Optimal. Leaf size=836 \[ -\frac{2 b c^5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^5}{25 \sqrt{c x-1} \sqrt{c x+1}}+\frac{22 b c^3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^3}{45 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2}{27} b^2 c^2 d^2 \sqrt{d-c^2 d x^2} x^2-\frac{2 b^2 c d^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) x}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{16 b c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x}{15 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 a b c d^2 \sqrt{d-c^2 d x^2} x}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{5} \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{3} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2+d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{2 i b d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{2 i b d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{2 i b^2 d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{2 i b^2 d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{125 (1-c x) (c x+1)}+\frac{68}{27} b^2 d^2 \sqrt{d-c^2 d x^2}+\frac{8 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{225 (1-c x) (c x+1)}+\frac{16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{75 (1-c x) (c x+1)} \]
[Out]
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Rubi [A] time = 1.77567, 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 5798
Rule 5745
Rule 5743
Rule 5761
Rule 4180
Rule 2531
Rule 2282
Rule 6589
Rule 5654
Rule 74
Rule 5680
Rule 12
Rule 460
Rule 194
Rule 520
Rule 1247
Rule 698
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 = 7.19022, 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{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )-\text{PolyLog}\left (2,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{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right ) \cosh ^{-1}(c x)-2 \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right ) \cosh ^{-1}(c x)+2 \text{PolyLog}\left (3,-i e^{-\cosh ^{-1}(c x)}\right )-2 \text{PolyLog}\left (3,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.
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Maple [F] time = 0.413, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{2}}{x} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{2}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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