Optimal. Leaf size=630 \[ -\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac {3 c^2 d \sqrt {d-c^2 d x^2} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {3 a b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^3 d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {3 i b^2 c^2 d \sqrt {d-c^2 d x^2} \text {Li}_3\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {3 i b^2 c^2 d \sqrt {d-c^2 d x^2} \text {Li}_3\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-2 b^2 c^2 d \sqrt {d-c^2 d x^2}+\frac {b^2 c^2 d \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b^2 c^3 d x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 1.37, antiderivative size = 642, normalized size of antiderivative = 1.02, number of steps used = 18, number of rules used = 15, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {5798, 5740, 5743, 5761, 4180, 2531, 2282, 6589, 5654, 74, 14, 5731, 460, 92, 205} \[ -\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {3 i b^2 c^2 d \sqrt {d-c^2 d x^2} \text {PolyLog}\left (3,-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {3 i b^2 c^2 d \sqrt {d-c^2 d x^2} \text {PolyLog}\left (3,i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {3 a b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^3 d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {d (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac {3 c^2 d \sqrt {d-c^2 d x^2} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {c x-1} \sqrt {c x+1}}-2 b^2 c^2 d \sqrt {d-c^2 d x^2}+\frac {b^2 c^2 d \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b^2 c^3 d x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 74
Rule 92
Rule 205
Rule 460
Rule 2282
Rule 2531
Rule 4180
Rule 5654
Rule 5731
Rule 5740
Rule 5743
Rule 5761
Rule 5798
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x^3} \, dx &=-\frac {\left (d \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^3} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-1+c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{x^2} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 c^2 d \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}{2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac {\left (3 c^2 d \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}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {1+c^2 x^2}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b c^3 d \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}}\\ &=b^2 c^2 d \sqrt {d-c^2 d x^2}+\frac {3 a b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac {\left (3 c^2 d \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 )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b^2 c^3 d \sqrt {d-c^2 d x^2}\right ) \int \cosh ^{-1}(c x) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=b^2 c^2 d \sqrt {d-c^2 d x^2}+\frac {3 a b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b^2 c^3 d x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac {3 c^2 d \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 (3 i b c^2 d \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 (3 i b c^2 d \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 (b^2 c^3 d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 b^2 c^4 d \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}}\\ &=-2 b^2 c^2 d \sqrt {d-c^2 d x^2}+\frac {3 a b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b^2 c^3 d x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac {3 c^2 d \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 {b^2 c^2 d \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 i b c^2 d \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 {3 i b c^2 d \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 (3 i b^2 c^2 d \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 (3 i b^2 c^2 d \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}}\\ &=-2 b^2 c^2 d \sqrt {d-c^2 d x^2}+\frac {3 a b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b^2 c^3 d x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac {3 c^2 d \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 {b^2 c^2 d \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 i b c^2 d \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 {3 i b c^2 d \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 (3 i b^2 c^2 d \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 (3 i b^2 c^2 d \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}}\\ &=-2 b^2 c^2 d \sqrt {d-c^2 d x^2}+\frac {3 a b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b^2 c^3 d x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac {3 c^2 d \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 {b^2 c^2 d \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 i b c^2 d \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 {3 i b c^2 d \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 {3 i b^2 c^2 d \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 {3 i b^2 c^2 d \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*}
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Mathematica [A] time = 176.12, size = 1129, normalized size = 1.79 \[ \frac {1}{2} d \sqrt {d-c^2 d x^2} \left (\frac {4 x^2 \cosh ^{-1}(c x) c^4}{(c x-1)^{3/2} \sqrt {c x+1}}-\frac {2 x \cosh ^{-1}(c x)^2 c^3}{c x-1}-\frac {4 x \cosh ^{-1}(c x) c^3}{(c x-1)^{3/2} \sqrt {c x+1}}-\frac {2 x \tan ^{-1}\left (\frac {1}{\sqrt {c^2 x^2-1}}\right ) c^3}{(c x-1) \sqrt {c^2 x^2-1}}-\frac {4 x c^3}{c x-1}+\frac {2 \cosh ^{-1}(c x)^2 c^2}{c x-1}-\frac {2 \cosh ^{-1}(c x) c^2}{(c x-1)^{3/2} \sqrt {c x+1}}+\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {c^2 x^2-1}}\right ) c^2}{(c x-1) \sqrt {c^2 x^2-1}}-\frac {3 i \sqrt {\frac {c x-1}{c x+1}} \cosh ^{-1}(c x)^2 \log \left (1-i e^{-\cosh ^{-1}(c x)}\right ) c^2}{c x-1}+\frac {3 i \sqrt {\frac {c x-1}{c x+1}} \cosh ^{-1}(c x)^2 \log \left (1+i e^{-\cosh ^{-1}(c x)}\right ) c^2}{c x-1}-\frac {6 i \sqrt {\frac {c x-1}{c x+1}} \cosh ^{-1}(c x) \text {Li}_2\left (-i e^{-\cosh ^{-1}(c x)}\right ) c^2}{c x-1}+\frac {6 i \sqrt {\frac {c x-1}{c x+1}} \cosh ^{-1}(c x) \text {Li}_2\left (i e^{-\cosh ^{-1}(c x)}\right ) c^2}{c x-1}-\frac {6 i \sqrt {\frac {c x-1}{c x+1}} \text {Li}_3\left (-i e^{-\cosh ^{-1}(c x)}\right ) c^2}{c x-1}+\frac {6 i \sqrt {\frac {c x-1}{c x+1}} \text {Li}_3\left (i e^{-\cosh ^{-1}(c x)}\right ) c^2}{c x-1}+\frac {4 c^2}{c x-1}+\frac {\cosh ^{-1}(c x)^2 c}{x-c x^2}+\frac {2 \cosh ^{-1}(c x) c}{x (c x-1)^{3/2} \sqrt {c x+1}}+\frac {\cosh ^{-1}(c x)^2}{x^2 (c x-1)}\right ) b^2-2 a c^2 d \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 ) b+\frac {i a c^2 d^2 \left (-\frac {i (c x-1) \cosh ^{-1}(c x) (c x+1)}{c^2 x^2}+\sqrt {\frac {c x-1}{c x+1}} \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right ) (c x+1)-\sqrt {\frac {c x-1}{c x+1}} \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right ) (c x+1)+\sqrt {\frac {c x-1}{c x+1}} \text {Li}_2\left (-i e^{-\cosh ^{-1}(c x)}\right ) (c x+1)-\sqrt {\frac {c x-1}{c x+1}} \text {Li}_2\left (i e^{-\cosh ^{-1}(c x)}\right ) (c x+1)-\frac {i \sqrt {\frac {c x-1}{c x+1}} (c x+1)}{c x}\right ) b}{\sqrt {-d (c x-1) (c x+1)}}-\frac {3}{2} a^2 c^2 d^{3/2} \log (x)+\frac {3}{2} a^2 c^2 d^{3/2} \log \left (d+\sqrt {-d \left (c^2 x^2-1\right )} \sqrt {d}\right )+\left (-c^2 d a^2-\frac {d a^2}{2 x^2}\right ) \sqrt {-d \left (c^2 x^2-1\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a^{2} c^{2} d x^{2} - a^{2} d + {\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, {\left (a b c^{2} d x^{2} - a b d\right )} \operatorname {arcosh}\left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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maple [F] time = 0.89, size = 0, normalized size = 0.00 \[ \int \frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, {\left (3 \, c^{2} d^{\frac {3}{2}} \log \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {d}}{{\left | x \right |}} + \frac {2 \, d}{{\left | x \right |}}\right ) - {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} - 3 \, \sqrt {-c^{2} d x^{2} + d} c^{2} d - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{2}}\right )} a^{2} + \int \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} b^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2}}{x^{3}} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a b \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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