Optimal. Leaf size=292 \[ \frac{i b d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{i b d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{3} \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )+d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{2 d \sqrt{d-c^2 d x^2} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{c x-1} \sqrt{c x+1}}-\frac{4 b c d x \sqrt{d-c^2 d x^2}}{3 \sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 0.789303, antiderivative size = 304, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {5798, 5745, 5743, 5761, 4180, 2279, 2391, 8} \[ \frac{i b d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{i b d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}+d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{2 d \sqrt{d-c^2 d x^2} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{c x-1} \sqrt{c x+1}}-\frac{4 b c d x \sqrt{d-c^2 d x^2}}{3 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5745
Rule 5743
Rule 5761
Rule 4180
Rule 2279
Rule 2391
Rule 8
Rubi steps
\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{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 )}{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 )+\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 )}{x} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right ) \, dx}{3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d x \sqrt{d-c^2 d x^2}}{3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d x^3 \sqrt{d-c^2 d x^2}}{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 )+\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 )-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int 1 \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{4 b c d x \sqrt{d-c^2 d x^2}}{3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d x^3 \sqrt{d-c^2 d x^2}}{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 )+\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 )-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \text{sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{4 b c d x \sqrt{d-c^2 d x^2}}{3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d x^3 \sqrt{d-c^2 d x^2}}{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 )+\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 )-\frac{2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (i b d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (i b d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{4 b c d x \sqrt{d-c^2 d x^2}}{3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d x^3 \sqrt{d-c^2 d x^2}}{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 )+\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 )-\frac{2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (i b d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (i b d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{4 b c d x \sqrt{d-c^2 d x^2}}{3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d x^3 \sqrt{d-c^2 d x^2}}{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 )+\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 )-\frac{2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{i b d \sqrt{d-c^2 d x^2} \text{Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{i b d \sqrt{d-c^2 d x^2} \text{Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 1.16786, size = 336, normalized size = 1.15 \[ \frac{b d \sqrt{d-c^2 d x^2} \left (i \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )-i \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right )-c x+c x \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)+\sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)+i \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-i \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )\right )}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}-a d^{3/2} \log \left (\sqrt{d} \sqrt{d-c^2 d x^2}+d\right )-\frac{1}{3} a d \left (c^2 x^2-4\right ) \sqrt{d-c^2 d x^2}+a d^{3/2} \log (x)-\frac{b d \sqrt{d-c^2 d x^2} \left (9 c x+12 \left (\frac{c x-1}{c x+1}\right )^{3/2} (c x+1)^3 \cosh ^{-1}(c x)-\cosh \left (3 \cosh ^{-1}(c x)\right )\right )}{36 \sqrt{\frac{c x-1}{c x+1}} (c x+1)} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.218, size = 499, normalized size = 1.7 \begin{align*}{\frac{a}{3} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-a{d}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,d+2\,\sqrt{d}\sqrt{-{c}^{2}d{x}^{2}+d} \right ) } \right ) +a\sqrt{-{c}^{2}d{x}^{2}+d}d-{\frac{4\,bd{\rm arccosh} \left (cx\right )}{ \left ( 3\,cx+3 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{bd{x}^{3}{c}^{3}}{9}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-{\frac{4\,bdxc}{3}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}+{ibd\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\it dilog} \left ( 1+i \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) \right ){\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}+{ib{\rm arccosh} \left (cx\right )d\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\ln \left ( 1+i \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) \right ){\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-{ib{\rm arccosh} \left (cx\right )d\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\ln \left ( 1-i \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) \right ){\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-{ibd\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\it dilog} \left ( 1-i \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) \right ){\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-{\frac{db{\rm arccosh} \left (cx\right ){x}^{4}{c}^{4}}{ \left ( 3\,cx+3 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{5\,db{\rm arccosh} \left (cx\right ){c}^{2}{x}^{2}}{ \left ( 3\,cx+3 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }} \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 c^{2} d x^{2} - a d +{\left (b c^{2} d x^{2} - b d\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] time = 0., size = 0, normalized size = 0. \begin{align*} \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 )}{x}\, dx \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{3}{2}}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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