Optimal. Leaf size=650 \[ -\frac {3 i b c^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 d \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{d x \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \sqrt {c x-1} \sqrt {c x+1} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}+\frac {4 b c^2 \sqrt {c x-1} \sqrt {c x+1} \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 i b^2 c^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_3\left (-i e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b^2 c^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_3\left (i e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {c x-1} \sqrt {c x+1} \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )}{d \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 1.49, antiderivative size = 650, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 15, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {5798, 5748, 5756, 5761, 4180, 2531, 2282, 6589, 5694, 4182, 2279, 2391, 5746, 92, 205} \[ -\frac {3 i b c^2 \sqrt {c x-1} \sqrt {c x+1} \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {c x-1} \sqrt {c x+1} \text {PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1} \text {PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1} \text {PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 i b^2 c^2 \sqrt {c x-1} \sqrt {c x+1} \text {PolyLog}\left (3,-i e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b^2 c^2 \sqrt {c x-1} \sqrt {c x+1} \text {PolyLog}\left (3,i e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 d \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{d x \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \sqrt {c x-1} \sqrt {c x+1} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}+\frac {4 b c^2 \sqrt {c x-1} \sqrt {c x+1} \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {c x-1} \sqrt {c x+1} \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )}{d \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 92
Rule 205
Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 4180
Rule 4182
Rule 5694
Rule 5746
Rule 5748
Rule 5756
Rule 5761
Rule 5798
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{x^3 (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x^2 \left (-1+c^2 x^2\right )} \, dx}{d \sqrt {d-c^2 d x^2}}-\frac {\left (3 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d \sqrt {d-c^2 d x^2}}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 d \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 d \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{d \sqrt {d-c^2 d x^2}}+\frac {\left (b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{d \sqrt {d-c^2 d x^2}}-\frac {\left (3 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 d \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \text {sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d \sqrt {d-c^2 d x^2}}+\frac {\left (b c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (3 b c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 d \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {4 b c^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (3 i b c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (3 i b c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (3 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (3 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 d \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {4 b c^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (3 i b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (3 i b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (3 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (3 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 d \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {4 b c^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (3 i b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (3 i b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 d \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {4 b c^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 i b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_3\left (-i e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_3\left (i e^{\cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 92.28, size = 979, normalized size = 1.51 \[ \frac {b^2 \sqrt {d-c^2 d x^2} \left (-2 \cosh ^2\left (\frac {1}{2} \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)^2+2 \sinh ^2\left (\frac {1}{2} \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)^2+\left (\frac {1}{c^2 x^2}-1\right ) \cosh ^{-1}(c x)^2+3 i \sqrt {\frac {c x-1}{c x+1}} (c x+1) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right ) \cosh ^{-1}(c x)^2-3 i \sqrt {\frac {c x-1}{c x+1}} (c x+1) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right ) \cosh ^{-1}(c x)^2-\frac {2 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)}{c x}+4 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \log \left (1-e^{-\cosh ^{-1}(c x)}\right ) \cosh ^{-1}(c x)-4 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \log \left (1+e^{-\cosh ^{-1}(c x)}\right ) \cosh ^{-1}(c x)+6 i \sqrt {\frac {c x-1}{c x+1}} (c x+1) \text {Li}_2\left (-i e^{-\cosh ^{-1}(c x)}\right ) \cosh ^{-1}(c x)-6 i \sqrt {\frac {c x-1}{c x+1}} (c x+1) \text {Li}_2\left (i e^{-\cosh ^{-1}(c x)}\right ) \cosh ^{-1}(c x)+4 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \tan ^{-1}\left (\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )+4 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \text {Li}_2\left (-e^{-\cosh ^{-1}(c x)}\right )-4 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \text {Li}_2\left (e^{-\cosh ^{-1}(c x)}\right )+6 i \sqrt {\frac {c x-1}{c x+1}} (c x+1) \text {Li}_3\left (-i e^{-\cosh ^{-1}(c x)}\right )-6 i \sqrt {\frac {c x-1}{c x+1}} (c x+1) \text {Li}_3\left (i e^{-\cosh ^{-1}(c x)}\right )\right ) c^2}{2 d^2 \left (c^2 x^2-1\right )}+\frac {a \left (3 a \sqrt {d} \log (x) c^2-3 a \sqrt {d} \log \left (d+\sqrt {d-c^2 d x^2} \sqrt {d}\right ) c^2-\frac {2 b d \left (-2 \cosh ^{-1}(c x) \cosh ^2\left (\frac {1}{2} \cosh ^{-1}(c x)\right )+2 \cosh ^{-1}(c x) \sinh ^2\left (\frac {1}{2} \cosh ^{-1}(c x)\right )-\frac {\sqrt {\frac {c x-1}{c x+1}} (c x+1)}{c x}+\left (\frac {1}{c^2 x^2}-1\right ) \cosh ^{-1}(c x)+3 i \sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-3 i \sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )+2 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \log \left (\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )+3 i \sqrt {\frac {c x-1}{c x+1}} (c x+1) \text {Li}_2\left (-i e^{-\cosh ^{-1}(c x)}\right )-3 i \sqrt {\frac {c x-1}{c x+1}} (c x+1) \text {Li}_2\left (i e^{-\cosh ^{-1}(c x)}\right )\right ) c^2}{\sqrt {d-c^2 d x^2}}-\frac {a \left (3 c^2 x^2-1\right ) \sqrt {d-c^2 d x^2}}{x^2 \left (c^2 x^2-1\right )}\right )}{2 d^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} d x^{2} + d} {\left (b^{2} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname {arcosh}\left (c x\right ) + a^{2}\right )}}{c^{4} d^{2} x^{7} - 2 \, c^{2} d^{2} x^{5} + d^{2} x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.92, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}{x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}\, dx \]
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 \[ -\frac {1}{2} \, {\left (\frac {3 \, c^{2} \log \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {d}}{{\left | x \right |}} + \frac {2 \, d}{{\left | x \right |}}\right )}{d^{\frac {3}{2}}} - \frac {3 \, c^{2}}{\sqrt {-c^{2} d x^{2} + d} d} + \frac {1}{\sqrt {-c^{2} d x^{2} + d} d x^{2}}\right )} a^{2} + \int \frac {b^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{3}} + \frac {2 \, a b \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{3}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{x^3\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{x^{3} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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