqrt {d-c^2 d x^2} c^3}{16 g^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) c^2}{2 g^4}+\frac {d^2 f x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) c^2}{8 g^2}-\frac {d^2 x^2 (1-c x) (c x+1) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) c^2}{5 g}+\frac {d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 c}{4 b g^4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 c}{2 b g^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {d^2 f \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 c}{16 b g^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} c}{g^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} c}{3 g^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b d^2 x \sqrt {d-c^2 d x^2} c}{15 g \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{g^5}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) (1-c x) (c x+1) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 g^3}-\frac {2 d^2 (1-c x) (c x+1) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 g}-\frac {a d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt {c^2 x^2-1} \sqrt {d-c^2 d x^2} \text {arctanh}\left (\frac {f x c^2+g}{\sqrt {c^2 f^2-g^2} \sqrt {c^2 x^2-1}}\right )}{g^6 (1-c x) (c x+1)}+\frac {b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt {d-c^2 d x^2} \text {arccosh}(c x) \log \left (\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{g^6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt {d-c^2 d x^2} \text {arccosh}(c x) \log \left (\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}+1\right )}{g^6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {a d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{g^5 (1-c x) (c x+1)}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b g^4 \sqrt {c x-1} \sqrt {c x+1} (f+g x) c}-\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b g^6 \sqrt {c x-1} \sqrt {c x+1} (f+g x) c} \]

[In]

Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(f + g*x),x]

[Out]

(2*b*c*d^2*x*Sqrt[d - c^2*d*x^2])/(15*g*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*d^2*(c^2*f^2 - 2*g^2)*x*Sqrt[d -
c^2*d*x^2])/(3*g^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^2*(c^2*f^2 - g^2)^2*x*Sqrt[d - c^2*d*x^2])/(g^5*Sqrt
[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^3*d^2*f*x^2*Sqrt[d - c^2*d*x^2])/(16*g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c
^3*d^2*f*(c^2*f^2 - 2*g^2)*x^2*Sqrt[d - c^2*d*x^2])/(4*g^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d^2*x^3*Sqrt
[d - c^2*d*x^2])/(45*g*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^3*d^2*(c^2*f^2 - 2*g^2)*x^3*Sqrt[d - c^2*d*x^2])/(
9*g^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^5*d^2*f*x^4*Sqrt[d - c^2*d*x^2])/(16*g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*
x]) - (b*c^5*d^2*x^5*Sqrt[d - c^2*d*x^2])/(25*g*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (a*d^2*(c^2*f^2 - g^2)^2*(1 -
c^2*x^2)*Sqrt[d - c^2*d*x^2])/(g^5*(1 - c*x)*(1 + c*x)) + (b*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2]*ArcCosh
[c*x])/g^5 + (c^2*d^2*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(8*g^2) - (c^2*d^2*f*(c^2*f^2 - 2*g^2)*x*S
qrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(2*g^4) - (c^4*d^2*f*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(4
*g^2) - (2*d^2*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(15*g) - (d^2*(c^2*f^2 - 2*g^2)*(
1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(3*g^3) - (c^2*d^2*x^2*(1 - c*x)*(1 + c*x)*Sqrt[d
- c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(5*g) + (c*d^2*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(16*b*g^2*Sqr
t[-1 + c*x]*Sqrt[1 + c*x]) + (c*d^2*f*(c^2*f^2 - 2*g^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(4*b*g^4*S
qrt[-1 + c*x]*Sqrt[1 + c*x]) - (c*d^2*(c^2*f^2 - g^2)^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*g^5
*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d^2*(c^2*f^2 - g^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*c*g^6
*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(f + g*x)) - (d^2*(c^2*f^2 - g^2)^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*Arc
Cosh[c*x])^2)/(2*b*c*g^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(f + g*x)) - (a*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[-1 + c^2*
x^2]*Sqrt[d - c^2*d*x^2]*ArcTanh[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[-1 + c^2*x^2])])/(g^6*(1 - c*x)*(1 +
c*x)) + (b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]*Log[1 + (E^ArcCosh[c*x]*g)/(c*f - Sqrt[c
^2*f^2 - g^2])])/(g^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcCosh
[c*x]*Log[1 + (E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*(c^
2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(g^6*Sqr
t[-1 + c*x]*Sqrt[1 + c*x]) - (b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^ArcCosh[c*x]*g)/
(c*f + Sqrt[c^2*f^2 - g^2]))])/(g^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 75

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 102

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 1))), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 697

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 1624

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Dist[(a
+ b*x)^FracPart[m]*((c + d*x)^FracPart[m]/(a*c + b*d*x^2)^FracPart[m]), Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] &&  !Intege
rQ[m]

Rule 1668

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x)^(m + q - 1)*((a + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3401

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol]
:> Dist[2, Int[((c + d*x)^m*(E^((-I)*e + f*fz*x)/(b + (2*a*E^((-I)*e + f*fz*x))/E^(I*Pi*(k - 1/2)) - (b*E^(2*(
(-I)*e + f*fz*x)))/E^(2*I*k*Pi))))/E^(I*Pi*(k - 1/2)), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && IntegerQ[
2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 5893

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*Arc
Cosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && NeQ[n
, -1]

Rule 5896

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)], x_Symbol] :
> Simp[x*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*((a + b*ArcCosh[c*x])^n/2), x] + (-Dist[(1/2)*Simp[Sqrt[d1 + e1*x]/Sq
rt[1 + c*x]]*Simp[Sqrt[d2 + e2*x]/Sqrt[-1 + c*x]], Int[(a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]),
x], x] - Dist[b*c*(n/2)*Simp[Sqrt[d1 + e1*x]/Sqrt[1 + c*x]]*Simp[Sqrt[d2 + e2*x]/Sqrt[-1 + c*x]], Int[x*(a + b
*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] &&
GtQ[n, 0]

Rule 5915

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sy
mbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Dist[b*
(n/(2*c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Int[(1 + c*x)^(p + 1/2)*(-
1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1, c
*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]

Rule 5923

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sym
bol] :> With[{u = IntHide[x^m*(d1 + e1*x)^p*(d2 + e2*x)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c*Simp
[Sqrt[d1 + e1*x]*(Sqrt[d2 + e2*x]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]))], Int[SimplifyIntegrand[u/(Sqrt[d1 + e1*x]*S
qrt[d2 + e2*x]), x], x], x]] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && In
tegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])

Rule 5927

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*
(x_)], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*((a + b*ArcCosh[c*x])^n/(f*(m + 2))), x
] + (-Dist[(1/(m + 2))*Simp[Sqrt[d1 + e1*x]/Sqrt[1 + c*x]]*Simp[Sqrt[d2 + e2*x]/Sqrt[-1 + c*x]], Int[(f*x)^m*(
(a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] - Dist[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d1 + e1*x]
/Sqrt[1 + c*x]]*Simp[Sqrt[d2 + e2*x]/Sqrt[-1 + c*x]], Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x])
/; FreeQ[{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[n, 0] && (IGtQ[m, -2
] || EqQ[n, 1])

Rule 5939

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1
*e2*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m + 2*p + 1))), Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)
^p*(a + b*ArcCosh[c*x])^n, x], x] - Dist[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 +
e2*x)^p/(-1 + c*x)^p], Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1)
, x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IG
tQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rule 5970

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.) + (g_.)*(x_) + (h_.)*(x_)^2)^(p_.))/((d_) + (e_.)*(x_))^2
, x_Symbol] :> With[{u = IntHide[(f + g*x + h*x^2)^p/(d + e*x)^2, x]}, Dist[(a + b*ArcCosh[c*x])^n, u, x] - Di
st[b*c*n, Int[SimplifyIntegrand[u*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])), x], x], x]] /
; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[n, 0] && IGtQ[p, 0] && EqQ[e*g - 2*d*h, 0]

Rule 5972

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol]
:> Dist[(-d)^IntPart[p]*((d + e*x^2)^FracPart[p]/((-1 + c*x)^FracPart[p]*(1 + c*x)^FracPart[p])), Int[(f + g*x
)^m*(-1 + c*x)^p*(1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d
+ e, 0] && IntegerQ[p - 1/2] && IntegerQ[m]

Rule 5976

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]*((f_) + (g_.
)*(x_))^(m_), x_Symbol] :> Simp[(f + g*x)^m*(d1*d2 + e1*e2*x^2)*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[(-d1)*
d2]*(n + 1))), x] - Dist[1/(b*c*Sqrt[(-d1)*d2]*(n + 1)), Int[(d1*d2*g*m + 2*e1*e2*f*x + e1*e2*g*(m + 2)*x^2)*(
f + g*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 -
c*d1, 0] && EqQ[e2 + c*d2, 0] && ILtQ[m, 0] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0]

Rule 5977

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_)*((f_) + (g
_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n, (f +
g*x)^m*(d1 + e1*x)^(p - 1/2)*(d2 + e2*x)^(p - 1/2), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && Eq
Q[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[m] && IGtQ[p + 1/2, 0] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n,
0]

Rule 5980

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_
) + (e2_.)*(x_)]), x_Symbol] :> Dist[1/(c^(m + 1)*Sqrt[(-d1)*d2]), Subst[Int[(a + b*x)^n*(c*f + g*Cosh[x])^m,
x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2
, 0] && IntegerQ[m] && GtQ[d1, 0] && LtQ[d2, 0] && (GtQ[m, 0] || IGtQ[n, 0])

Rule 5992

Int[ArcCosh[(c_.)*(x_)]^(n_.)*(RFx_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> With[
{u = ExpandIntegrand[(d1 + e1*x)^p*(d2 + e2*x)^p*ArcCosh[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{c,
d1, e1, d2, e2}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && In
tegerQ[p - 1/2]

Rule 5993

Int[(ArcCosh[(c_.)*(x_)]*(b_.) + (a_))^(n_.)*(RFx_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_S
ymbol] :> Int[ExpandIntegrand[(d1 + e1*x)^p*(d2 + e2*x)^p, RFx*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b,
c, d1, e1, d2, e2}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] &&
IntegerQ[p - 1/2]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x)^{5/2} (1+c x)^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-\frac {c^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{g^4}+\frac {c^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{g^3}-\frac {c^4 f x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{g^2}+\frac {c^4 x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{g}+\frac {\left (-c^2 f^2+g^2\right )^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{g^4 (f+g x)}\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (c^4 d^2 f \sqrt {d-c^2 d x^2}\right ) \int x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \, dx}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \, dx}{g \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (c^2 d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \, dx}{g^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \, dx}{g^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{f+g x} \, dx}{g^4 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{4 g^2}-\frac {2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 g^3}-\frac {c^2 d^2 x^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 g}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c g^4 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}+\frac {\left (c^4 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c^5 d^2 f \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{4 g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c^5 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {-2-c^2 x^2+3 c^4 x^4}{15 c^4} \, dx}{g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c^2 d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 g^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{2 g^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int (-1+c x) (1+c x) \, dx}{3 g^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (g+2 c^2 f x+c^2 g x^2\right ) (a+b \text {arccosh}(c x))^2}{(f+g x)^2} \, dx}{2 b c g^4 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2}}{4 g^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{4 g^2}-\frac {2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 g^3}-\frac {c^2 d^2 x^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 g}+\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b g^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b g^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c g^6 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c g^4 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}+\frac {\left (c^2 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c^3 d^2 f \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{8 g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-2-c^2 x^2+3 c^4 x^4\right ) \, dx}{15 g \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right ) \, dx}{3 g^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (\frac {1}{f+g x}-\frac {c^2 \left (g x+\frac {f^2}{f+g x}\right )}{g^2}\right ) (-a-b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{g^4 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {2 b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{3 g^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2}}{4 g^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 g \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2}}{9 g^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{4 g^2}-\frac {2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 g^3}-\frac {c^2 d^2 x^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 g}+\frac {c d^2 f \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b g^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b g^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c g^6 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c g^4 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}+\frac {\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {a \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right )}{g^2 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}+\frac {b \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \text {arccosh}(c x)}{g^2 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}\right ) \, dx}{g^4 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {2 b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{3 g^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2}}{4 g^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 g \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2}}{9 g^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{4 g^2}-\frac {2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 g^3}-\frac {c^2 d^2 x^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 g}+\frac {c d^2 f \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b g^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b g^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c g^6 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c g^4 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}+\frac {\left (a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} (f+g x)} \, dx}{g^6 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x} (f+g x)} \, dx}{g^6 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {2 b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{3 g^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2}}{4 g^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 g \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2}}{9 g^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{4 g^2}-\frac {2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 g^3}-\frac {c^2 d^2 x^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 g}+\frac {c d^2 f \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b g^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b g^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c g^6 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c g^4 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}+\frac {\left (b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {c^2 g x \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c^2 f^2-g^2\right ) \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x} (f+g x)}\right ) \, dx}{g^6 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2}{(f+g x) \sqrt {-1+c^2 x^2}} \, dx}{g^6 (-1+c x) (1+c x)} \\ & = \frac {2 b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{3 g^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2}}{4 g^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 g \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2}}{9 g^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{g^5 (1-c x) (1+c x)}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{4 g^2}-\frac {2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 g^3}-\frac {c^2 d^2 x^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 g}+\frac {c d^2 f \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b g^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b g^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c g^6 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c g^4 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}+\frac {\left (b c^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{g^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x} (f+g x)} \, dx}{g^6 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {c^4 f^2 g^2-c^2 g^4}{(f+g x) \sqrt {-1+c^2 x^2}} \, dx}{c^2 g^8 (-1+c x) (1+c x)} \\ & = \frac {2 b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{3 g^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2}}{4 g^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 g \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2}}{9 g^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{g^5 (1-c x) (1+c x)}+\frac {b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{g^5}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{4 g^2}-\frac {2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 g^3}-\frac {c^2 d^2 x^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 g}+\frac {c d^2 f \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b g^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b g^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c g^6 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c g^4 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}-\frac {\left (b c d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int 1 \, dx}{g^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x}{c f+g \cosh (x)} \, dx,x,\text {arccosh}(c x)\right )}{g^6 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{(f+g x) \sqrt {-1+c^2 x^2}} \, dx}{g^6 (-1+c x) (1+c x)} \\ & = \frac {2 b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{3 g^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2}}{g^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2}}{4 g^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 g \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2}}{9 g^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{g^5 (1-c x) (1+c x)}+\frac {b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{g^5}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{4 g^2}-\frac {2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 g^3}-\frac {c^2 d^2 x^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 g}+\frac {c d^2 f \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b g^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b g^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c g^6 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c g^4 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}+\frac {\left (2 b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^x x}{2 c e^x f+g+e^{2 x} g} \, dx,x,\text {arccosh}(c x)\right )}{g^6 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {1}{c^2 f^2-g^2-x^2} \, dx,x,\frac {-g-c^2 f x}{\sqrt {-1+c^2 x^2}}\right )}{g^6 (-1+c x) (1+c x)} \\ & = \frac {2 b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{3 g^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2}}{g^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2}}{4 g^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 g \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2}}{9 g^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{g^5 (1-c x) (1+c x)}+\frac {b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{g^5}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{4 g^2}-\frac {2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 g^3}-\frac {c^2 d^2 x^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 g}+\frac {c d^2 f \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b g^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b g^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c g^6 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c g^4 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}-\frac {a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2} \text {arctanh}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {-1+c^2 x^2}}\right )}{g^6 (1-c x) (1+c x)}+\frac {\left (2 b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^x x}{2 c f+2 e^x g-2 \sqrt {c^2 f^2-g^2}} \, dx,x,\text {arccosh}(c x)\right )}{g^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^x x}{2 c f+2 e^x g+2 \sqrt {c^2 f^2-g^2}} \, dx,x,\text {arccosh}(c x)\right )}{g^5 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 16.31 (sec) , antiderivative size = 6244, normalized size of antiderivative = 3.58 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\text {Result too large to show} \]

[In]

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(f + g*x),x]

[Out]

Result too large to show

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3768\) vs. \(2(1608)=3216\).

Time = 2.14 (sec) , antiderivative size = 3769, normalized size of antiderivative = 2.16

method	result	size
default	\(\text {Expression too large to display}\)	\(3769\)
parts	\(\text {Expression too large to display}\)	\(3769\)

[In]

int((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/(g*x+f),x,method=_RETURNVERBOSE)

[Out]

1/5*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/(c*x-1)/g*arccosh(c*x)*x^6*c^6+34/15*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*
x+1)/(c*x-1)/g*arccosh(c*x)*x^2*c^2+7/3*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/(c*x-1)/g^3*arccosh(c*x)*c^2*f^2+
b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^2*ln((-(c*x+(c*x-1)^(1/2)*(c*x+
1)^(1/2))*g-c*f+(c^2*f^2-g^2)^(1/2))/(-c*f+(c^2*f^2-g^2)^(1/2)))*arccosh(c*x)-b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^
2*f^2-g^2)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^2*ln(((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g+c*f+(c^2*f^2-g^2)^(1/
2))/(c*f+(c^2*f^2-g^2)^(1/2)))*arccosh(c*x)-1/2*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*f^5*arcco
sh(c*x)^2*d^2*c^5/g^6+a/g*(1/5*(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(5/2)+c^2*d*f/g*(-1/
8*(-2*c^2*d*(x+f/g)+2*c^2*d*f/g)/c^2/d*(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(3/2)-3/16*(
4*c^2*d^2*(c^2*f^2-g^2)/g^2-4*c^4*d^2*f^2/g^2)/c^2/d*(-1/4*(-2*c^2*d*(x+f/g)+2*c^2*d*f/g)/c^2/d*(-(x+f/g)^2*c^
2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2)-1/8*(4*c^2*d^2*(c^2*f^2-g^2)/g^2-4*c^4*d^2*f^2/g^2)/c^2/d/(
c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2))))-d*(c^2
*f^2-g^2)/g^2*(1/3*(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(3/2)+c^2*d*f/g*(-1/4*(-2*c^2*d*
(x+f/g)+2*c^2*d*f/g)/c^2/d*(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2)-1/8*(4*c^2*d^2*(c^
2*f^2-g^2)/g^2-4*c^4*d^2*f^2/g^2)/c^2/d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+
f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2)))-d*(c^2*f^2-g^2)/g^2*((-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g
^2)^(1/2)+c^2*d*f/g/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)
/g^2)^(1/2))+d*(c^2*f^2-g^2)/g^2/(-d*(c^2*f^2-g^2)/g^2)^(1/2)*ln((-2*d*(c^2*f^2-g^2)/g^2+2*c^2*d*f/g*(x+f/g)+2
*(-d*(c^2*f^2-g^2)/g^2)^(1/2)*(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2))/(x+f/g)))))-23
/15*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/(c*x-1)/g*arccosh(c*x)+33/128*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c/(c*x+1
)^(1/2)/(c*x-1)^(1/2)/g^2-1/8*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d^2*c^3/(c*x+1)^(1/2)/(c*x-1)^(1/2)/g^4-1/25*b*(-d*
(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)/g*x^5*c^5+11/45*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)/
(c*x-1)^(1/2)/g*x^3*c^3-23/15*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)/g*x*c+b*d^2*(-d*(c^2*x^
2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^2*dilog((-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g-c*
f+(c^2*f^2-g^2)^(1/2))/(-c*f+(c^2*f^2-g^2)^(1/2)))-b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)/(c*x-1)^(1
/2)/(c*x+1)^(1/2)/g^2*dilog(((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g+c*f+(c^2*f^2-g^2)^(1/2))/(c*f+(c^2*f^2-g^2)^(
1/2)))+b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^6*ln((-(c*x+(c*x-1)^(1/2
)*(c*x+1)^(1/2))*g-c*f+(c^2*f^2-g^2)^(1/2))/(-c*f+(c^2*f^2-g^2)^(1/2)))*arccosh(c*x)*c^4*f^4-b*d^2*(-d*(c^2*x^
2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^6*ln(((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g+c*f+(c
^2*f^2-g^2)^(1/2))/(c*f+(c^2*f^2-g^2)^(1/2)))*arccosh(c*x)*c^4*f^4-2*b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2
)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^4*ln((-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g-c*f+(c^2*f^2-g^2)^(1/2))/(-c*
f+(c^2*f^2-g^2)^(1/2)))*arccosh(c*x)*c^2*f^2+2*b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)/(c*x-1)^(1/2)/
(c*x+1)^(1/2)/g^4*ln(((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g+c*f+(c^2*f^2-g^2)^(1/2))/(c*f+(c^2*f^2-g^2)^(1/2)))*
arccosh(c*x)*c^2*f^2-b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/(c*x-1)/g^5*arccosh(c*x)*c^4*f^4+1/4*b*(-d*(c^2*x^2-
1))^(1/2)*f^3*d^2*c^5/(c*x+1)^(1/2)/(c*x-1)^(1/2)/g^4*x^2-b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)/(c*x-1)^(
1/2)/g^5*x*c^5*f^4+1/16*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c^5/(c*x+1)^(1/2)/(c*x-1)^(1/2)/g^2*x^4-9/16*b*(-d*(c^2
*x^2-1))^(1/2)*f*d^2*c^3/(c*x+1)^(1/2)/(c*x-1)^(1/2)/g^2*x^2-1/9*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)/(c
*x-1)^(1/2)/g^3*x^3*c^5*f^2+7/3*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)/g^3*x*c^3*f^2+5/4*b*(
-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*f^3*arccosh(c*x)^2*d^2*c^3/g^4-15/16*b*(-d*(c^2*x^2-1))^(1/2
)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*f*arccosh(c*x)^2*d^2*c/g^2-14/15*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/(c*x-1)/g*
arccosh(c*x)*x^4*c^4-b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^6*dilog(((
c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g+c*f+(c^2*f^2-g^2)^(1/2))/(c*f+(c^2*f^2-g^2)^(1/2)))*c^4*f^4-2*b*d^2*(-d*(c^
2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^4*dilog((-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*
g-c*f+(c^2*f^2-g^2)^(1/2))/(-c*f+(c^2*f^2-g^2)^(1/2)))*c^2*f^2+2*b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1
/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^4*dilog(((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g+c*f+(c^2*f^2-g^2)^(1/2))/(c*f+(
c^2*f^2-g^2)^(1/2)))*c^2*f^2+1/3*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/(c*x-1)/g^3*arccosh(c*x)*x^4*c^6*f^2-8/3
*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/(c*x-1)/g^3*arccosh(c*x)*x^2*c^4*f^2-1/2*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d^
2*c^6/(c*x+1)/(c*x-1)/g^4*arccosh(c*x)*x^3+1/2*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d^2*c^4/(c*x+1)/(c*x-1)/g^4*arccos
h(c*x)*x-1/4*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c^6/(c*x+1)/(c*x-1)/g^2*arccosh(c*x)*x^5+11/8*b*(-d*(c^2*x^2-1))^(
1/2)*f*d^2*c^4/(c*x+1)/(c*x-1)/g^2*arccosh(c*x)*x^3-9/8*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c^2/(c*x+1)/(c*x-1)/g^2
*arccosh(c*x)*x+b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^6*dilog((-(c*x+
(c*x-1)^(1/2)*(c*x+1)^(1/2))*g-c*f+(c^2*f^2-g^2)^(1/2))/(-c*f+(c^2*f^2-g^2)^(1/2)))*c^4*f^4+b*(-d*(c^2*x^2-1))
^(1/2)*d^2/(c*x+1)/(c*x-1)/g^5*arccosh(c*x)*x^2*c^6*f^4

Fricas [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{g x + f} \,d x } \]

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/(g*x+f),x, algorithm="fricas")

[Out]

integral((a*c^4*d^2*x^4 - 2*a*c^2*d^2*x^2 + a*d^2 + (b*c^4*d^2*x^4 - 2*b*c^2*d^2*x^2 + b*d^2)*arccosh(c*x))*sq
rt(-c^2*d*x^2 + d)/(g*x + f), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\text {Timed out} \]

[In]

integrate((-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x))/(g*x+f),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/(g*x+f),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(g-c*f>0)', see `assume?` for m
ore details)

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/(g*x+f),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{f+g\,x} \,d x \]

[In]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2))/(f + g*x),x)

[Out]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2))/(f + g*x), x)