Integrand size = 20, antiderivative size = 387 \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {1}{4 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )}+\frac {\text {arccosh}(a x)^2}{4 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {9 \text {arccosh}(a x)^2}{8 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arccosh}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac {5 \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c^3}+\frac {3 \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {5 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{2 a c^3}+\frac {9 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{8 a c^3}+\frac {5 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{2 a c^3}-\frac {9 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{8 a c^3}-\frac {9 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {9 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {9 \operatorname {PolyLog}\left (4,-e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {9 \operatorname {PolyLog}\left (4,e^{\text {arccosh}(a x)}\right )}{4 a c^3} \]
-1/4*x*arccosh(a*x)/c^3/(-a^2*x^2+1)+1/4*arccosh(a*x)^2/a/c^3/(a*x-1)^(3/2 )/(a*x+1)^(3/2)+1/4*x*arccosh(a*x)^3/c^3/(-a^2*x^2+1)^2+3/8*x*arccosh(a*x) ^3/c^3/(-a^2*x^2+1)-5*arccosh(a*x)*arctanh(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2) )/a/c^3+3/4*arccosh(a*x)^3*arctanh(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3- 5/2*polylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3+9/8*arccosh(a*x)^2*p olylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3+5/2*polylog(2,a*x+(a*x-1) ^(1/2)*(a*x+1)^(1/2))/a/c^3-9/8*arccosh(a*x)^2*polylog(2,a*x+(a*x-1)^(1/2) *(a*x+1)^(1/2))/a/c^3-9/4*arccosh(a*x)*polylog(3,-a*x-(a*x-1)^(1/2)*(a*x+1 )^(1/2))/a/c^3+9/4*arccosh(a*x)*polylog(3,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)) /a/c^3+9/4*polylog(4,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3-9/4*polylog(4 ,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3+1/4/a/c^3/(a*x-1)^(1/2)/(a*x+1)^(1 /2)-9/8*arccosh(a*x)^2/a/c^3/(a*x-1)^(1/2)/(a*x+1)^(1/2)
Time = 6.83 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.18 \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^3} \, dx=-\frac {3 \pi ^4-6 \text {arccosh}(a x)^4-8 \coth \left (\frac {1}{2} \text {arccosh}(a x)\right )+40 \text {arccosh}(a x)^2 \coth \left (\frac {1}{2} \text {arccosh}(a x)\right )-4 \text {arccosh}(a x) \text {csch}^2\left (\frac {1}{2} \text {arccosh}(a x)\right )+6 \text {arccosh}(a x)^3 \text {csch}^2\left (\frac {1}{2} \text {arccosh}(a x)\right )-\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x)^2 \text {csch}^4\left (\frac {1}{2} \text {arccosh}(a x)\right )-\text {arccosh}(a x)^3 \text {csch}^4\left (\frac {1}{2} \text {arccosh}(a x)\right )-160 \text {arccosh}(a x) \log \left (1-e^{-\text {arccosh}(a x)}\right )+160 \text {arccosh}(a x) \log \left (1+e^{-\text {arccosh}(a x)}\right )-24 \text {arccosh}(a x)^3 \log \left (1+e^{-\text {arccosh}(a x)}\right )+24 \text {arccosh}(a x)^3 \log \left (1-e^{\text {arccosh}(a x)}\right )+8 \left (-20+9 \text {arccosh}(a x)^2\right ) \operatorname {PolyLog}\left (2,-e^{-\text {arccosh}(a x)}\right )+160 \operatorname {PolyLog}\left (2,e^{-\text {arccosh}(a x)}\right )+72 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )+144 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{-\text {arccosh}(a x)}\right )-144 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )+144 \operatorname {PolyLog}\left (4,-e^{-\text {arccosh}(a x)}\right )+144 \operatorname {PolyLog}\left (4,e^{\text {arccosh}(a x)}\right )-4 \text {arccosh}(a x) \text {sech}^2\left (\frac {1}{2} \text {arccosh}(a x)\right )+6 \text {arccosh}(a x)^3 \text {sech}^2\left (\frac {1}{2} \text {arccosh}(a x)\right )+\text {arccosh}(a x)^3 \text {sech}^4\left (\frac {1}{2} \text {arccosh}(a x)\right )-\frac {16 \text {arccosh}(a x)^2 \sinh ^4\left (\frac {1}{2} \text {arccosh}(a x)\right )}{\left (\frac {-1+a x}{1+a x}\right )^{3/2} (1+a x)^3}+8 \tanh \left (\frac {1}{2} \text {arccosh}(a x)\right )-40 \text {arccosh}(a x)^2 \tanh \left (\frac {1}{2} \text {arccosh}(a x)\right )}{64 a c^3} \]
-1/64*(3*Pi^4 - 6*ArcCosh[a*x]^4 - 8*Coth[ArcCosh[a*x]/2] + 40*ArcCosh[a*x ]^2*Coth[ArcCosh[a*x]/2] - 4*ArcCosh[a*x]*Csch[ArcCosh[a*x]/2]^2 + 6*ArcCo sh[a*x]^3*Csch[ArcCosh[a*x]/2]^2 - Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Ar cCosh[a*x]^2*Csch[ArcCosh[a*x]/2]^4 - ArcCosh[a*x]^3*Csch[ArcCosh[a*x]/2]^ 4 - 160*ArcCosh[a*x]*Log[1 - E^(-ArcCosh[a*x])] + 160*ArcCosh[a*x]*Log[1 + E^(-ArcCosh[a*x])] - 24*ArcCosh[a*x]^3*Log[1 + E^(-ArcCosh[a*x])] + 24*Ar cCosh[a*x]^3*Log[1 - E^ArcCosh[a*x]] + 8*(-20 + 9*ArcCosh[a*x]^2)*PolyLog[ 2, -E^(-ArcCosh[a*x])] + 160*PolyLog[2, E^(-ArcCosh[a*x])] + 72*ArcCosh[a* x]^2*PolyLog[2, E^ArcCosh[a*x]] + 144*ArcCosh[a*x]*PolyLog[3, -E^(-ArcCosh [a*x])] - 144*ArcCosh[a*x]*PolyLog[3, E^ArcCosh[a*x]] + 144*PolyLog[4, -E^ (-ArcCosh[a*x])] + 144*PolyLog[4, E^ArcCosh[a*x]] - 4*ArcCosh[a*x]*Sech[Ar cCosh[a*x]/2]^2 + 6*ArcCosh[a*x]^3*Sech[ArcCosh[a*x]/2]^2 + ArcCosh[a*x]^3 *Sech[ArcCosh[a*x]/2]^4 - (16*ArcCosh[a*x]^2*Sinh[ArcCosh[a*x]/2]^4)/(((-1 + a*x)/(1 + a*x))^(3/2)*(1 + a*x)^3) + 8*Tanh[ArcCosh[a*x]/2] - 40*ArcCos h[a*x]^2*Tanh[ArcCosh[a*x]/2])/(a*c^3)
Result contains complex when optimal does not.
Time = 4.78 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.07, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {6316, 27, 6316, 6318, 3042, 26, 4670, 3011, 6330, 25, 6304, 6316, 83, 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.
| \(\displaystyle \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 6316 |
| \(\displaystyle \frac {3 \int \frac {\text {arccosh}(a x)^3}{c^2 \left (1-a^2 x^2\right )^2}dx}{4 c}-\frac {3 a \int \frac {x \text {arccosh}(a x)^2}{(a x-1)^{5/2} (a x+1)^{5/2}}dx}{4 c^3}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \int \frac {\text {arccosh}(a x)^3}{\left (1-a^2 x^2\right )^2}dx}{4 c^3}-\frac {3 a \int \frac {x \text {arccosh}(a x)^2}{(a x-1)^{5/2} (a x+1)^{5/2}}dx}{4 c^3}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 6316 |
| \(\displaystyle \frac {3 \left (\frac {1}{2} \int \frac {\text {arccosh}(a x)^3}{1-a^2 x^2}dx+\frac {3}{2} a \int \frac {x \text {arccosh}(a x)^2}{(a x-1)^{3/2} (a x+1)^{3/2}}dx+\frac {x \text {arccosh}(a x)^3}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {3 a \int \frac {x \text {arccosh}(a x)^2}{(a x-1)^{5/2} (a x+1)^{5/2}}dx}{4 c^3}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 6318 |
| \(\displaystyle \frac {3 \left (\frac {3}{2} a \int \frac {x \text {arccosh}(a x)^2}{(a x-1)^{3/2} (a x+1)^{3/2}}dx-\frac {\int \frac {\text {arccosh}(a x)^3}{\sqrt {\frac {a x-1}{a x+1}} (a x+1)}d\text {arccosh}(a x)}{2 a}+\frac {x \text {arccosh}(a x)^3}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {3 a \int \frac {x \text {arccosh}(a x)^2}{(a x-1)^{5/2} (a x+1)^{5/2}}dx}{4 c^3}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {3}{2} a \int \frac {x \text {arccosh}(a x)^2}{(a x-1)^{3/2} (a x+1)^{3/2}}dx-\frac {\int i \text {arccosh}(a x)^3 \csc (i \text {arccosh}(a x))d\text {arccosh}(a x)}{2 a}+\frac {x \text {arccosh}(a x)^3}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {3 a \int \frac {x \text {arccosh}(a x)^2}{(a x-1)^{5/2} (a x+1)^{5/2}}dx}{4 c^3}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {3 \left (\frac {3}{2} a \int \frac {x \text {arccosh}(a x)^2}{(a x-1)^{3/2} (a x+1)^{3/2}}dx-\frac {i \int \text {arccosh}(a x)^3 \csc (i \text {arccosh}(a x))d\text {arccosh}(a x)}{2 a}+\frac {x \text {arccosh}(a x)^3}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {3 a \int \frac {x \text {arccosh}(a x)^2}{(a x-1)^{5/2} (a x+1)^{5/2}}dx}{4 c^3}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {3 \left (-\frac {i \left (3 i \int \text {arccosh}(a x)^2 \log \left (1-e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-3 i \int \text {arccosh}(a x)^2 \log \left (1+e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)+2 i \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+\frac {3}{2} a \int \frac {x \text {arccosh}(a x)^2}{(a x-1)^{3/2} (a x+1)^{3/2}}dx+\frac {x \text {arccosh}(a x)^3}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {3 a \int \frac {x \text {arccosh}(a x)^2}{(a x-1)^{5/2} (a x+1)^{5/2}}dx}{4 c^3}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {3 \left (-\frac {i \left (-3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+\frac {3}{2} a \int \frac {x \text {arccosh}(a x)^2}{(a x-1)^{3/2} (a x+1)^{3/2}}dx+\frac {x \text {arccosh}(a x)^3}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {3 a \int \frac {x \text {arccosh}(a x)^2}{(a x-1)^{5/2} (a x+1)^{5/2}}dx}{4 c^3}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 6330 |
| \(\displaystyle \frac {3 \left (\frac {3}{2} a \left (\frac {2 \int -\frac {\text {arccosh}(a x)}{(1-a x) (a x+1)}dx}{a}-\frac {\text {arccosh}(a x)^2}{a^2 \sqrt {a x-1} \sqrt {a x+1}}\right )-\frac {i \left (-3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+\frac {x \text {arccosh}(a x)^3}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {3 a \left (\frac {2 \int \frac {\text {arccosh}(a x)}{(1-a x)^2 (a x+1)^2}dx}{3 a}-\frac {\text {arccosh}(a x)^2}{3 a^2 (a x-1)^{3/2} (a x+1)^{3/2}}\right )}{4 c^3}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 \left (\frac {3}{2} a \left (-\frac {2 \int \frac {\text {arccosh}(a x)}{(1-a x) (a x+1)}dx}{a}-\frac {\text {arccosh}(a x)^2}{a^2 \sqrt {a x-1} \sqrt {a x+1}}\right )-\frac {i \left (-3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+\frac {x \text {arccosh}(a x)^3}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {3 a \left (\frac {2 \int \frac {\text {arccosh}(a x)}{(1-a x)^2 (a x+1)^2}dx}{3 a}-\frac {\text {arccosh}(a x)^2}{3 a^2 (a x-1)^{3/2} (a x+1)^{3/2}}\right )}{4 c^3}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 6304 |
| \(\displaystyle \frac {3 \left (\frac {3}{2} a \left (-\frac {2 \int \frac {\text {arccosh}(a x)}{1-a^2 x^2}dx}{a}-\frac {\text {arccosh}(a x)^2}{a^2 \sqrt {a x-1} \sqrt {a x+1}}\right )-\frac {i \left (-3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+\frac {x \text {arccosh}(a x)^3}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {3 a \left (\frac {2 \int \frac {\text {arccosh}(a x)}{\left (1-a^2 x^2\right )^2}dx}{3 a}-\frac {\text {arccosh}(a x)^2}{3 a^2 (a x-1)^{3/2} (a x+1)^{3/2}}\right )}{4 c^3}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 6316 |
| \(\displaystyle \frac {3 \left (\frac {3}{2} a \left (-\frac {2 \int \frac {\text {arccosh}(a x)}{1-a^2 x^2}dx}{a}-\frac {\text {arccosh}(a x)^2}{a^2 \sqrt {a x-1} \sqrt {a x+1}}\right )-\frac {i \left (-3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+\frac {x \text {arccosh}(a x)^3}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {3 a \left (\frac {2 \left (\frac {1}{2} \int \frac {\text {arccosh}(a x)}{1-a^2 x^2}dx+\frac {1}{2} a \int \frac {x}{(a x-1)^{3/2} (a x+1)^{3/2}}dx+\frac {x \text {arccosh}(a x)}{2 \left (1-a^2 x^2\right )}\right )}{3 a}-\frac {\text {arccosh}(a x)^2}{3 a^2 (a x-1)^{3/2} (a x+1)^{3/2}}\right )}{4 c^3}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle \frac {3 \left (\frac {3}{2} a \left (-\frac {2 \int \frac {\text {arccosh}(a x)}{1-a^2 x^2}dx}{a}-\frac {\text {arccosh}(a x)^2}{a^2 \sqrt {a x-1} \sqrt {a x+1}}\right )-\frac {i \left (-3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+\frac {x \text {arccosh}(a x)^3}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {3 a \left (\frac {2 \left (\frac {1}{2} \int \frac {\text {arccosh}(a x)}{1-a^2 x^2}dx+\frac {x \text {arccosh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{2 a \sqrt {a x-1} \sqrt {a x+1}}\right )}{3 a}-\frac {\text {arccosh}(a x)^2}{3 a^2 (a x-1)^{3/2} (a x+1)^{3/2}}\right )}{4 c^3}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 6318 |
| \(\displaystyle \frac {3 \left (\frac {3}{2} a \left (\frac {2 \int \frac {\text {arccosh}(a x)}{\sqrt {\frac {a x-1}{a x+1}} (a x+1)}d\text {arccosh}(a x)}{a^2}-\frac {\text {arccosh}(a x)^2}{a^2 \sqrt {a x-1} \sqrt {a x+1}}\right )-\frac {i \left (-3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+\frac {x \text {arccosh}(a x)^3}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {3 a \left (\frac {2 \left (-\frac {\int \frac {\text {arccosh}(a x)}{\sqrt {\frac {a x-1}{a x+1}} (a x+1)}d\text {arccosh}(a x)}{2 a}+\frac {x \text {arccosh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{2 a \sqrt {a x-1} \sqrt {a x+1}}\right )}{3 a}-\frac {\text {arccosh}(a x)^2}{3 a^2 (a x-1)^{3/2} (a x+1)^{3/2}}\right )}{4 c^3}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {3}{2} a \left (-\frac {\text {arccosh}(a x)^2}{a^2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {2 \int i \text {arccosh}(a x) \csc (i \text {arccosh}(a x))d\text {arccosh}(a x)}{a^2}\right )-\frac {i \left (-3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+\frac {x \text {arccosh}(a x)^3}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {3 a \left (-\frac {\text {arccosh}(a x)^2}{3 a^2 (a x-1)^{3/2} (a x+1)^{3/2}}+\frac {2 \left (-\frac {\int i \text {arccosh}(a x) \csc (i \text {arccosh}(a x))d\text {arccosh}(a x)}{2 a}+\frac {x \text {arccosh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{2 a \sqrt {a x-1} \sqrt {a x+1}}\right )}{3 a}\right )}{4 c^3}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {3 \left (\frac {3}{2} a \left (-\frac {\text {arccosh}(a x)^2}{a^2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {2 i \int \text {arccosh}(a x) \csc (i \text {arccosh}(a x))d\text {arccosh}(a x)}{a^2}\right )-\frac {i \left (-3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+\frac {x \text {arccosh}(a x)^3}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {3 a \left (-\frac {\text {arccosh}(a x)^2}{3 a^2 (a x-1)^{3/2} (a x+1)^{3/2}}+\frac {2 \left (-\frac {i \int \text {arccosh}(a x) \csc (i \text {arccosh}(a x))d\text {arccosh}(a x)}{2 a}+\frac {x \text {arccosh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{2 a \sqrt {a x-1} \sqrt {a x+1}}\right )}{3 a}\right )}{4 c^3}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {3 \left (\frac {3}{2} a \left (-\frac {\text {arccosh}(a x)^2}{a^2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {2 i \left (i \int \log \left (1-e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-i \int \log \left (1+e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)+2 i \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{a^2}\right )-\frac {i \left (-3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+\frac {x \text {arccosh}(a x)^3}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {3 a \left (-\frac {\text {arccosh}(a x)^2}{3 a^2 (a x-1)^{3/2} (a x+1)^{3/2}}+\frac {2 \left (-\frac {i \left (i \int \log \left (1-e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-i \int \log \left (1+e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)+2 i \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+\frac {x \text {arccosh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{2 a \sqrt {a x-1} \sqrt {a x+1}}\right )}{3 a}\right )}{4 c^3}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {3 \left (\frac {3}{2} a \left (-\frac {\text {arccosh}(a x)^2}{a^2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {2 i \left (i \int e^{-\text {arccosh}(a x)} \log \left (1-e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-i \int e^{-\text {arccosh}(a x)} \log \left (1+e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}+2 i \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{a^2}\right )-\frac {i \left (-3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+\frac {x \text {arccosh}(a x)^3}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {3 a \left (-\frac {\text {arccosh}(a x)^2}{3 a^2 (a x-1)^{3/2} (a x+1)^{3/2}}+\frac {2 \left (-\frac {i \left (i \int e^{-\text {arccosh}(a x)} \log \left (1-e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-i \int e^{-\text {arccosh}(a x)} \log \left (1+e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}+2 i \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+\frac {x \text {arccosh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{2 a \sqrt {a x-1} \sqrt {a x+1}}\right )}{3 a}\right )}{4 c^3}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {3 \left (-\frac {i \left (-3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+\frac {3}{2} a \left (-\frac {\text {arccosh}(a x)^2}{a^2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {2 i \left (2 i \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )-i \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )}{a^2}\right )+\frac {x \text {arccosh}(a x)^3}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {3 a \left (-\frac {\text {arccosh}(a x)^2}{3 a^2 (a x-1)^{3/2} (a x+1)^{3/2}}+\frac {2 \left (\frac {x \text {arccosh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {i \left (2 i \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )-i \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )}{2 a}-\frac {1}{2 a \sqrt {a x-1} \sqrt {a x+1}}\right )}{3 a}\right )}{4 c^3}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle \frac {3 \left (-\frac {i \left (-3 i \left (2 \left (\text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )-\int \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)\right )-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+3 i \left (2 \left (\text {arccosh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )-\int \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)\right )-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+\frac {3}{2} a \left (-\frac {\text {arccosh}(a x)^2}{a^2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {2 i \left (2 i \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )-i \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )}{a^2}\right )+\frac {x \text {arccosh}(a x)^3}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {3 a \left (-\frac {\text {arccosh}(a x)^2}{3 a^2 (a x-1)^{3/2} (a x+1)^{3/2}}+\frac {2 \left (\frac {x \text {arccosh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {i \left (2 i \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )-i \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )}{2 a}-\frac {1}{2 a \sqrt {a x-1} \sqrt {a x+1}}\right )}{3 a}\right )}{4 c^3}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {3 \left (-\frac {i \left (-3 i \left (2 \left (\text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )-\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}\right )-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+3 i \left (2 \left (\text {arccosh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )-\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}\right )-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+\frac {3}{2} a \left (-\frac {\text {arccosh}(a x)^2}{a^2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {2 i \left (2 i \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )-i \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )}{a^2}\right )+\frac {x \text {arccosh}(a x)^3}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {3 a \left (-\frac {\text {arccosh}(a x)^2}{3 a^2 (a x-1)^{3/2} (a x+1)^{3/2}}+\frac {2 \left (\frac {x \text {arccosh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {i \left (2 i \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )-i \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )}{2 a}-\frac {1}{2 a \sqrt {a x-1} \sqrt {a x+1}}\right )}{3 a}\right )}{4 c^3}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {3 a \left (-\frac {\text {arccosh}(a x)^2}{3 a^2 (a x-1)^{3/2} (a x+1)^{3/2}}+\frac {2 \left (\frac {x \text {arccosh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {i \left (2 i \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )-i \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )}{2 a}-\frac {1}{2 a \sqrt {a x-1} \sqrt {a x+1}}\right )}{3 a}\right )}{4 c^3}+\frac {3 \left (\frac {3}{2} a \left (-\frac {\text {arccosh}(a x)^2}{a^2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {2 i \left (2 i \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )-i \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )}{a^2}\right )+\frac {x \text {arccosh}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {i \left (2 i \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )-3 i \left (2 \left (\text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )-\operatorname {PolyLog}\left (4,-e^{\text {arccosh}(a x)}\right )\right )-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+3 i \left (2 \left (\text {arccosh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )-\operatorname {PolyLog}\left (4,e^{\text {arccosh}(a x)}\right )\right )-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )\right )}{2 a}\right )}{4 c^3}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
(x*ArcCosh[a*x]^3)/(4*c^3*(1 - a^2*x^2)^2) - (3*a*(-1/3*ArcCosh[a*x]^2/(a^ 2*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)) + (2*(-1/2*1/(a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (x*ArcCosh[a*x])/(2*(1 - a^2*x^2)) - ((I/2)*((2*I)*ArcCosh[a*x]* ArcTanh[E^ArcCosh[a*x]] + I*PolyLog[2, -E^ArcCosh[a*x]] - I*PolyLog[2, E^A rcCosh[a*x]]))/a))/(3*a)))/(4*c^3) + (3*((x*ArcCosh[a*x]^3)/(2*(1 - a^2*x^ 2)) + (3*a*(-(ArcCosh[a*x]^2/(a^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])) + ((2*I)* ((2*I)*ArcCosh[a*x]*ArcTanh[E^ArcCosh[a*x]] + I*PolyLog[2, -E^ArcCosh[a*x] ] - I*PolyLog[2, E^ArcCosh[a*x]]))/a^2))/2 - ((I/2)*((2*I)*ArcCosh[a*x]^3* ArcTanh[E^ArcCosh[a*x]] - (3*I)*(-(ArcCosh[a*x]^2*PolyLog[2, -E^ArcCosh[a* x]]) + 2*(ArcCosh[a*x]*PolyLog[3, -E^ArcCosh[a*x]] - PolyLog[4, -E^ArcCosh [a*x]])) + (3*I)*(-(ArcCosh[a*x]^2*PolyLog[2, E^ArcCosh[a*x]]) + 2*(ArcCos h[a*x]*PolyLog[3, E^ArcCosh[a*x]] - PolyLog[4, E^ArcCosh[a*x]]))))/a))/(4* c^3)
3.3.45.3.1 Defintions of rubi rules used
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[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> 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]
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 = 0.55 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.36
| method | result | size |
| derivativedivides | \(\frac {-\frac {3 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{3}+9 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}-5 a x \operatorname {arccosh}\left (a x \right )^{3}-11 \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}-2 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )-2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+2 a x \,\operatorname {arccosh}\left (a x \right )+2 \sqrt {a x -1}\, \sqrt {a x +1}}{8 \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right ) c^{3}}-\frac {5 \,\operatorname {arccosh}\left (a x \right ) \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{3}}-\frac {5 \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{3}}+\frac {5 \,\operatorname {arccosh}\left (a x \right ) \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{3}}+\frac {5 \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{3}}+\frac {3 \operatorname {arccosh}\left (a x \right )^{3} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {9 \operatorname {arccosh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}-\frac {9 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}+\frac {9 \operatorname {polylog}\left (4, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}-\frac {3 \operatorname {arccosh}\left (a x \right )^{3} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}-\frac {9 \operatorname {arccosh}\left (a x \right )^{2} \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {9 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}-\frac {9 \operatorname {polylog}\left (4, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}}{a}\) | \(527\) |
| default | \(\frac {-\frac {3 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{3}+9 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}-5 a x \operatorname {arccosh}\left (a x \right )^{3}-11 \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}-2 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )-2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+2 a x \,\operatorname {arccosh}\left (a x \right )+2 \sqrt {a x -1}\, \sqrt {a x +1}}{8 \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right ) c^{3}}-\frac {5 \,\operatorname {arccosh}\left (a x \right ) \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{3}}-\frac {5 \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{3}}+\frac {5 \,\operatorname {arccosh}\left (a x \right ) \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{3}}+\frac {5 \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{3}}+\frac {3 \operatorname {arccosh}\left (a x \right )^{3} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {9 \operatorname {arccosh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}-\frac {9 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}+\frac {9 \operatorname {polylog}\left (4, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}-\frac {3 \operatorname {arccosh}\left (a x \right )^{3} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}-\frac {9 \operatorname {arccosh}\left (a x \right )^{2} \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {9 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}-\frac {9 \operatorname {polylog}\left (4, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}}{a}\) | \(527\) |
1/a*(-1/8*(3*a^3*x^3*arccosh(a*x)^3+9*a^2*x^2*arccosh(a*x)^2*(a*x-1)^(1/2) *(a*x+1)^(1/2)-5*a*x*arccosh(a*x)^3-11*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1 )^(1/2)-2*a^3*x^3*arccosh(a*x)-2*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+2*a*x *arccosh(a*x)+2*(a*x-1)^(1/2)*(a*x+1)^(1/2))/(a^4*x^4-2*a^2*x^2+1)/c^3-5/2 /c^3*arccosh(a*x)*ln(1+a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))-5/2/c^3*polylog(2, -a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))+5/2/c^3*arccosh(a*x)*ln(1-a*x-(a*x-1)^(1 /2)*(a*x+1)^(1/2))+5/2/c^3*polylog(2,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))+3/8/ c^3*arccosh(a*x)^3*ln(1+a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))+9/8/c^3*arccosh(a *x)^2*polylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))-9/4/c^3*arccosh(a*x)*pol ylog(3,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))+9/4/c^3*polylog(4,-a*x-(a*x-1)^(1 /2)*(a*x+1)^(1/2))-3/8/c^3*arccosh(a*x)^3*ln(1-a*x-(a*x-1)^(1/2)*(a*x+1)^( 1/2))-9/8/c^3*arccosh(a*x)^2*polylog(2,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))+9/ 4/c^3*arccosh(a*x)*polylog(3,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))-9/4/c^3*poly log(4,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))
\[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^3} \, dx=\int { -\frac {\operatorname {arcosh}\left (a x\right )^{3}}{{\left (a^{2} c x^{2} - c\right )}^{3}} \,d x } \]
\[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^3} \, dx=- \frac {\int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx}{c^{3}} \]
\[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^3} \, dx=\int { -\frac {\operatorname {arcosh}\left (a x\right )^{3}}{{\left (a^{2} c x^{2} - c\right )}^{3}} \,d x } \]
-1/16*(6*a^3*x^3 - 10*a*x - 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1) + 3*( a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^3/(a^5*c^3*x^4 - 2*a^3*c^3*x^2 + a*c^3) - integrate(-3/16*(6*a^5*x^5 - 16*a^3*x^3 + (6*a^4*x^4 - 10*a^2*x^2 - 3*(a^5*x^5 - 2*a^3*x^3 + a*x)*log( a*x + 1) + 3*(a^5*x^5 - 2*a^3*x^3 + a*x)*log(a*x - 1))*sqrt(a*x + 1)*sqrt( a*x - 1) + 10*a*x - 3*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x + 1) + 3*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2/(a^7*c^3*x^7 - 3*a^5*c^3*x^5 + 3*a^3*c^3*x^3 - a*c^3 *x + (a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3)*sqrt(a*x + 1)*sqr t(a*x - 1)), x)
\[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^3} \, dx=\int { -\frac {\operatorname {arcosh}\left (a x\right )^{3}}{{\left (a^{2} c x^{2} - c\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^3} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{{\left (c-a^2\,c\,x^2\right )}^3} \,d x \]