Optimal. Leaf size=115 \[ \frac{b c^2 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt{\pi }}-\frac{b c^2 \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt{\pi }}-\frac{\sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{2 \pi x^2}+\frac{c^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{\pi }}-\frac{b c}{2 \sqrt{\pi } x} \]
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Rubi [A] time = 0.213172, antiderivative size = 137, normalized size of antiderivative = 1.19, number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {5747, 5760, 4182, 2279, 2391, 30} \[ \frac{b c^2 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt{\pi }}-\frac{b c^2 \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt{\pi }}-\frac{\sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{2 \pi x^2}+\frac{c^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{\pi }}-\frac{b c \sqrt{c^2 x^2+1}}{2 x \sqrt{\pi c^2 x^2+\pi }} \]
Antiderivative was successfully verified.
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Rule 5747
Rule 5760
Rule 4182
Rule 2279
Rule 2391
Rule 30
Rubi steps
\begin{align*} \int \frac{a+b \sinh ^{-1}(c x)}{x^3 \sqrt{\pi +c^2 \pi x^2}} \, dx &=-\frac{\sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \pi x^2}-\frac{1}{2} c^2 \int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{\pi +c^2 \pi x^2}} \, dx+\frac{\left (b c \sqrt{1+c^2 x^2}\right ) \int \frac{1}{x^2} \, dx}{2 \sqrt{\pi +c^2 \pi x^2}}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{2 x \sqrt{\pi +c^2 \pi x^2}}-\frac{\sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \pi x^2}-\frac{c^2 \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt{\pi }}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{2 x \sqrt{\pi +c^2 \pi x^2}}-\frac{\sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \pi x^2}+\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{\pi }}+\frac{\left (b c^2\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt{\pi }}-\frac{\left (b c^2\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt{\pi }}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{2 x \sqrt{\pi +c^2 \pi x^2}}-\frac{\sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \pi x^2}+\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{\pi }}+\frac{\left (b c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt{\pi }}-\frac{\left (b c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt{\pi }}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{2 x \sqrt{\pi +c^2 \pi x^2}}-\frac{\sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \pi x^2}+\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{\pi }}+\frac{b c^2 \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt{\pi }}-\frac{b c^2 \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt{\pi }}\\ \end{align*}
Mathematica [A] time = 2.71724, size = 185, normalized size = 1.61 \[ \frac{b c^2 \left (-4 \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )+4 \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )-4 \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )+4 \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )+2 \tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-2 \coth \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-\sinh ^{-1}(c x) \text{csch}^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right )-\sinh ^{-1}(c x) \text{sech}^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )-\frac{4 a \sqrt{c^2 x^2+1}}{x^2}+4 a c^2 \log \left (\pi \left (\sqrt{c^2 x^2+1}+1\right )\right )-4 a c^2 \log (x)}{8 \sqrt{\pi }} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.079, size = 225, normalized size = 2. \begin{align*} -{\frac{a}{2\,\pi \,{x}^{2}}\sqrt{\pi \,{c}^{2}{x}^{2}+\pi }}+{\frac{a{c}^{2}}{2\,\sqrt{\pi }}{\it Artanh} \left ({\sqrt{\pi }{\frac{1}{\sqrt{\pi \,{c}^{2}{x}^{2}+\pi }}}} \right ) }-{\frac{b{\it Arcsinh} \left ( cx \right ){c}^{2}}{2\,\sqrt{\pi }}{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}-{\frac{bc}{2\,x\sqrt{\pi }}}-{\frac{b{\it Arcsinh} \left ( cx \right ) }{2\,\sqrt{\pi }{x}^{2}}{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}+{\frac{b{\it Arcsinh} \left ( cx \right ){c}^{2}}{2\,\sqrt{\pi }}\ln \left ( 1+cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) }+{\frac{b{c}^{2}}{2\,\sqrt{\pi }}{\it polylog} \left ( 2,-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) }-{\frac{b{\it Arcsinh} \left ( cx \right ){c}^{2}}{2\,\sqrt{\pi }}\ln \left ( 1-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) }-{\frac{b{c}^{2}}{2\,\sqrt{\pi }}{\it polylog} \left ( 2,cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \,{\left (\frac{c^{2} \operatorname{arsinh}\left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right )}{\sqrt{\pi }} - \frac{\sqrt{\pi + \pi c^{2} x^{2}}}{\pi x^{2}}\right )} a + b \int \frac{\log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{\sqrt{\pi + \pi c^{2} x^{2}} x^{3}}\,{d x} \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{\pi + \pi c^{2} x^{2}}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}}{\pi c^{2} x^{5} + \pi x^{3}}, 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*} \frac{\int \frac{a}{x^{3} \sqrt{c^{2} x^{2} + 1}}\, dx + \int \frac{b \operatorname{asinh}{\left (c x \right )}}{x^{3} \sqrt{c^{2} x^{2} + 1}}\, dx}{\sqrt{\pi }} \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{arsinh}\left (c x\right ) + a}{\sqrt{\pi + \pi c^{2} x^{2}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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