Integrand size = 20, antiderivative size = 258 \[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^3} \, dx=-\frac {x}{12 c^3 \left (1-a^2 x^2\right )}+\frac {\text {arccosh}(a x)}{6 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {3 \text {arccosh}(a x)}{4 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arccosh}(a x)^2}{8 c^3 \left (1-a^2 x^2\right )}+\frac {3 \text {arccosh}(a x)^2 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {5 \text {arctanh}(a x)}{6 a c^3}+\frac {3 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {3 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {3 \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {3 \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )}{4 a c^3} \]
-1/12*x/c^3/(-a^2*x^2+1)+1/6*arccosh(a*x)/a/c^3/(a*x-1)^(3/2)/(a*x+1)^(3/2 )+1/4*x*arccosh(a*x)^2/c^3/(-a^2*x^2+1)^2+3/8*x*arccosh(a*x)^2/c^3/(-a^2*x ^2+1)+3/4*arccosh(a*x)^2*arctanh(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3-5/ 6*arctanh(a*x)/a/c^3+3/4*arccosh(a*x)*polylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1) ^(1/2))/a/c^3-3/4*arccosh(a*x)*polylog(2,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/ a/c^3-3/4*polylog(3,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3+3/4*polylog(3, a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3-3/4*arccosh(a*x)/a/c^3/(a*x-1)^(1/2 )/(a*x+1)^(1/2)
Time = 3.30 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.24 \[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^3} \, dx=-\frac {80 \text {arccosh}(a x) \coth \left (\frac {1}{2} \text {arccosh}(a x)\right )+2 \left (-2+9 \text {arccosh}(a x)^2\right ) \text {csch}^2\left (\frac {1}{2} \text {arccosh}(a x)\right )-2 \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x) \text {csch}^4\left (\frac {1}{2} \text {arccosh}(a x)\right )-3 \text {arccosh}(a x)^2 \text {csch}^4\left (\frac {1}{2} \text {arccosh}(a x)\right )-160 \log \left (\tanh \left (\frac {1}{2} \text {arccosh}(a x)\right )\right )+72 \left (\text {arccosh}(a x)^2 \log \left (1-e^{-\text {arccosh}(a x)}\right )-\text {arccosh}(a x)^2 \log \left (1+e^{-\text {arccosh}(a x)}\right )+2 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{-\text {arccosh}(a x)}\right )-2 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{-\text {arccosh}(a x)}\right )+2 \operatorname {PolyLog}\left (3,-e^{-\text {arccosh}(a x)}\right )-2 \operatorname {PolyLog}\left (3,e^{-\text {arccosh}(a x)}\right )\right )+2 \left (-2+9 \text {arccosh}(a x)^2\right ) \text {sech}^2\left (\frac {1}{2} \text {arccosh}(a x)\right )+3 \text {arccosh}(a x)^2 \text {sech}^4\left (\frac {1}{2} \text {arccosh}(a x)\right )-\frac {32 \text {arccosh}(a x) \sinh ^4\left (\frac {1}{2} \text {arccosh}(a x)\right )}{\left (\frac {-1+a x}{1+a x}\right )^{3/2} (1+a x)^3}-80 \text {arccosh}(a x) \tanh \left (\frac {1}{2} \text {arccosh}(a x)\right )}{192 a c^3} \]
-1/192*(80*ArcCosh[a*x]*Coth[ArcCosh[a*x]/2] + 2*(-2 + 9*ArcCosh[a*x]^2)*C sch[ArcCosh[a*x]/2]^2 - 2*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh[a*x ]*Csch[ArcCosh[a*x]/2]^4 - 3*ArcCosh[a*x]^2*Csch[ArcCosh[a*x]/2]^4 - 160*L og[Tanh[ArcCosh[a*x]/2]] + 72*(ArcCosh[a*x]^2*Log[1 - E^(-ArcCosh[a*x])] - ArcCosh[a*x]^2*Log[1 + E^(-ArcCosh[a*x])] + 2*ArcCosh[a*x]*PolyLog[2, -E^ (-ArcCosh[a*x])] - 2*ArcCosh[a*x]*PolyLog[2, E^(-ArcCosh[a*x])] + 2*PolyLo g[3, -E^(-ArcCosh[a*x])] - 2*PolyLog[3, E^(-ArcCosh[a*x])]) + 2*(-2 + 9*Ar cCosh[a*x]^2)*Sech[ArcCosh[a*x]/2]^2 + 3*ArcCosh[a*x]^2*Sech[ArcCosh[a*x]/ 2]^4 - (32*ArcCosh[a*x]*Sinh[ArcCosh[a*x]/2]^4)/(((-1 + a*x)/(1 + a*x))^(3 /2)*(1 + a*x)^3) - 80*ArcCosh[a*x]*Tanh[ArcCosh[a*x]/2])/(a*c^3)
Result contains complex when optimal does not.
Time = 2.99 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.99, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6316, 27, 6316, 6318, 3042, 26, 4670, 3011, 2720, 6330, 25, 39, 215, 219, 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.
| \(\displaystyle \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 6316 |
| \(\displaystyle \frac {3 \int \frac {\text {arccosh}(a x)^2}{c^2 \left (1-a^2 x^2\right )^2}dx}{4 c}-\frac {a \int \frac {x \text {arccosh}(a x)}{(a x-1)^{5/2} (a x+1)^{5/2}}dx}{2 c^3}+\frac {x \text {arccosh}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \int \frac {\text {arccosh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx}{4 c^3}-\frac {a \int \frac {x \text {arccosh}(a x)}{(a x-1)^{5/2} (a x+1)^{5/2}}dx}{2 c^3}+\frac {x \text {arccosh}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 6316 |
| \(\displaystyle \frac {3 \left (\frac {1}{2} \int \frac {\text {arccosh}(a x)^2}{1-a^2 x^2}dx+a \int \frac {x \text {arccosh}(a x)}{(a x-1)^{3/2} (a x+1)^{3/2}}dx+\frac {x \text {arccosh}(a x)^2}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {a \int \frac {x \text {arccosh}(a x)}{(a x-1)^{5/2} (a x+1)^{5/2}}dx}{2 c^3}+\frac {x \text {arccosh}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 6318 |
| \(\displaystyle \frac {3 \left (a \int \frac {x \text {arccosh}(a x)}{(a x-1)^{3/2} (a x+1)^{3/2}}dx-\frac {\int \frac {\text {arccosh}(a x)^2}{\sqrt {\frac {a x-1}{a x+1}} (a x+1)}d\text {arccosh}(a x)}{2 a}+\frac {x \text {arccosh}(a x)^2}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {a \int \frac {x \text {arccosh}(a x)}{(a x-1)^{5/2} (a x+1)^{5/2}}dx}{2 c^3}+\frac {x \text {arccosh}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (a \int \frac {x \text {arccosh}(a x)}{(a x-1)^{3/2} (a x+1)^{3/2}}dx-\frac {\int i \text {arccosh}(a x)^2 \csc (i \text {arccosh}(a x))d\text {arccosh}(a x)}{2 a}+\frac {x \text {arccosh}(a x)^2}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {a \int \frac {x \text {arccosh}(a x)}{(a x-1)^{5/2} (a x+1)^{5/2}}dx}{2 c^3}+\frac {x \text {arccosh}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {3 \left (a \int \frac {x \text {arccosh}(a x)}{(a x-1)^{3/2} (a x+1)^{3/2}}dx-\frac {i \int \text {arccosh}(a x)^2 \csc (i \text {arccosh}(a x))d\text {arccosh}(a x)}{2 a}+\frac {x \text {arccosh}(a x)^2}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {a \int \frac {x \text {arccosh}(a x)}{(a x-1)^{5/2} (a x+1)^{5/2}}dx}{2 c^3}+\frac {x \text {arccosh}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {3 \left (-\frac {i \left (2 i \int \text {arccosh}(a x) \log \left (1-e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-2 i \int \text {arccosh}(a x) \log \left (1+e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)+2 i \text {arccosh}(a x)^2 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+a \int \frac {x \text {arccosh}(a x)}{(a x-1)^{3/2} (a x+1)^{3/2}}dx+\frac {x \text {arccosh}(a x)^2}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {a \int \frac {x \text {arccosh}(a x)}{(a x-1)^{5/2} (a x+1)^{5/2}}dx}{2 c^3}+\frac {x \text {arccosh}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {3 \left (-\frac {i \left (-2 i \left (\int \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+2 i \left (\int \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^2 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+a \int \frac {x \text {arccosh}(a x)}{(a x-1)^{3/2} (a x+1)^{3/2}}dx+\frac {x \text {arccosh}(a x)^2}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {a \int \frac {x \text {arccosh}(a x)}{(a x-1)^{5/2} (a x+1)^{5/2}}dx}{2 c^3}+\frac {x \text {arccosh}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {3 \left (-\frac {i \left (-2 i \left (\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+2 i \left (\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^2 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+a \int \frac {x \text {arccosh}(a x)}{(a x-1)^{3/2} (a x+1)^{3/2}}dx+\frac {x \text {arccosh}(a x)^2}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {a \int \frac {x \text {arccosh}(a x)}{(a x-1)^{5/2} (a x+1)^{5/2}}dx}{2 c^3}+\frac {x \text {arccosh}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 6330 |
| \(\displaystyle \frac {3 \left (a \left (\frac {\int -\frac {1}{(1-a x) (a x+1)}dx}{a}-\frac {\text {arccosh}(a x)}{a^2 \sqrt {a x-1} \sqrt {a x+1}}\right )-\frac {i \left (-2 i \left (\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+2 i \left (\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^2 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+\frac {x \text {arccosh}(a x)^2}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {a \left (\frac {\int \frac {1}{(1-a x)^2 (a x+1)^2}dx}{3 a}-\frac {\text {arccosh}(a x)}{3 a^2 (a x-1)^{3/2} (a x+1)^{3/2}}\right )}{2 c^3}+\frac {x \text {arccosh}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 \left (a \left (-\frac {\int \frac {1}{(1-a x) (a x+1)}dx}{a}-\frac {\text {arccosh}(a x)}{a^2 \sqrt {a x-1} \sqrt {a x+1}}\right )-\frac {i \left (-2 i \left (\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+2 i \left (\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^2 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+\frac {x \text {arccosh}(a x)^2}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {a \left (\frac {\int \frac {1}{(1-a x)^2 (a x+1)^2}dx}{3 a}-\frac {\text {arccosh}(a x)}{3 a^2 (a x-1)^{3/2} (a x+1)^{3/2}}\right )}{2 c^3}+\frac {x \text {arccosh}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 39 |
| \(\displaystyle \frac {3 \left (a \left (-\frac {\int \frac {1}{1-a^2 x^2}dx}{a}-\frac {\text {arccosh}(a x)}{a^2 \sqrt {a x-1} \sqrt {a x+1}}\right )-\frac {i \left (-2 i \left (\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+2 i \left (\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^2 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+\frac {x \text {arccosh}(a x)^2}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {a \left (\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2}dx}{3 a}-\frac {\text {arccosh}(a x)}{3 a^2 (a x-1)^{3/2} (a x+1)^{3/2}}\right )}{2 c^3}+\frac {x \text {arccosh}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {3 \left (a \left (-\frac {\int \frac {1}{1-a^2 x^2}dx}{a}-\frac {\text {arccosh}(a x)}{a^2 \sqrt {a x-1} \sqrt {a x+1}}\right )-\frac {i \left (-2 i \left (\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+2 i \left (\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^2 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+\frac {x \text {arccosh}(a x)^2}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {a \left (\frac {\frac {1}{2} \int \frac {1}{1-a^2 x^2}dx+\frac {x}{2 \left (1-a^2 x^2\right )}}{3 a}-\frac {\text {arccosh}(a x)}{3 a^2 (a x-1)^{3/2} (a x+1)^{3/2}}\right )}{2 c^3}+\frac {x \text {arccosh}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3 \left (-\frac {i \left (-2 i \left (\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+2 i \left (\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^2 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+a \left (-\frac {\text {arccosh}(a x)}{a^2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arctanh}(a x)}{a^2}\right )+\frac {x \text {arccosh}(a x)^2}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}-\frac {a \left (\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{3 a}-\frac {\text {arccosh}(a x)}{3 a^2 (a x-1)^{3/2} (a x+1)^{3/2}}\right )}{2 c^3}+\frac {x \text {arccosh}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {3 \left (a \left (-\frac {\text {arccosh}(a x)}{a^2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arctanh}(a x)}{a^2}\right )+\frac {x \text {arccosh}(a x)^2}{2 \left (1-a^2 x^2\right )}-\frac {i \left (2 i \text {arccosh}(a x)^2 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )-2 i \left (\operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+2 i \left (\operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )\right )}{2 a}\right )}{4 c^3}-\frac {a \left (\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{3 a}-\frac {\text {arccosh}(a x)}{3 a^2 (a x-1)^{3/2} (a x+1)^{3/2}}\right )}{2 c^3}+\frac {x \text {arccosh}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}\) |
(x*ArcCosh[a*x]^2)/(4*c^3*(1 - a^2*x^2)^2) - (a*(-1/3*ArcCosh[a*x]/(a^2*(- 1 + a*x)^(3/2)*(1 + a*x)^(3/2)) + (x/(2*(1 - a^2*x^2)) + ArcTanh[a*x]/(2*a ))/(3*a)))/(2*c^3) + (3*((x*ArcCosh[a*x]^2)/(2*(1 - a^2*x^2)) + a*(-(ArcCo sh[a*x]/(a^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])) - ArcTanh[a*x]/a^2) - ((I/2)*( (2*I)*ArcCosh[a*x]^2*ArcTanh[E^ArcCosh[a*x]] - (2*I)*(-(ArcCosh[a*x]*PolyL og[2, -E^ArcCosh[a*x]]) + PolyLog[3, -E^ArcCosh[a*x]]) + (2*I)*(-(ArcCosh[ a*x]*PolyLog[2, E^ArcCosh[a*x]]) + PolyLog[3, E^ArcCosh[a*x]])))/a))/(4*c^ 3)
3.2.69.3.1 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_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[( a*c + b*d*x^2)^m, x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && ( IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 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[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_.)*((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]
Time = 0.50 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.24
| method | result | size |
| derivativedivides | \(\frac {-\frac {9 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{2}+18 a^{2} x^{2} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}-15 a x \operatorname {arccosh}\left (a x \right )^{2}-22 \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )-2 a^{3} x^{3}+2 a x}{24 \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right ) c^{3}}-\frac {5 \,\operatorname {arctanh}\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{3 c^{3}}+\frac {3 \operatorname {arccosh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {3 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}-\frac {3 \operatorname {polylog}\left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}-\frac {3 \operatorname {arccosh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}-\frac {3 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}+\frac {3 \operatorname {polylog}\left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}}{a}\) | \(320\) |
| default | \(\frac {-\frac {9 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{2}+18 a^{2} x^{2} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}-15 a x \operatorname {arccosh}\left (a x \right )^{2}-22 \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )-2 a^{3} x^{3}+2 a x}{24 \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right ) c^{3}}-\frac {5 \,\operatorname {arctanh}\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{3 c^{3}}+\frac {3 \operatorname {arccosh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {3 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}-\frac {3 \operatorname {polylog}\left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}-\frac {3 \operatorname {arccosh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}-\frac {3 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}+\frac {3 \operatorname {polylog}\left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}}{a}\) | \(320\) |
1/a*(-1/24*(9*a^3*x^3*arccosh(a*x)^2+18*a^2*x^2*arccosh(a*x)*(a*x-1)^(1/2) *(a*x+1)^(1/2)-15*a*x*arccosh(a*x)^2-22*(a*x-1)^(1/2)*(a*x+1)^(1/2)*arccos h(a*x)-2*a^3*x^3+2*a*x)/(a^4*x^4-2*a^2*x^2+1)/c^3-5/3/c^3*arctanh(a*x+(a*x -1)^(1/2)*(a*x+1)^(1/2))+3/8/c^3*arccosh(a*x)^2*ln(1+a*x+(a*x-1)^(1/2)*(a* x+1)^(1/2))+3/4/c^3*arccosh(a*x)*polylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2 ))-3/4/c^3*polylog(3,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))-3/8/c^3*arccosh(a*x )^2*ln(1-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))-3/4/c^3*arccosh(a*x)*polylog(2,a *x+(a*x-1)^(1/2)*(a*x+1)^(1/2))+3/4/c^3*polylog(3,a*x+(a*x-1)^(1/2)*(a*x+1 )^(1/2)))
\[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^3} \, dx=\int { -\frac {\operatorname {arcosh}\left (a x\right )^{2}}{{\left (a^{2} c x^{2} - c\right )}^{3}} \,d x } \]
\[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^3} \, dx=- \frac {\int \frac {\operatorname {acosh}^{2}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx}{c^{3}} \]
\[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^3} \, dx=\int { -\frac {\operatorname {arcosh}\left (a x\right )^{2}}{{\left (a^{2} c x^{2} - c\right )}^{3}} \,d x } \]
-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))^2/(a^5*c^3*x^4 - 2*a^3*c^3*x^2 + a*c^3) - integrate(-1/8*(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))/(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)*sqrt(a *x - 1)), x)
\[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^3} \, dx=\int { -\frac {\operatorname {arcosh}\left (a x\right )^{2}}{{\left (a^{2} c x^{2} - c\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^3} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^2}{{\left (c-a^2\,c\,x^2\right )}^3} \,d x \]