Integrand size = 31, antiderivative size = 478 \[ \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {3 b f^2 g x \sqrt {-1+c x} \sqrt {1+c x}}{c \sqrt {d-c^2 d x^2}}-\frac {2 b g^3 x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^3 \sqrt {d-c^2 d x^2}}-\frac {3 b f g^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{4 c \sqrt {d-c^2 d x^2}}-\frac {b g^3 x^3 \sqrt {-1+c x} \sqrt {1+c x}}{9 c \sqrt {d-c^2 d x^2}}-\frac {3 f^2 g (1-c x) (1+c x) (a+b \text {arccosh}(c x))}{c^2 \sqrt {d-c^2 d x^2}}-\frac {2 g^3 (1-c x) (1+c x) (a+b \text {arccosh}(c x))}{3 c^4 \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 x (1-c x) (1+c x) (a+b \text {arccosh}(c x))}{2 c^2 \sqrt {d-c^2 d x^2}}-\frac {g^3 x^2 (1-c x) (1+c x) (a+b \text {arccosh}(c x))}{3 c^2 \sqrt {d-c^2 d x^2}}+\frac {f^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{2 b c \sqrt {d-c^2 d x^2}}+\frac {3 f g^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{4 b c^3 \sqrt {d-c^2 d x^2}} \] Output:
-3*b*f^2*g*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/(-c^2*d*x^2+d)^(1/2)-2/3*b*g^3* x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3/(-c^2*d*x^2+d)^(1/2)-3/4*b*f*g^2*x^2*(c* x-1)^(1/2)*(c*x+1)^(1/2)/c/(-c^2*d*x^2+d)^(1/2)-1/9*b*g^3*x^3*(c*x-1)^(1/2 )*(c*x+1)^(1/2)/c/(-c^2*d*x^2+d)^(1/2)-3*f^2*g*(-c*x+1)*(c*x+1)*(a+b*arcco sh(c*x))/c^2/(-c^2*d*x^2+d)^(1/2)-2/3*g^3*(-c*x+1)*(c*x+1)*(a+b*arccosh(c* x))/c^4/(-c^2*d*x^2+d)^(1/2)-3/2*f*g^2*x*(-c*x+1)*(c*x+1)*(a+b*arccosh(c*x ))/c^2/(-c^2*d*x^2+d)^(1/2)-1/3*g^3*x^2*(-c*x+1)*(c*x+1)*(a+b*arccosh(c*x) )/c^2/(-c^2*d*x^2+d)^(1/2)+1/2*f^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccos h(c*x))^2/b/c/(-c^2*d*x^2+d)^(1/2)+3/4*f*g^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*( a+b*arccosh(c*x))^2/b/c^3/(-c^2*d*x^2+d)^(1/2)
Time = 1.53 (sec) , antiderivative size = 371, normalized size of antiderivative = 0.78 \[ \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {18 b c \sqrt {d} f \left (2 c^2 f^2+3 g^2\right ) (-1+c x) \text {arccosh}(c x)^2-4 \left (\sqrt {d} g \left (-1+c^2 x^2\right ) \left (2 b \left (7 g^2-c g^2 x+c^2 \left (27 f^2+g^2 x^2\right )\right )-3 a \sqrt {\frac {-1+c x}{1+c x}} \left (4 g^2+c^2 \left (18 f^2+9 f g x+2 g^2 x^2\right )\right )\right )+9 a c f \left (2 c^2 f^2+3 g^2\right ) \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )\right )-27 b c \sqrt {d} f g^2 (-1+c x) \cosh (2 \text {arccosh}(c x))+6 b \sqrt {d} g (-1+c x) \text {arccosh}(c x) \left (4 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (2 g^2+c^2 \left (9 f^2+g^2 x^2\right )\right )+9 c f g \sinh (2 \text {arccosh}(c x))\right )}{72 c^4 \sqrt {d} \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}} \] Input:
Integrate[((f + g*x)^3*(a + b*ArcCosh[c*x]))/Sqrt[d - c^2*d*x^2],x]
Output:
(18*b*c*Sqrt[d]*f*(2*c^2*f^2 + 3*g^2)*(-1 + c*x)*ArcCosh[c*x]^2 - 4*(Sqrt[ d]*g*(-1 + c^2*x^2)*(2*b*(7*g^2 - c*g^2*x + c^2*(27*f^2 + g^2*x^2)) - 3*a* Sqrt[(-1 + c*x)/(1 + c*x)]*(4*g^2 + c^2*(18*f^2 + 9*f*g*x + 2*g^2*x^2))) + 9*a*c*f*(2*c^2*f^2 + 3*g^2)*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2 ]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))]) - 27*b*c*Sqr t[d]*f*g^2*(-1 + c*x)*Cosh[2*ArcCosh[c*x]] + 6*b*Sqrt[d]*g*(-1 + c*x)*ArcC osh[c*x]*(4*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(2*g^2 + c^2*(9*f^2 + g^2 *x^2)) + 9*c*f*g*Sinh[2*ArcCosh[c*x]]))/(72*c^4*Sqrt[d]*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2])
Time = 1.93 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.59, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {6387, 6390, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx\) |
| \(\Big \downarrow \) 6387 |
| \(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6390 |
| \(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \int \left (\frac {(a+b \text {arccosh}(c x)) f^3}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {3 g x (a+b \text {arccosh}(c x)) f^2}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {3 g^2 x^2 (a+b \text {arccosh}(c x)) f}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {g^3 x^3 (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}\right )dx}{\sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \left (\frac {2 g^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{3 c^4}+\frac {3 f g^2 (a+b \text {arccosh}(c x))^2}{4 b c^3}+\frac {3 f^2 g \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c^2}+\frac {3 f g^2 x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 c^2}+\frac {g^3 x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{3 c^2}+\frac {f^3 (a+b \text {arccosh}(c x))^2}{2 b c}-\frac {2 b g^3 x}{3 c^3}-\frac {3 b f^2 g x}{c}-\frac {3 b f g^2 x^2}{4 c}-\frac {b g^3 x^3}{9 c}\right )}{\sqrt {d-c^2 d x^2}}\) |
Input:
Int[((f + g*x)^3*(a + b*ArcCosh[c*x]))/Sqrt[d - c^2*d*x^2],x]
Output:
(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((-3*b*f^2*g*x)/c - (2*b*g^3*x)/(3*c^3) - (3 *b*f*g^2*x^2)/(4*c) - (b*g^3*x^3)/(9*c) + (3*f^2*g*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/c^2 + (2*g^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*c^4) + (3*f*g^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b *ArcCosh[c*x]))/(2*c^2) + (g^3*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*Arc Cosh[c*x]))/(3*c^2) + (f^3*(a + b*ArcCosh[c*x])^2)/(2*b*c) + (3*f*g^2*(a + b*ArcCosh[c*x])^2)/(4*b*c^3)))/Sqrt[d - c^2*d*x^2]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d _) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-d)^IntPart[p]*((d + e*x^2)^Fra cPart[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]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_)*(( d2_) + (e2_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(m_.), x_Symbol] :> Int[Expand Integrand[(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, (f + g*x)^m, 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] && IntegerQ[p + 1/2] && GtQ[d1, 0] && LtQ[ d2, 0] && IGtQ[n, 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] || EqQ[m, 1 ] || (EqQ[m, 2] && LtQ[p, -2]))
Time = 0.59 (sec) , antiderivative size = 831, normalized size of antiderivative = 1.74
| method | result | size |
| default | \(a \left (\frac {f^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}+g^{3} \left (-\frac {x^{2} \sqrt {-c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {-c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )+3 f \,g^{2} \left (-\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2 c^{2} d}+\frac {\arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{2} \sqrt {c^{2} d}}\right )-\frac {3 f^{2} g \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2} f \left (2 c^{2} f^{2}+3 g^{2}\right )}{4 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) g^{3} \left (-1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) f \,g^{2} \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) g \left (4 \,\operatorname {arccosh}\left (c x \right ) c^{2} f^{2}-4 c^{2} f^{2}+\operatorname {arccosh}\left (c x \right ) g^{2}-g^{2}\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) g \left (4 \,\operatorname {arccosh}\left (c x \right ) c^{2} f^{2}+4 c^{2} f^{2}+\operatorname {arccosh}\left (c x \right ) g^{2}+g^{2}\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) f \,g^{2} \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) g^{3} \left (1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}-1\right )}\right )\) | \(831\) |
| parts | \(a \left (\frac {f^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}+g^{3} \left (-\frac {x^{2} \sqrt {-c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {-c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )+3 f \,g^{2} \left (-\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2 c^{2} d}+\frac {\arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{2} \sqrt {c^{2} d}}\right )-\frac {3 f^{2} g \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2} f \left (2 c^{2} f^{2}+3 g^{2}\right )}{4 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) g^{3} \left (-1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) f \,g^{2} \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) g \left (4 \,\operatorname {arccosh}\left (c x \right ) c^{2} f^{2}-4 c^{2} f^{2}+\operatorname {arccosh}\left (c x \right ) g^{2}-g^{2}\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) g \left (4 \,\operatorname {arccosh}\left (c x \right ) c^{2} f^{2}+4 c^{2} f^{2}+\operatorname {arccosh}\left (c x \right ) g^{2}+g^{2}\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) f \,g^{2} \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) g^{3} \left (1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}-1\right )}\right )\) | \(831\) |
Input:
int((g*x+f)^3*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERB OSE)
Output:
a*(f^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+g^3*(-1/ 3*x^2/c^2/d*(-c^2*d*x^2+d)^(1/2)-2/3/d/c^4*(-c^2*d*x^2+d)^(1/2))+3*f*g^2*( -1/2*x/c^2/d*(-c^2*d*x^2+d)^(1/2)+1/2/c^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/ 2)*x/(-c^2*d*x^2+d)^(1/2)))-3*f^2*g/c^2/d*(-c^2*d*x^2+d)^(1/2))+b*(-1/4*(- d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/c^3/(c^2*x^2-1)*arccosh (c*x)^2*f*(2*c^2*f^2+3*g^2)-1/72*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x ^2+4*c^3*x^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)-3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x +1)*g^3*(-1+3*arccosh(c*x))/c^4/d/(c^2*x^2-1)-3/16*(-d*(c^2*x^2-1))^(1/2)* (2*c^3*x^3-2*c*x+2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+ 1)^(1/2))*f*g^2*(-1+2*arccosh(c*x))/d/c^3/(c^2*x^2-1)-3/8*(-d*(c^2*x^2-1)) ^(1/2)*((c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*g*(4*arccosh(c*x)*c^2*f ^2-4*c^2*f^2+arccosh(c*x)*g^2-g^2)/c^4/d/(c^2*x^2-1)-3/8*(-d*(c^2*x^2-1))^ (1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*g*(4*arccosh(c*x)*c^2*f ^2+4*c^2*f^2+arccosh(c*x)*g^2+g^2)/c^4/d/(c^2*x^2-1)-3/16*(-d*(c^2*x^2-1)) ^(1/2)*(-2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+2*c^3*x^3+(c*x-1)^(1/2)*(c* x+1)^(1/2)-2*c*x)*f*g^2*(1+2*arccosh(c*x))/d/c^3/(c^2*x^2-1)-1/72*(-d*(c^2 *x^2-1))^(1/2)*(-4*c^3*x^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c^4*x^4+3*(c*x-1) ^(1/2)*(c*x+1)^(1/2)*c*x-5*c^2*x^2+1)*g^3*(1+3*arccosh(c*x))/c^4/d/(c^2*x^ 2-1))
\[ \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \] Input:
integrate((g*x+f)^3*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm=" fricas")
Output:
integral(-(a*g^3*x^3 + 3*a*f*g^2*x^2 + 3*a*f^2*g*x + a*f^3 + (b*g^3*x^3 + 3*b*f*g^2*x^2 + 3*b*f^2*g*x + b*f^3)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d)/(c ^2*d*x^2 - d), x)
\[ \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (f + g x\right )^{3}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \] Input:
integrate((g*x+f)**3*(a+b*acosh(c*x))/(-c**2*d*x**2+d)**(1/2),x)
Output:
Integral((a + b*acosh(c*x))*(f + g*x)**3/sqrt(-d*(c*x - 1)*(c*x + 1)), x)
\[ \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \] Input:
integrate((g*x+f)^3*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm=" maxima")
Output:
-1/3*a*g^3*(sqrt(-c^2*d*x^2 + d)*x^2/(c^2*d) + 2*sqrt(-c^2*d*x^2 + d)/(c^4 *d)) - 3/2*a*f*g^2*(sqrt(-c^2*d*x^2 + d)*x/(c^2*d) - arcsin(c*x)/(c^3*sqrt (d))) + a*f^3*arcsin(c*x)/(c*sqrt(d)) - 3*sqrt(-c^2*d*x^2 + d)*a*f^2*g/(c^ 2*d) + integrate(b*g^3*x^3*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/sqrt(-c^ 2*d*x^2 + d) + 3*b*f*g^2*x^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/sqrt(- c^2*d*x^2 + d) + 3*b*f^2*g*x*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/sqrt(- c^2*d*x^2 + d) + b*f^3*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/sqrt(-c^2*d* x^2 + d), x)
\[ \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \] Input:
integrate((g*x+f)^3*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm=" giac")
Output:
integrate((g*x + f)^3*(b*arccosh(c*x) + a)/sqrt(-c^2*d*x^2 + d), x)
Timed out. \[ \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {{\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{\sqrt {d-c^2\,d\,x^2}} \,d x \] Input:
int(((f + g*x)^3*(a + b*acosh(c*x)))/(d - c^2*d*x^2)^(1/2),x)
Output:
int(((f + g*x)^3*(a + b*acosh(c*x)))/(d - c^2*d*x^2)^(1/2), x)
\[ \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {6 \mathit {asin} \left (c x \right ) a \,c^{3} f^{3}+9 \mathit {asin} \left (c x \right ) a c f \,g^{2}-18 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f^{2} g -9 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f \,g^{2} x -2 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} g^{3} x^{2}-4 \sqrt {-c^{2} x^{2}+1}\, a \,g^{3}+6 \left (\int \frac {\mathit {acosh} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{4} f^{3}+6 \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{3}}{\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{4} g^{3}+18 \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{2}}{\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{4} f \,g^{2}+18 \left (\int \frac {\mathit {acosh} \left (c x \right ) x}{\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{4} f^{2} g}{6 \sqrt {d}\, c^{4}} \] Input:
int((g*x+f)^3*(a+b*acosh(c*x))/(-c^2*d*x^2+d)^(1/2),x)
Output:
(6*asin(c*x)*a*c**3*f**3 + 9*asin(c*x)*a*c*f*g**2 - 18*sqrt( - c**2*x**2 + 1)*a*c**2*f**2*g - 9*sqrt( - c**2*x**2 + 1)*a*c**2*f*g**2*x - 2*sqrt( - c **2*x**2 + 1)*a*c**2*g**3*x**2 - 4*sqrt( - c**2*x**2 + 1)*a*g**3 + 6*int(a cosh(c*x)/sqrt( - c**2*x**2 + 1),x)*b*c**4*f**3 + 6*int((acosh(c*x)*x**3)/ sqrt( - c**2*x**2 + 1),x)*b*c**4*g**3 + 18*int((acosh(c*x)*x**2)/sqrt( - c **2*x**2 + 1),x)*b*c**4*f*g**2 + 18*int((acosh(c*x)*x)/sqrt( - c**2*x**2 + 1),x)*b*c**4*f**2*g)/(6*sqrt(d)*c**4)