Optimal. Leaf size=329 \[ -\frac{3 i b c^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{2 d \sqrt{d-c^2 d x^2}}+\frac{3 i b c^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{2 d \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{2 d x^2 \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{2 d x \sqrt{d-c^2 d x^2}}+\frac{b c^2 \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}(c x)}{d \sqrt{d-c^2 d x^2}} \]
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Rubi [A] time = 0.862479, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5798, 5748, 5756, 5761, 4180, 2279, 2391, 207, 325} \[ -\frac{3 i b c^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{2 d \sqrt{d-c^2 d x^2}}+\frac{3 i b c^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{2 d \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{2 d x^2 \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{2 d x \sqrt{d-c^2 d x^2}}+\frac{b c^2 \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}(c x)}{d \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5748
Rule 5756
Rule 5761
Rule 4180
Rule 2279
Rule 2391
Rule 207
Rule 325
Rubi steps
\begin{align*} \int \frac{a+b \cosh ^{-1}(c x)}{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{a+b \cosh ^{-1}(c x)}{x^3 (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=-\frac{a+b \cosh ^{-1}(c x)}{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{1}{x^2 \left (-1+c^2 x^2\right )} \, dx}{2 d \sqrt{d-c^2 d x^2}}-\frac{\left (3 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{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}}{2 d x \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{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{a+b \cosh ^{-1}(c x)}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 d \sqrt{d-c^2 d x^2}}+\frac{\left (b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{-1+c^2 x^2} \, dx}{2 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{1}{-1+c^2 x^2} \, dx}{2 d \sqrt{d-c^2 d x^2}}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{2 d x \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{2 d x^2 \sqrt{d-c^2 d x^2}}+\frac{b c^2 \sqrt{-1+c x} \sqrt{1+c x} \tanh ^{-1}(c x)}{d \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) \text{sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d \sqrt{d-c^2 d x^2}}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{2 d x \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{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 ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{b c^2 \sqrt{-1+c x} \sqrt{1+c x} \tanh ^{-1}(c x)}{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 \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 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 \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d \sqrt{d-c^2 d x^2}}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{2 d x \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{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 ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{b c^2 \sqrt{-1+c x} \sqrt{1+c x} \tanh ^{-1}(c x)}{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 \frac{\log (1-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 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 \frac{\log (1+i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 d \sqrt{d-c^2 d x^2}}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{2 d x \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{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 ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{b c^2 \sqrt{-1+c x} \sqrt{1+c x} \tanh ^{-1}(c x)}{d \sqrt{d-c^2 d x^2}}-\frac{3 i b c^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{2 d \sqrt{d-c^2 d x^2}}+\frac{3 i b c^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{2 d \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 4.29032, size = 405, normalized size = 1.23 \[ \frac{1}{2} \left (-\frac{b c^2 \left (3 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )-3 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right )+\left (\frac{1}{c^2 x^2}-1\right ) \cosh ^{-1}(c x)-\frac{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}{c x}-2 \cosh ^{-1}(c x) \cosh ^2\left (\frac{1}{2} \cosh ^{-1}(c x)\right )+3 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-3 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )+2 \cosh ^{-1}(c x) \sinh ^2\left (\frac{1}{2} \cosh ^{-1}(c x)\right )+2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left (\tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )\right )}{d \sqrt{d-c^2 d x^2}}-\frac{a \left (3 c^2 x^2-1\right ) \sqrt{d-c^2 d x^2}}{d^2 x^2 \left (c^2 x^2-1\right )}-\frac{3 a c^2 \log \left (\sqrt{d} \sqrt{d-c^2 d x^2}+d\right )}{d^{3/2}}+\frac{3 a c^2 \log (x)}{d^{3/2}}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.227, size = 648, normalized size = 2. \begin{align*} -{\frac{a}{2\,d{x}^{2}}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}}+{\frac{3\,a{c}^{2}}{2\,d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}}-{\frac{3\,a{c}^{2}}{2}\ln \left ({\frac{1}{x} \left ( 2\,d+2\,\sqrt{d}\sqrt{-{c}^{2}d{x}^{2}+d} \right ) } \right ){d}^{-{\frac{3}{2}}}}-{\frac{3\,b{\rm arccosh} \left (cx\right ){c}^{2}}{2\,{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{bc}{2\,{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) x}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}}+{\frac{b{\rm arccosh} \left (cx\right )}{2\,{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ){x}^{2}}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{b{c}^{2}}{{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}\ln \left ( cx+\sqrt{cx-1}\sqrt{cx+1}-1 \right ) }-{\frac{b{c}^{2}}{{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}\ln \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{{\frac{3\,i}{2}}b{\rm arccosh} \left (cx\right ){c}^{2}}{{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}\ln \left ( 1+i \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) \right ) }-{\frac{{\frac{3\,i}{2}}b{\rm arccosh} \left (cx\right ){c}^{2}}{{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}\ln \left ( 1-i \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) \right ) }+{\frac{{\frac{3\,i}{2}}b{c}^{2}}{{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}{\it dilog} \left ( 1+i \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) \right ) }-{\frac{{\frac{3\,i}{2}}b{c}^{2}}{{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}{\it dilog} \left ( 1-i \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) \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{\sqrt{-c^{2} d x^{2} + d}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}{c^{4} d^{2} x^{7} - 2 \, c^{2} d^{2} x^{5} + d^{2} x^{3}}, 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{b \operatorname{arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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