Optimal. Leaf size=573 \[ \frac {2 i b d \sqrt {d-c^2 d x^2} \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {2 i b d \sqrt {d-c^2 d x^2} \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {2 a b c d x \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2+d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {2 d \sqrt {d-c^2 d x^2} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b c^3 d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 i b^2 d \sqrt {d-c^2 d x^2} \text {Li}_3\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {2 i b^2 d \sqrt {d-c^2 d x^2} \text {Li}_3\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {2}{27} b^2 c^2 d x^2 \sqrt {d-c^2 d x^2}+\frac {68}{27} b^2 d \sqrt {d-c^2 d x^2}-\frac {2 b^2 c d x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 1.25, antiderivative size = 585, normalized size of antiderivative = 1.02, number of steps used = 18, number of rules used = 13, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {5798, 5745, 5743, 5761, 4180, 2531, 2282, 6589, 5654, 74, 5680, 12, 460} \[ \frac {2 i b d \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {2 i b d \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {2 i b^2 d \sqrt {d-c^2 d x^2} \text {PolyLog}\left (3,-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {2 i b^2 d \sqrt {d-c^2 d x^2} \text {PolyLog}\left (3,i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {2 a b c d x \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b c^3 d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} d (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {2 d \sqrt {d-c^2 d x^2} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {2}{27} b^2 c^2 d x^2 \sqrt {d-c^2 d x^2}+\frac {68}{27} b^2 d \sqrt {d-c^2 d x^2}-\frac {2 b^2 c d x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 74
Rule 460
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 )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x} \, 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} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{3} d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {\left (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}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b c d \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 {2 b c d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {-1+c x} \sqrt {1+c x}}+d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {\left (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}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b c 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}}-\frac {\left (2 b^2 c^2 d \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 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {-1+c x} \sqrt {1+c x}}+d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {\left (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 )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b^2 c d \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 \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 {2}{27} b^2 c^2 d x^2 \sqrt {d-c^2 d x^2}-\frac {2 a b c d x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b^2 c d x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {-1+c x} \sqrt {1+c x}}+d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {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 (2 i b 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 (2 i b 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 (14 b^2 c^2 d \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 \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 {68}{27} b^2 d \sqrt {d-c^2 d x^2}-\frac {2}{27} b^2 c^2 d x^2 \sqrt {d-c^2 d x^2}-\frac {2 a b c d x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b^2 c d x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {-1+c x} \sqrt {1+c x}}+d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {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 {2 i b 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 {2 i b 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 (2 i b^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 (2 i b^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 {68}{27} b^2 d \sqrt {d-c^2 d x^2}-\frac {2}{27} b^2 c^2 d x^2 \sqrt {d-c^2 d x^2}-\frac {2 a b c d x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b^2 c d x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {-1+c x} \sqrt {1+c x}}+d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {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 {2 i b 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 {2 i b 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 (2 i b^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 (2 i b^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 {68}{27} b^2 d \sqrt {d-c^2 d x^2}-\frac {2}{27} b^2 c^2 d x^2 \sqrt {d-c^2 d x^2}-\frac {2 a b c d x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b^2 c d x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {-1+c x} \sqrt {1+c x}}+d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {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 {2 i b 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 {2 i b 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 {2 i b^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 {2 i b^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 = 2.91, size = 650, normalized size = 1.13 \[ -a^2 d^{3/2} \log \left (\sqrt {d} \sqrt {d-c^2 d x^2}+d\right )-\frac {1}{3} a^2 d \left (c^2 x^2-4\right ) \sqrt {d-c^2 d x^2}+a^2 d^{3/2} \log (c x)+\frac {2 a b d \sqrt {d-c^2 d x^2} \left (i \text {Li}_2\left (-i e^{-\cosh ^{-1}(c x)}\right )-i \text {Li}_2\left (i e^{-\cosh ^{-1}(c x)}\right )-c x+c x \sqrt {\frac {c x-1}{c x+1}} \cosh ^{-1}(c x)+\sqrt {\frac {c x-1}{c x+1}} \cosh ^{-1}(c x)+i \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-i \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )\right )}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}-\frac {a b d \sqrt {d-c^2 d x^2} \left (9 c x+12 \left (\frac {c x-1}{c x+1}\right )^{3/2} (c x+1)^3 \cosh ^{-1}(c x)-\cosh \left (3 \cosh ^{-1}(c x)\right )\right )}{18 \sqrt {\frac {c x-1}{c x+1}} (c x+1)}+b^2 d \sqrt {d-c^2 d x^2} \left (\frac {i \left (2 \cosh ^{-1}(c x) \text {Li}_2\left (-i e^{-\cosh ^{-1}(c x)}\right )-2 \cosh ^{-1}(c x) \text {Li}_2\left (i e^{-\cosh ^{-1}(c x)}\right )+2 \text {Li}_3\left (-i e^{-\cosh ^{-1}(c x)}\right )-2 \text {Li}_3\left (i e^{-\cosh ^{-1}(c x)}\right )+\cosh ^{-1}(c x)^2 \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-\cosh ^{-1}(c x)^2 \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )\right )}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}+\cosh ^{-1}(c x)^2+\frac {2 c x \sqrt {\frac {c x-1}{c x+1}} \cosh ^{-1}(c x)}{1-c x}+2\right )-\frac {1}{54} b^2 d \sqrt {d-c^2 d x^2} \left (9 \left (\cosh \left (2 \cosh ^{-1}(c x)\right )-1\right ) \cosh ^{-1}(c x)^2+\frac {3 \sqrt {\frac {c x-1}{c x+1}} \left (9 c x-\cosh \left (3 \cosh ^{-1}(c x)\right )\right ) \cosh ^{-1}(c x)}{c x-1}+2 \left (\cosh \left (2 \cosh ^{-1}(c x)\right )-13\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a^{2} c^{2} d x^{2} - a^{2} d + {\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, {\left (a b c^{2} d x^{2} - a b d\right )} \operatorname {arcosh}\left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{x}, x\right ) \]
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 = 0.65, size = 0, normalized size = 0.00 \[ \int \frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}{x}\, dx \]
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 \[ -\frac {1}{3} \, {\left (3 \, d^{\frac {3}{2}} \log \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {d}}{{\left | x \right |}} + \frac {2 \, d}{{\left | x \right |}}\right ) - {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {-c^{2} d x^{2} + d} d\right )} a^{2} + \int \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} b^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2}}{x} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a b \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{x}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x} \,d x \]
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 \[ \int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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