Optimal. Leaf size=321 \[ \frac{3 i b c^4 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 i b c^4 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 c^2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac{3 c^4 d \sqrt{d-c^2 d x^2} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b c^3 d \sqrt{d-c^2 d x^2}}{8 x \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 0.836621, antiderivative size = 333, normalized size of antiderivative = 1.04, 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, 5740, 5738, 30, 5761, 4180, 2279, 2391, 14} \[ \frac{3 i b c^4 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 i b c^4 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 c^2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac{d (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac{3 c^4 d \sqrt{d-c^2 d x^2} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b c^3 d \sqrt{d-c^2 d x^2}}{8 x \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5740
Rule 5738
Rule 30
Rule 5761
Rule 4180
Rule 2279
Rule 2391
Rule 14
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^5} \, 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^5} \, 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 )}{4 x^4}-\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int \frac{-1+c^2 x^2}{x^4} \, dx}{4 \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 )}{x^3} \, dx}{4 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{3 c^2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int \left (-\frac{1}{x^4}+\frac{c^2}{x^2}\right ) \, dx}{4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 b c^3 d \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{x^2} \, dx}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 c^4 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}{8 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 b c^3 d \sqrt{d-c^2 d x^2}}{8 x \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 c^2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac{\left (3 c^4 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 )}{8 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 b c^3 d \sqrt{d-c^2 d x^2}}{8 x \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 c^2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac{3 c^4 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 )}{4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 i b c^4 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 )}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 i b c^4 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 )}{8 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 b c^3 d \sqrt{d-c^2 d x^2}}{8 x \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 c^2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac{3 c^4 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 )}{4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 i b c^4 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 )}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 i b c^4 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 )}{8 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 b c^3 d \sqrt{d-c^2 d x^2}}{8 x \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 c^2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac{3 c^4 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 )}{4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 i b c^4 d \sqrt{d-c^2 d x^2} \text{Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{3 i b c^4 d \sqrt{d-c^2 d x^2} \text{Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 1.18366, size = 574, normalized size = 1.79 \[ \frac{-9 i b c^4 d^2 x^4 (c x-1) \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )+9 i b c^4 d^2 x^4 (c x-1) \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right )-15 a c^4 d^2 x^4 \sqrt{\frac{c x-1}{c x+1}}+21 a c^2 d^2 x^2 \sqrt{\frac{c x-1}{c x+1}}+9 a c^4 d^{3/2} x^4 \sqrt{\frac{c x-1}{c x+1}} \log (x) \sqrt{d-c^2 d x^2}-9 a c^4 d^{3/2} x^4 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} \log \left (\sqrt{d} \sqrt{d-c^2 d x^2}+d\right )-6 a d^2 \sqrt{\frac{c x-1}{c x+1}}-15 b c^4 d^2 x^4+15 b c^3 d^2 x^3+2 b c^2 d^2 x^2-15 b c^4 d^2 x^4 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)+21 b c^2 d^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)-9 i b c^5 d^2 x^5 \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )+9 i b c^5 d^2 x^5 \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )+9 i b c^4 d^2 x^4 \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-9 i b c^4 d^2 x^4 \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )-2 b c d^2 x-6 b d^2 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)}{24 x^4 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.241, size = 570, normalized size = 1.8 \begin{align*} -{\frac{a}{4\,d{x}^{4}} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{5}{2}}}}+{\frac{a{c}^{2}}{8\,d{x}^{2}} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{5}{2}}}}+{\frac{a{c}^{4}}{8} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-{\frac{3\,a{c}^{4}}{8}{d}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,d+2\,\sqrt{d}\sqrt{-{c}^{2}d{x}^{2}+d} \right ) } \right ) }+{\frac{3\,a{c}^{4}d}{8}\sqrt{-{c}^{2}d{x}^{2}+d}}+{\frac{5\,bd{\rm arccosh} \left (cx\right ){c}^{4}}{ \left ( 8\,cx+8 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{5\,bd{c}^{3}}{8\,x}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-{\frac{7\,b{c}^{2}d{\rm arccosh} \left (cx\right )}{ \left ( 8\,cx+8 \right ){x}^{2} \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{bdc}{12\,{x}^{3}}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}+{\frac{bd{\rm arccosh} \left (cx\right )}{ \left ( 4\,cx+4 \right ){x}^{4} \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{{\frac{3\,i}{8}}b{\rm arccosh} \left (cx\right )d{c}^{4}\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}}}}-{{\frac{3\,i}{8}}b{\rm arccosh} \left (cx\right )d{c}^{4}\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}}}}+{{\frac{3\,i}{8}}bd{c}^{4}\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{3\,i}{8}}bd{c}^{4}\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}}}} \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^{5}}, 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{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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