\(\int \frac {(f+g x) (a+b \text {arccosh}(c x))}{(d-c^2 d x^2)^{3/2}} \, dx\) [73]

Optimal result

Integrand size = 29, antiderivative size = 142 \[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {\left (g+c^2 f x\right ) (a+b \text {arccosh}(c x))}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {b (c f-g) \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {b f \sqrt {-1+c x} \sqrt {1+c x} \log (1-c x)}{c d \sqrt {d-c^2 d x^2}} \]

[Out]

(c^2*f*x+g)*(a+b*arccosh(c*x))/c^2/d/(-c^2*d*x^2+d)^(1/2)-b*(c*f-g)*arctanh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c
^2/d/(-c^2*d*x^2+d)^(1/2)-b*f*ln(-c*x+1)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/d/(-c^2*d*x^2+d)^(1/2)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.25, number of steps used = 5, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {5972, 79, 37, 5974, 35, 212} \[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {(c f-g) (a+b \text {arccosh}(c x))}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {f (c x+1) (a+b \text {arccosh}(c x))}{c d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}(c x) (c f-g)}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {b f \sqrt {c x-1} \sqrt {c x+1} \log (1-c x)}{c d \sqrt {d-c^2 d x^2}} \]

[In]

Int[((f + g*x)*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(3/2),x]

[Out]

-(((c*f - g)*(a + b*ArcCosh[c*x]))/(c^2*d*Sqrt[d - c^2*d*x^2])) + (f*(1 + c*x)*(a + b*ArcCosh[c*x]))/(c*d*Sqrt
[d - c^2*d*x^2]) - (b*(c*f - g)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTanh[c*x])/(c^2*d*Sqrt[d - c^2*d*x^2]) - (b*f*
Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 - c*x])/(c*d*Sqrt[d - c^2*d*x^2])

Rule 35

Int[1/(((a_) + (b_.)*(x_))*((c_) + (d_.)*(x_))), x_Symbol] :> Int[1/(a*c + b*d*x^2), x] /; FreeQ[{a, b, c, d},
x] && EqQ[b*c + a*d, 0]

Rule 37

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

Rule 79

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

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 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 5974

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_)*((f_) + (g_.)*(x
_))^(m_.), x_Symbol] :> With[{u = IntHide[(f + g*x)^m*(d1 + e1*x)^p*(d2 + e2*x)^p, x]}, Dist[a + b*ArcCosh[c*x
], u, x] - Dist[b*c, Int[Dist[1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), u, x], x], x]] /; FreeQ[{a, b, c, d1, e1, d2,
e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[m, 0] && ILtQ[p + 1/2, 0] && GtQ[d1, 0] && LtQ
[d2, 0] && (GtQ[m, 3] || LtQ[m, -2*p - 1])

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}} \\ & = -\frac {(c f-g) (a+b \text {arccosh}(c x))}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {f (1+c x) (a+b \text {arccosh}(c x))}{c d \sqrt {d-c^2 d x^2}}+\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \left (\frac {f}{c (1-c x)}-\frac {c f-g}{c^2 (1-c x) (1+c x)}\right ) \, dx}{d \sqrt {d-c^2 d x^2}} \\ & = -\frac {(c f-g) (a+b \text {arccosh}(c x))}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {f (1+c x) (a+b \text {arccosh}(c x))}{c d \sqrt {d-c^2 d x^2}}-\frac {b f \sqrt {-1+c x} \sqrt {1+c x} \log (1-c x)}{c d \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f-g) \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{(1-c x) (1+c x)} \, dx}{c d \sqrt {d-c^2 d x^2}} \\ & = -\frac {(c f-g) (a+b \text {arccosh}(c x))}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {f (1+c x) (a+b \text {arccosh}(c x))}{c d \sqrt {d-c^2 d x^2}}-\frac {b f \sqrt {-1+c x} \sqrt {1+c x} \log (1-c x)}{c d \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f-g) \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{1-c^2 x^2} \, dx}{c d \sqrt {d-c^2 d x^2}} \\ & = -\frac {(c f-g) (a+b \text {arccosh}(c x))}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {f (1+c x) (a+b \text {arccosh}(c x))}{c d \sqrt {d-c^2 d x^2}}-\frac {b (c f-g) \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {b f \sqrt {-1+c x} \sqrt {1+c x} \log (1-c x)}{c d \sqrt {d-c^2 d x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.85 \[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (-\frac {2 a \left (g+c^2 f x\right )}{-1+c^2 x^2}-\frac {2 b \left (g+c^2 f x\right ) \text {arccosh}(c x)}{-1+c^2 x^2}+\frac {b ((c f+g) \log (-1+c x)+(c f-g) \log (1+c x))}{\sqrt {-1+c x} \sqrt {1+c x}}\right )}{2 c^2 d^2} \]

[In]

Integrate[((f + g*x)*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(3/2),x]

[Out]

(Sqrt[d - c^2*d*x^2]*((-2*a*(g + c^2*f*x))/(-1 + c^2*x^2) - (2*b*(g + c^2*f*x)*ArcCosh[c*x])/(-1 + c^2*x^2) +
(b*((c*f + g)*Log[-1 + c*x] + (c*f - g)*Log[1 + c*x]))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/(2*c^2*d^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(498\) vs. \(2(128)=256\).

Time = 2.66 (sec) , antiderivative size = 499, normalized size of antiderivative = 3.51

method	result	size
default	\(a \left (\frac {f x}{d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {g}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, f \,\operatorname {arccosh}\left (c x \right )}{d^{2} c \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \left (c x +1\right ) \left (c x -1\right ) g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x^{2} g}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x f}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) f}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) f}{d^{2} c \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}\)	\(499\)
parts	\(a \left (\frac {f x}{d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {g}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, f \,\operatorname {arccosh}\left (c x \right )}{d^{2} c \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \left (c x +1\right ) \left (c x -1\right ) g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x^{2} g}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x f}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) f}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) f}{d^{2} c \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}\)	\(499\)

[In]

int((g*x+f)*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)

[Out]

a*(f/d*x/(-c^2*d*x^2+d)^(1/2)+g/c^2/d/(-c^2*d*x^2+d)^(1/2))-b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/
2)/d^2/c/(c^2*x^2-1)*f*arccosh(c*x)+b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d^2/c^2/(c^2*x^2-1)*(c*x+1)*(c*x-1)*
g-b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d^2/(c^2*x^2-1)*x^2*g-b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d^2/(c^2*x
^2-1)*x*f+b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/d^2/c/(c^
2*x^2-1)*f-b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/d^2/c^2/
(c^2*x^2-1)*g+b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/c/(c^2*x^2-1)*ln((c*x-1)^(1/2)*(c*x+1)^
(1/2)+c*x-1)*f+b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/c^2/(c^2*x^2-1)*ln((c*x-1)^(1/2)*(c*x+
1)^(1/2)+c*x-1)*g

Fricas [F]

\[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((g*x+f)*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (f + g x\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((g*x+f)*(a+b*acosh(c*x))/(-c**2*d*x**2+d)**(3/2),x)

[Out]

Integral((a + b*acosh(c*x))*(f + g*x)/(-d*(c*x - 1)*(c*x + 1))**(3/2), x)

Maxima [F]

\[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((g*x+f)*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="maxima")

[Out]

-1/2*b*c*f*sqrt(-1/(c^4*d))*log(x^2 - 1/c^2)/d + b*g*(((c*sqrt(d)*x + sqrt(c*x + 1)*sqrt(c*x - 1)*sqrt(d))*log
(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/sqrt(-c*x + 1) + sqrt(c*x + 1)*sqrt(c*x - 1)*sqrt(d)/sqrt(-c*x + 1))/(sqrt
(c*x + 1)*c^3*d^2*x + (c*x + 1)*sqrt(c*x - 1)*c^2*d^2) - integrate((c^2*x^3 + c*x^2*e^(1/2*log(c*x + 1) + 1/2*
log(c*x - 1)) - x)/(sqrt(-c*x + 1)*((c^2*d^(3/2)*x^2 - d^(3/2))*e^(3/2*log(c*x + 1) + log(c*x - 1)) + 2*(c^3*d
^(3/2)*x^3 - c*d^(3/2)*x)*e^(log(c*x + 1) + 1/2*log(c*x - 1)) + (c^4*d^(3/2)*x^4 - c^2*d^(3/2)*x^2)*sqrt(c*x +
1))), x)) + b*f*x*arccosh(c*x)/(sqrt(-c^2*d*x^2 + d)*d) + a*f*x/(sqrt(-c^2*d*x^2 + d)*d) + a*g/(sqrt(-c^2*d*x
^2 + d)*c^2*d)

Giac [F]

\[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((g*x+f)*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="giac")

[Out]

integrate((g*x + f)*(b*arccosh(c*x) + a)/(-c^2*d*x^2 + d)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (f+g\,x\right )\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]

[In]

int(((f + g*x)*(a + b*acosh(c*x)))/(d - c^2*d*x^2)^(3/2),x)

[Out]

int(((f + g*x)*(a + b*acosh(c*x)))/(d - c^2*d*x^2)^(3/2), x)