Integrand size = 24, antiderivative size = 316 \[ \int \frac {(a+b \text {arccosh}(c x))^3}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {3 b (a+b \text {arccosh}(c x))^2}{2 c d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x (a+b \text {arccosh}(c x))^3}{2 d^2 \left (1-c^2 x^2\right )}-\frac {6 b^2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c d^2}+\frac {(a+b \text {arccosh}(c x))^3 \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c d^2}-\frac {3 b^3 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{c d^2}+\frac {3 b (a+b \text {arccosh}(c x))^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{2 c d^2}+\frac {3 b^3 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{c d^2}-\frac {3 b (a+b \text {arccosh}(c x))^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{2 c d^2}-\frac {3 b^2 (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(c x)}\right )}{c d^2}+\frac {3 b^2 (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(c x)}\right )}{c d^2}+\frac {3 b^3 \operatorname {PolyLog}\left (4,-e^{\text {arccosh}(c x)}\right )}{c d^2}-\frac {3 b^3 \operatorname {PolyLog}\left (4,e^{\text {arccosh}(c x)}\right )}{c d^2} \] Output:
-3/2*b*(a+b*arccosh(c*x))^2/c/d^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/2*x*(a+b*a rccosh(c*x))^3/d^2/(-c^2*x^2+1)-6*b^2*(a+b*arccosh(c*x))*arctanh(c*x+(c*x- 1)^(1/2)*(c*x+1)^(1/2))/c/d^2+(a+b*arccosh(c*x))^3*arctanh(c*x+(c*x-1)^(1/ 2)*(c*x+1)^(1/2))/c/d^2-3*b^3*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/ c/d^2+3/2*b*(a+b*arccosh(c*x))^2*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2 ))/c/d^2+3*b^3*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c/d^2-3/2*b*(a+b *arccosh(c*x))^2*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c/d^2-3*b^2*(a +b*arccosh(c*x))*polylog(3,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/c/d^2+3*b^2*( a+b*arccosh(c*x))*polylog(3,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c/d^2+3*b^3*p olylog(4,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/c/d^2-3*b^3*polylog(4,c*x+(c*x- 1)^(1/2)*(c*x+1)^(1/2))/c/d^2
Leaf count is larger than twice the leaf count of optimal. \(708\) vs. \(2(316)=632\).
Time = 7.52 (sec) , antiderivative size = 708, normalized size of antiderivative = 2.24 \[ \int \frac {(a+b \text {arccosh}(c x))^3}{\left (d-c^2 d x^2\right )^2} \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*ArcCosh[c*x])^3/(d - c^2*d*x^2)^2,x]
Output:
-1/2*(a^3*x)/(d^2*(-1 + c^2*x^2)) - (a^3*Log[1 - c*x])/(4*c*d^2) + (a^3*Lo g[1 + c*x])/(4*c*d^2) + (3*a^2*b*((-(Sqrt[1 + c*x]/Sqrt[-1 + c*x]) - ArcCo sh[c*x]/(-1 + c*x))/4 + (Sqrt[-1 + c*x]/Sqrt[1 + c*x] - ArcCosh[c*x]/(1 + c*x))/4 + (-1/2*ArcCosh[c*x]^2 + 2*ArcCosh[c*x]*Log[1 + E^ArcCosh[c*x]] + 2*PolyLog[2, -E^ArcCosh[c*x]])/4 + (ArcCosh[c*x]^2/2 - 2*ArcCosh[c*x]*Log[ 1 - E^ArcCosh[c*x]] - 2*PolyLog[2, E^ArcCosh[c*x]])/4))/(c*d^2) + (3*a*b^2 *(-4*ArcCosh[c*x]*Coth[ArcCosh[c*x]/2] - ArcCosh[c*x]^2*Csch[ArcCosh[c*x]/ 2]^2 - 4*ArcCosh[c*x]^2*Log[1 - E^(-ArcCosh[c*x])] + 4*ArcCosh[c*x]^2*Log[ 1 + E^(-ArcCosh[c*x])] + 8*Log[Tanh[ArcCosh[c*x]/2]] - 8*ArcCosh[c*x]*Poly Log[2, -E^(-ArcCosh[c*x])] + 8*ArcCosh[c*x]*PolyLog[2, E^(-ArcCosh[c*x])] - 8*PolyLog[3, -E^(-ArcCosh[c*x])] + 8*PolyLog[3, E^(-ArcCosh[c*x])] - Arc Cosh[c*x]^2*Sech[ArcCosh[c*x]/2]^2 + 4*ArcCosh[c*x]*Tanh[ArcCosh[c*x]/2])) /(8*c*d^2) + (b^3*(-Pi^4 + 2*ArcCosh[c*x]^4 - 12*ArcCosh[c*x]^2*Coth[ArcCo sh[c*x]/2] - 2*ArcCosh[c*x]^3*Csch[ArcCosh[c*x]/2]^2 + 48*ArcCosh[c*x]*Log [1 - E^(-ArcCosh[c*x])] - 48*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])] + 8*A rcCosh[c*x]^3*Log[1 + E^(-ArcCosh[c*x])] - 8*ArcCosh[c*x]^3*Log[1 - E^ArcC osh[c*x]] - 24*(-2 + ArcCosh[c*x]^2)*PolyLog[2, -E^(-ArcCosh[c*x])] - 48*P olyLog[2, E^(-ArcCosh[c*x])] - 24*ArcCosh[c*x]^2*PolyLog[2, E^ArcCosh[c*x] ] - 48*ArcCosh[c*x]*PolyLog[3, -E^(-ArcCosh[c*x])] + 48*ArcCosh[c*x]*PolyL og[3, E^ArcCosh[c*x]] - 48*PolyLog[4, -E^(-ArcCosh[c*x])] - 48*PolyLog[...
Result contains complex when optimal does not.
Time = 3.10 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.91, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.792, Rules used = {6316, 27, 6318, 3042, 26, 4670, 3011, 6330, 25, 6304, 6318, 3042, 26, 4670, 2715, 2838, 7163, 2720, 7143}
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.
| \(\displaystyle \int \frac {(a+b \text {arccosh}(c x))^3}{\left (d-c^2 d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 6316 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arccosh}(c x))^3}{d \left (1-c^2 x^2\right )}dx}{2 d}+\frac {3 b c \int \frac {x (a+b \text {arccosh}(c x))^2}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{2 d^2}+\frac {x (a+b \text {arccosh}(c x))^3}{2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arccosh}(c x))^3}{1-c^2 x^2}dx}{2 d^2}+\frac {3 b c \int \frac {x (a+b \text {arccosh}(c x))^2}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{2 d^2}+\frac {x (a+b \text {arccosh}(c x))^3}{2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 6318 |
| \(\displaystyle \frac {3 b c \int \frac {x (a+b \text {arccosh}(c x))^2}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{2 d^2}-\frac {\int \frac {(a+b \text {arccosh}(c x))^3}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{2 c d^2}+\frac {x (a+b \text {arccosh}(c x))^3}{2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 b c \int \frac {x (a+b \text {arccosh}(c x))^2}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{2 d^2}-\frac {\int i (a+b \text {arccosh}(c x))^3 \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{2 c d^2}+\frac {x (a+b \text {arccosh}(c x))^3}{2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {3 b c \int \frac {x (a+b \text {arccosh}(c x))^2}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{2 d^2}-\frac {i \int (a+b \text {arccosh}(c x))^3 \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{2 c d^2}+\frac {x (a+b \text {arccosh}(c x))^3}{2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {i \left (3 i b \int (a+b \text {arccosh}(c x))^2 \log \left (1-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-3 i b \int (a+b \text {arccosh}(c x))^2 \log \left (1+e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^3\right )}{2 c d^2}+\frac {3 b c \int \frac {x (a+b \text {arccosh}(c x))^2}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{2 d^2}+\frac {x (a+b \text {arccosh}(c x))^3}{2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {i \left (-3 i b \left (2 b \int (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+3 i b \left (2 b \int (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^3\right )}{2 c d^2}+\frac {3 b c \int \frac {x (a+b \text {arccosh}(c x))^2}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{2 d^2}+\frac {x (a+b \text {arccosh}(c x))^3}{2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 6330 |
| \(\displaystyle -\frac {i \left (-3 i b \left (2 b \int (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+3 i b \left (2 b \int (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^3\right )}{2 c d^2}+\frac {3 b c \left (\frac {2 b \int -\frac {a+b \text {arccosh}(c x)}{(1-c x) (c x+1)}dx}{c}-\frac {(a+b \text {arccosh}(c x))^2}{c^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 d^2}+\frac {x (a+b \text {arccosh}(c x))^3}{2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {i \left (-3 i b \left (2 b \int (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+3 i b \left (2 b \int (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^3\right )}{2 c d^2}+\frac {3 b c \left (-\frac {2 b \int \frac {a+b \text {arccosh}(c x)}{(1-c x) (c x+1)}dx}{c}-\frac {(a+b \text {arccosh}(c x))^2}{c^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 d^2}+\frac {x (a+b \text {arccosh}(c x))^3}{2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 6304 |
| \(\displaystyle -\frac {i \left (-3 i b \left (2 b \int (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+3 i b \left (2 b \int (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^3\right )}{2 c d^2}+\frac {3 b c \left (-\frac {2 b \int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx}{c}-\frac {(a+b \text {arccosh}(c x))^2}{c^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 d^2}+\frac {x (a+b \text {arccosh}(c x))^3}{2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 6318 |
| \(\displaystyle -\frac {i \left (-3 i b \left (2 b \int (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+3 i b \left (2 b \int (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^3\right )}{2 c d^2}+\frac {3 b c \left (\frac {2 b \int \frac {a+b \text {arccosh}(c x)}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{c^2}-\frac {(a+b \text {arccosh}(c x))^2}{c^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 d^2}+\frac {x (a+b \text {arccosh}(c x))^3}{2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {i \left (-3 i b \left (2 b \int (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+3 i b \left (2 b \int (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^3\right )}{2 c d^2}+\frac {3 b c \left (-\frac {(a+b \text {arccosh}(c x))^2}{c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b \int i (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{c^2}\right )}{2 d^2}+\frac {x (a+b \text {arccosh}(c x))^3}{2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \left (-3 i b \left (2 b \int (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+3 i b \left (2 b \int (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^3\right )}{2 c d^2}+\frac {3 b c \left (-\frac {(a+b \text {arccosh}(c x))^2}{c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 i b \int (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{c^2}\right )}{2 d^2}+\frac {x (a+b \text {arccosh}(c x))^3}{2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {3 b c \left (-\frac {(a+b \text {arccosh}(c x))^2}{c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 i b \left (i b \int \log \left (1-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-i b \int \log \left (1+e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{c^2}\right )}{2 d^2}-\frac {i \left (-3 i b \left (2 b \int (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+3 i b \left (2 b \int (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^3\right )}{2 c d^2}+\frac {x (a+b \text {arccosh}(c x))^3}{2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {3 b c \left (-\frac {(a+b \text {arccosh}(c x))^2}{c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 i b \left (i b \int e^{-\text {arccosh}(c x)} \log \left (1-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-i b \int e^{-\text {arccosh}(c x)} \log \left (1+e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{c^2}\right )}{2 d^2}-\frac {i \left (-3 i b \left (2 b \int (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+3 i b \left (2 b \int (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^3\right )}{2 c d^2}+\frac {x (a+b \text {arccosh}(c x))^3}{2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {i \left (-3 i b \left (2 b \int (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+3 i b \left (2 b \int (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^3\right )}{2 c d^2}+\frac {3 b c \left (-\frac {(a+b \text {arccosh}(c x))^2}{c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 i b \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{c^2}\right )}{2 d^2}+\frac {x (a+b \text {arccosh}(c x))^3}{2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle -\frac {i \left (-3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-b \int \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)\right )-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-b \int \operatorname {PolyLog}\left (3,e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)\right )-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^3\right )}{2 c d^2}+\frac {3 b c \left (-\frac {(a+b \text {arccosh}(c x))^2}{c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 i b \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{c^2}\right )}{2 d^2}+\frac {x (a+b \text {arccosh}(c x))^3}{2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {i \left (-3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}\right )-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (3,e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}\right )-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^3\right )}{2 c d^2}+\frac {3 b c \left (-\frac {(a+b \text {arccosh}(c x))^2}{c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 i b \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{c^2}\right )}{2 d^2}+\frac {x (a+b \text {arccosh}(c x))^3}{2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {3 b c \left (-\frac {(a+b \text {arccosh}(c x))^2}{c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 i b \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{c^2}\right )}{2 d^2}-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^3-3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-b \operatorname {PolyLog}\left (4,-e^{\text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )+3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-b \operatorname {PolyLog}\left (4,e^{\text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )\right )}{2 c d^2}+\frac {x (a+b \text {arccosh}(c x))^3}{2 d^2 \left (1-c^2 x^2\right )}\) |
Input:
Int[(a + b*ArcCosh[c*x])^3/(d - c^2*d*x^2)^2,x]
Output:
(x*(a + b*ArcCosh[c*x])^3)/(2*d^2*(1 - c^2*x^2)) + (3*b*c*(-((a + b*ArcCos h[c*x])^2/(c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + ((2*I)*b*((2*I)*(a + b*Arc Cosh[c*x])*ArcTanh[E^ArcCosh[c*x]] + I*b*PolyLog[2, -E^ArcCosh[c*x]] - I*b *PolyLog[2, E^ArcCosh[c*x]]))/c^2))/(2*d^2) - ((I/2)*((2*I)*(a + b*ArcCosh [c*x])^3*ArcTanh[E^ArcCosh[c*x]] - (3*I)*b*(-((a + b*ArcCosh[c*x])^2*PolyL og[2, -E^ArcCosh[c*x]]) + 2*b*((a + b*ArcCosh[c*x])*PolyLog[3, -E^ArcCosh[ c*x]] - b*PolyLog[4, -E^ArcCosh[c*x]])) + (3*I)*b*(-((a + b*ArcCosh[c*x])^ 2*PolyLog[2, E^ArcCosh[c*x]]) + 2*b*((a + b*ArcCosh[c*x])*PolyLog[3, E^Arc Cosh[c*x]] - b*PolyLog[4, E^ArcCosh[c*x]]))))/(c*d^2)
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[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]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
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]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*( (d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(d1*d2 + e1*e2*x^2)^p*(a + b*A rcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*d*(p + 1))), x] + (Simp[(2*p + 3)/(2*d*(p + 1)) Int[(d + e*x^2)^(p + 1)*(a + b* ArcCosh[c*x])^n, x], x] - Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[x*(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, d, e}, x] && EqQ[c^2* d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[-(c*d)^(-1) Subst[Int[(a + b*x)^n*Csch[x], x], x, ArcCosh[c*x ]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p _)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Simp[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] && E qQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
Time = 0.27 (sec) , antiderivative size = 811, normalized size of antiderivative = 2.57
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{3} \left (-\frac {1}{4 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b^{3} \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2} \left (c x \,\operatorname {arccosh}\left (c x \right )+3 \sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 \left (c^{2} x^{2}-1\right )}-\frac {\operatorname {arccosh}\left (c x \right )^{3} \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}-\frac {3 \operatorname {arccosh}\left (c x \right )^{2} \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+3 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (3, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-3 \operatorname {polylog}\left (4, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\frac {\operatorname {arccosh}\left (c x \right )^{3} \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {3 \operatorname {arccosh}\left (c x \right )^{2} \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}-3 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (3, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {polylog}\left (4, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}+\frac {3 a \,b^{2} \left (-\frac {\operatorname {arccosh}\left (c x \right ) \left (c x \,\operatorname {arccosh}\left (c x \right )+2 \sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 \left (c^{2} x^{2}-1\right )}-\frac {\operatorname {arccosh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}-\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (3, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\frac {\operatorname {arccosh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {polylog}\left (3, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \,\operatorname {arctanh}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}+\frac {3 a^{2} b \left (-\frac {c x \,\operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}}{2 \left (c^{2} x^{2}-1\right )}-\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}-\frac {\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}\right )}{d^{2}}}{c}\) | \(811\) |
| default | \(\frac {\frac {a^{3} \left (-\frac {1}{4 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b^{3} \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2} \left (c x \,\operatorname {arccosh}\left (c x \right )+3 \sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 \left (c^{2} x^{2}-1\right )}-\frac {\operatorname {arccosh}\left (c x \right )^{3} \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}-\frac {3 \operatorname {arccosh}\left (c x \right )^{2} \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+3 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (3, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-3 \operatorname {polylog}\left (4, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\frac {\operatorname {arccosh}\left (c x \right )^{3} \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {3 \operatorname {arccosh}\left (c x \right )^{2} \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}-3 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (3, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {polylog}\left (4, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}+\frac {3 a \,b^{2} \left (-\frac {\operatorname {arccosh}\left (c x \right ) \left (c x \,\operatorname {arccosh}\left (c x \right )+2 \sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 \left (c^{2} x^{2}-1\right )}-\frac {\operatorname {arccosh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}-\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (3, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\frac {\operatorname {arccosh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {polylog}\left (3, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \,\operatorname {arctanh}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}+\frac {3 a^{2} b \left (-\frac {c x \,\operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}}{2 \left (c^{2} x^{2}-1\right )}-\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}-\frac {\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}\right )}{d^{2}}}{c}\) | \(811\) |
| parts | \(\frac {a^{3} \left (-\frac {1}{4 c \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4 c}-\frac {1}{4 c \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{4 c}\right )}{d^{2}}+\frac {b^{3} \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2} \left (c x \,\operatorname {arccosh}\left (c x \right )+3 \sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 \left (c^{2} x^{2}-1\right )}-\frac {\operatorname {arccosh}\left (c x \right )^{3} \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}-\frac {3 \operatorname {arccosh}\left (c x \right )^{2} \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+3 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (3, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-3 \operatorname {polylog}\left (4, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\frac {\operatorname {arccosh}\left (c x \right )^{3} \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {3 \operatorname {arccosh}\left (c x \right )^{2} \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}-3 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (3, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {polylog}\left (4, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2} c}+\frac {3 a \,b^{2} \left (-\frac {\operatorname {arccosh}\left (c x \right ) \left (c x \,\operatorname {arccosh}\left (c x \right )+2 \sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 \left (c^{2} x^{2}-1\right )}-\frac {\operatorname {arccosh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}-\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (3, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\frac {\operatorname {arccosh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {polylog}\left (3, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \,\operatorname {arctanh}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2} c}+\frac {3 a^{2} b \left (-\frac {c x \,\operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}}{2 \left (c^{2} x^{2}-1\right )}-\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}-\frac {\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}\right )}{d^{2} c}\) | \(828\) |
Input:
int((a+b*arccosh(c*x))^3/(-c^2*d*x^2+d)^2,x,method=_RETURNVERBOSE)
Output:
1/c*(a^3/d^2*(-1/4/(c*x-1)-1/4*ln(c*x-1)-1/4/(c*x+1)+1/4*ln(c*x+1))+b^3/d^ 2*(-1/2/(c^2*x^2-1)*arccosh(c*x)^2*(c*x*arccosh(c*x)+3*(c*x-1)^(1/2)*(c*x+ 1)^(1/2))-1/2*arccosh(c*x)^3*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))-3/2*arc cosh(c*x)^2*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+3*arccosh(c*x)*poly log(3,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-3*polylog(4,c*x+(c*x-1)^(1/2)*(c*x+ 1)^(1/2))+1/2*arccosh(c*x)^3*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+3/2*arc cosh(c*x)^2*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))-3*arccosh(c*x)*pol ylog(3,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))+3*polylog(4,-c*x-(c*x-1)^(1/2)*(c *x+1)^(1/2))+3*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))+3*polylo g(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-3*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2) *(c*x+1)^(1/2))-3*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2)))+3*a*b^2/d^2 *(-1/2/(c^2*x^2-1)*arccosh(c*x)*(c*x*arccosh(c*x)+2*(c*x-1)^(1/2)*(c*x+1)^ (1/2))-1/2*arccosh(c*x)^2*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))-arccosh(c* x)*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+polylog(3,c*x+(c*x-1)^(1/2)* (c*x+1)^(1/2))+1/2*arccosh(c*x)^2*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+ar ccosh(c*x)*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))-polylog(3,-c*x-(c*x -1)^(1/2)*(c*x+1)^(1/2))-2*arctanh(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))+3*a^2 *b/d^2*(-1/2*(c*x*arccosh(c*x)+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c^2*x^2-1)-1/ 2*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))-1/2*polylog(2,c*x+(c* x-1)^(1/2)*(c*x+1)^(1/2))+1/2*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+...
\[ \int \frac {(a+b \text {arccosh}(c x))^3}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{3}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \] Input:
integrate((a+b*arccosh(c*x))^3/(-c^2*d*x^2+d)^2,x, algorithm="fricas")
Output:
integral((b^3*arccosh(c*x)^3 + 3*a*b^2*arccosh(c*x)^2 + 3*a^2*b*arccosh(c* x) + a^3)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)
\[ \int \frac {(a+b \text {arccosh}(c x))^3}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a^{3}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b^{3} \operatorname {acosh}^{3}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {3 a b^{2} \operatorname {acosh}^{2}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {3 a^{2} b \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \] Input:
integrate((a+b*acosh(c*x))**3/(-c**2*d*x**2+d)**2,x)
Output:
(Integral(a**3/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(b**3*acosh(c*x )**3/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(3*a*b**2*acosh(c*x)**2/( c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(3*a**2*b*acosh(c*x)/(c**4*x**4 - 2*c**2*x**2 + 1), x))/d**2
\[ \int \frac {(a+b \text {arccosh}(c x))^3}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{3}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \] Input:
integrate((a+b*arccosh(c*x))^3/(-c^2*d*x^2+d)^2,x, algorithm="maxima")
Output:
-1/4*a^3*(2*x/(c^2*d^2*x^2 - d^2) - log(c*x + 1)/(c*d^2) + log(c*x - 1)/(c *d^2)) - 1/4*(2*b^3*c*x - (b^3*c^2*x^2 - b^3)*log(c*x + 1) + (b^3*c^2*x^2 - b^3)*log(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^3/(c^3*d^2*x^2 - c*d^2) - integrate(-3/4*((2*b^3*c^3*x^3 + (2*b^3*c^2*x^2 + 4*a*b^2 - (b ^3*c^3*x^3 - b^3*c*x)*log(c*x + 1) + (b^3*c^3*x^3 - b^3*c*x)*log(c*x - 1)) *sqrt(c*x + 1)*sqrt(c*x - 1) + 2*(2*a*b^2*c - b^3*c)*x - (b^3*c^4*x^4 - 2* b^3*c^2*x^2 + b^3)*log(c*x + 1) + (b^3*c^4*x^4 - 2*b^3*c^2*x^2 + b^3)*log( c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2 + 4*(a^2*b*c*x + sqrt(c *x + 1)*sqrt(c*x - 1)*a^2*b)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^5* d^2*x^5 - 2*c^3*d^2*x^3 + c*d^2*x + (c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2)*sq rt(c*x + 1)*sqrt(c*x - 1)), x)
\[ \int \frac {(a+b \text {arccosh}(c x))^3}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{3}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \] Input:
integrate((a+b*arccosh(c*x))^3/(-c^2*d*x^2+d)^2,x, algorithm="giac")
Output:
integrate((b*arccosh(c*x) + a)^3/(c^2*d*x^2 - d)^2, x)
Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^3}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^3}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \] Input:
int((a + b*acosh(c*x))^3/(d - c^2*d*x^2)^2,x)
Output:
int((a + b*acosh(c*x))^3/(d - c^2*d*x^2)^2, x)
\[ \int \frac {(a+b \text {arccosh}(c x))^3}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {12 \left (\int \frac {\mathit {acosh} \left (c x \right )}{c^{4} x^{4}-2 c^{2} x^{2}+1}d x \right ) a^{2} b \,c^{3} x^{2}-12 \left (\int \frac {\mathit {acosh} \left (c x \right )}{c^{4} x^{4}-2 c^{2} x^{2}+1}d x \right ) a^{2} b c +4 \left (\int \frac {\mathit {acosh} \left (c x \right )^{3}}{c^{4} x^{4}-2 c^{2} x^{2}+1}d x \right ) b^{3} c^{3} x^{2}-4 \left (\int \frac {\mathit {acosh} \left (c x \right )^{3}}{c^{4} x^{4}-2 c^{2} x^{2}+1}d x \right ) b^{3} c +12 \left (\int \frac {\mathit {acosh} \left (c x \right )^{2}}{c^{4} x^{4}-2 c^{2} x^{2}+1}d x \right ) a \,b^{2} c^{3} x^{2}-12 \left (\int \frac {\mathit {acosh} \left (c x \right )^{2}}{c^{4} x^{4}-2 c^{2} x^{2}+1}d x \right ) a \,b^{2} c -\mathrm {log}\left (c^{2} x -c \right ) a^{3} c^{2} x^{2}+\mathrm {log}\left (c^{2} x -c \right ) a^{3}+\mathrm {log}\left (c^{2} x +c \right ) a^{3} c^{2} x^{2}-\mathrm {log}\left (c^{2} x +c \right ) a^{3}-2 a^{3} c x}{4 c \,d^{2} \left (c^{2} x^{2}-1\right )} \] Input:
int((a+b*acosh(c*x))^3/(-c^2*d*x^2+d)^2,x)
Output:
(12*int(acosh(c*x)/(c**4*x**4 - 2*c**2*x**2 + 1),x)*a**2*b*c**3*x**2 - 12* int(acosh(c*x)/(c**4*x**4 - 2*c**2*x**2 + 1),x)*a**2*b*c + 4*int(acosh(c*x )**3/(c**4*x**4 - 2*c**2*x**2 + 1),x)*b**3*c**3*x**2 - 4*int(acosh(c*x)**3 /(c**4*x**4 - 2*c**2*x**2 + 1),x)*b**3*c + 12*int(acosh(c*x)**2/(c**4*x**4 - 2*c**2*x**2 + 1),x)*a*b**2*c**3*x**2 - 12*int(acosh(c*x)**2/(c**4*x**4 - 2*c**2*x**2 + 1),x)*a*b**2*c - log(c**2*x - c)*a**3*c**2*x**2 + log(c**2 *x - c)*a**3 + log(c**2*x + c)*a**3*c**2*x**2 - log(c**2*x + c)*a**3 - 2*a **3*c*x)/(4*c*d**2*(c**2*x**2 - 1))