Optimal. Leaf size=836 \[ -\frac {2 b c^5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^5}{25 \sqrt {c x-1} \sqrt {c x+1}}+\frac {22 b c^3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^3}{45 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2}{27} b^2 c^2 d^2 \sqrt {d-c^2 d x^2} x^2-\frac {2 b^2 c d^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x) x}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {16 b c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x}{15 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 a b c d^2 \sqrt {d-c^2 d x^2} x}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2+d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {2 i b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {2 i b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {2 i b^2 d^2 \sqrt {d-c^2 d x^2} \text {Li}_3\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {2 i b^2 d^2 \sqrt {d-c^2 d x^2} \text {Li}_3\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{125 (1-c x) (c x+1)}+\frac {68}{27} b^2 d^2 \sqrt {d-c^2 d x^2}+\frac {8 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{225 (1-c x) (c x+1)}+\frac {16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{75 (1-c x) (c x+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.78, antiderivative size = 867, normalized size of antiderivative = 1.04, number of steps used = 25, number of rules used = 17, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.586, Rules used = {5798, 5745, 5743, 5761, 4180, 2531, 2282, 6589, 5654, 74, 5680, 12, 460, 194, 520, 1247, 698} \[ -\frac {2 b c^5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^5}{25 \sqrt {c x-1} \sqrt {c x+1}}+\frac {22 b c^3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^3}{45 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2}{27} b^2 c^2 d^2 \sqrt {d-c^2 d x^2} x^2-\frac {2 b^2 c d^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x) x}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {16 b c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x}{15 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 a b c d^2 \sqrt {d-c^2 d x^2} x}{\sqrt {c x-1} \sqrt {c x+1}}+d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{5} d^2 (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} d^2 (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {2 i b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {2 i b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {2 i b^2 d^2 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (3,-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {2 i b^2 d^2 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (3,i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{125 (1-c x) (c x+1)}+\frac {68}{27} b^2 d^2 \sqrt {d-c^2 d x^2}+\frac {8 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{225 (1-c x) (c x+1)}+\frac {16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{75 (1-c x) (c x+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 74
Rule 194
Rule 460
Rule 520
Rule 698
Rule 1247
Rule 2282
Rule 2531
Rule 4180
Rule 5654
Rule 5680
Rule 5743
Rule 5745
Rule 5761
Rule 5798
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x} \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{5} d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{5 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {2 b c d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{3} d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{5} d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{15 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{5 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {16 b c d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {22 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{45 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt {-1+c x} \sqrt {1+c x}}+d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{5} d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{75 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (-3+c^2 x^2\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {2 a b c d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {22 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{45 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt {-1+c x} \sqrt {1+c x}}+d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{5} d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \text {sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b^2 c d^2 \sqrt {d-c^2 d x^2}\right ) \int \cosh ^{-1}(c x) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (-3+c^2 x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{9 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b^2 c^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{\sqrt {-1+c^2 x^2}} \, dx}{75 (-1+c x) (1+c x)}\\ &=-\frac {2}{27} b^2 c^2 d^2 x^2 \sqrt {d-c^2 d x^2}-\frac {2 a b c d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b^2 c d^2 x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {22 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{45 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt {-1+c x} \sqrt {1+c x}}+d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{5} d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 i b d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 i b d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (14 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{27 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 c^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {15-10 c^2 x+3 c^4 x^2}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{75 (-1+c x) (1+c x)}\\ &=\frac {68}{27} b^2 d^2 \sqrt {d-c^2 d x^2}-\frac {2}{27} b^2 c^2 d^2 x^2 \sqrt {d-c^2 d x^2}-\frac {2 a b c d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b^2 c d^2 x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {22 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{45 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt {-1+c x} \sqrt {1+c x}}+d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{5} d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 i b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 i b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 i b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 i b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 c^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {8}{\sqrt {-1+c^2 x}}-4 \sqrt {-1+c^2 x}+3 \left (-1+c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{75 (-1+c x) (1+c x)}\\ &=\frac {68}{27} b^2 d^2 \sqrt {d-c^2 d x^2}-\frac {2}{27} b^2 c^2 d^2 x^2 \sqrt {d-c^2 d x^2}-\frac {2 a b c d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{75 (1-c x) (1+c x)}+\frac {8 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{225 (1-c x) (1+c x)}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{125 (1-c x) (1+c x)}-\frac {2 b^2 c d^2 x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {22 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{45 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt {-1+c x} \sqrt {1+c x}}+d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{5} d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 i b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 i b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 i b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 i b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {68}{27} b^2 d^2 \sqrt {d-c^2 d x^2}-\frac {2}{27} b^2 c^2 d^2 x^2 \sqrt {d-c^2 d x^2}-\frac {2 a b c d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{75 (1-c x) (1+c x)}+\frac {8 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{225 (1-c x) (1+c x)}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{125 (1-c x) (1+c x)}-\frac {2 b^2 c d^2 x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {22 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{45 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt {-1+c x} \sqrt {1+c x}}+d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{5} d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 i b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 i b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 i b^2 d^2 \sqrt {d-c^2 d x^2} \text {Li}_3\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 i b^2 d^2 \sqrt {d-c^2 d x^2} \text {Li}_3\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 7.25, size = 1031, normalized size = 1.23 \[ a^2 \log (c x) d^{5/2}-a^2 \log \left (d+\sqrt {-d \left (c^2 x^2-1\right )} \sqrt {d}\right ) d^{5/2}+\frac {a b \sqrt {-d (c x-1) (c x+1)} \left (-12 \left (\frac {c x-1}{c x+1}\right )^{3/2} \cosh ^{-1}(c x) (c x+1)^3-9 c x+\cosh \left (3 \cosh ^{-1}(c x)\right )\right ) d^2}{9 \sqrt {\frac {c x-1}{c x+1}} (c x+1)}-\frac {1}{27} b^2 \sqrt {-d (c x-1) (c x+1)} \left (-9 \cosh ^{-1}(c x)^2-\frac {3 \cosh \left (3 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}+\frac {27 c x \cosh ^{-1}(c x)}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}+\left (9 \cosh ^{-1}(c x)^2+2\right ) \cosh \left (2 \cosh ^{-1}(c x)\right )-26\right ) d^2+2 a b \sqrt {-d (c x-1) (c x+1)} \left (-\frac {c x}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}+\cosh ^{-1}(c x)+\frac {i \cosh ^{-1}(c x) \left (\log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-\log \left (1+i e^{-\cosh ^{-1}(c x)}\right )\right )}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}+\frac {i \left (\text {Li}_2\left (-i e^{-\cosh ^{-1}(c x)}\right )-\text {Li}_2\left (i e^{-\cosh ^{-1}(c x)}\right )\right )}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}\right ) d^2+b^2 \sqrt {-d (c x-1) (c x+1)} \left (\cosh ^{-1}(c x)^2-\frac {2 c x \cosh ^{-1}(c x)}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}+\frac {i \left (\log \left (1-i e^{-\cosh ^{-1}(c x)}\right ) \cosh ^{-1}(c x)^2-\log \left (1+i e^{-\cosh ^{-1}(c x)}\right ) \cosh ^{-1}(c x)^2+2 \text {Li}_2\left (-i e^{-\cosh ^{-1}(c x)}\right ) \cosh ^{-1}(c x)-2 \text {Li}_2\left (i e^{-\cosh ^{-1}(c x)}\right ) \cosh ^{-1}(c x)+2 \text {Li}_3\left (-i e^{-\cosh ^{-1}(c x)}\right )-2 \text {Li}_3\left (i e^{-\cosh ^{-1}(c x)}\right )\right )}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}+2\right ) d^2-\frac {a b \sqrt {-d (c x-1) (c x+1)} \left (-450 c x+450 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)+25 \cosh \left (3 \cosh ^{-1}(c x)\right )+9 \cosh \left (5 \cosh ^{-1}(c x)\right )-75 \cosh ^{-1}(c x) \sinh \left (3 \cosh ^{-1}(c x)\right )-45 \cosh ^{-1}(c x) \sinh \left (5 \cosh ^{-1}(c x)\right )\right ) d^2}{1800 \sqrt {\frac {c x-1}{c x+1}} (c x+1)}-\frac {b^2 \sqrt {-d (c x-1) (c x+1)} \left (6750 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)^2-1125 \sinh \left (3 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)^2-675 \sinh \left (5 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)^2-13500 c x \cosh ^{-1}(c x)+750 \cosh \left (3 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)+270 \cosh \left (5 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)+13500 \sqrt {\frac {c x-1}{c x+1}} (c x+1)-250 \sinh \left (3 \cosh ^{-1}(c x)\right )-54 \sinh \left (5 \cosh ^{-1}(c x)\right )\right ) d^2}{54000 \sqrt {\frac {c x-1}{c x+1}} (c x+1)}+\sqrt {-d \left (c^2 x^2-1\right )} \left (\frac {1}{5} a^2 c^4 d^2 x^4-\frac {11}{15} a^2 c^2 d^2 x^2+\frac {23 a^2 d^2}{15}\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} c^{4} d^{2} x^{4} - 2 \, a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} + {\left (b^{2} c^{4} d^{2} x^{4} - 2 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, {\left (a b c^{4} d^{2} x^{4} - 2 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \operatorname {arcosh}\left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.77, size = 0, normalized size = 0.00 \[ \int \frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{15} \, {\left (15 \, d^{\frac {5}{2}} \log \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {d}}{{\left | x \right |}} + \frac {2 \, d}{{\left | x \right |}}\right ) - 3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} - 5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d - 15 \, \sqrt {-c^{2} d x^{2} + d} d^{2}\right )} a^{2} + \int \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} b^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2}}{x} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} a b \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________