Optimal. Leaf size=94 \[ -\frac{b \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}}+\frac{b \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}}+\frac{a+b \sinh ^{-1}(c x)}{\pi \sqrt{\pi c^2 x^2+\pi }}-\frac{2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\pi ^{3/2}}-\frac{b \tan ^{-1}(c x)}{\pi ^{3/2}} \]
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Rubi [A] time = 0.221959, antiderivative size = 119, normalized size of antiderivative = 1.27, 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 = {5755, 5760, 4182, 2279, 2391, 203} \[ -\frac{b \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}}+\frac{b \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}}+\frac{a+b \sinh ^{-1}(c x)}{\pi \sqrt{\pi c^2 x^2+\pi }}-\frac{2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\pi ^{3/2}}-\frac{b \sqrt{c^2 x^2+1} \tan ^{-1}(c x)}{\pi \sqrt{\pi c^2 x^2+\pi }} \]
Antiderivative was successfully verified.
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Rule 5755
Rule 5760
Rule 4182
Rule 2279
Rule 2391
Rule 203
Rubi steps
\begin{align*} \int \frac{a+b \sinh ^{-1}(c x)}{x \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx &=\frac{a+b \sinh ^{-1}(c x)}{\pi \sqrt{\pi +c^2 \pi x^2}}+\frac{\int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{\pi +c^2 \pi x^2}} \, dx}{\pi }-\frac{\left (b c \sqrt{1+c^2 x^2}\right ) \int \frac{1}{1+c^2 x^2} \, dx}{\pi \sqrt{\pi +c^2 \pi x^2}}\\ &=\frac{a+b \sinh ^{-1}(c x)}{\pi \sqrt{\pi +c^2 \pi x^2}}-\frac{b \sqrt{1+c^2 x^2} \tan ^{-1}(c x)}{\pi \sqrt{\pi +c^2 \pi x^2}}+\frac{\operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{\pi ^{3/2}}\\ &=\frac{a+b \sinh ^{-1}(c x)}{\pi \sqrt{\pi +c^2 \pi x^2}}-\frac{b \sqrt{1+c^2 x^2} \tan ^{-1}(c x)}{\pi \sqrt{\pi +c^2 \pi x^2}}-\frac{2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}}-\frac{b \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\pi ^{3/2}}+\frac{b \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\pi ^{3/2}}\\ &=\frac{a+b \sinh ^{-1}(c x)}{\pi \sqrt{\pi +c^2 \pi x^2}}-\frac{b \sqrt{1+c^2 x^2} \tan ^{-1}(c x)}{\pi \sqrt{\pi +c^2 \pi x^2}}-\frac{2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}}-\frac{b \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}}+\frac{b \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}}\\ &=\frac{a+b \sinh ^{-1}(c x)}{\pi \sqrt{\pi +c^2 \pi x^2}}-\frac{b \sqrt{1+c^2 x^2} \tan ^{-1}(c x)}{\pi \sqrt{\pi +c^2 \pi x^2}}-\frac{2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}}-\frac{b \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}}+\frac{b \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.318419, size = 143, normalized size = 1.52 \[ \frac{b \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-b \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+\frac{a}{\sqrt{c^2 x^2+1}}-a \log \left (\pi \left (\sqrt{c^2 x^2+1}+1\right )\right )+a \log (x)+\frac{b \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+b \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-b \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )-2 b \tan ^{-1}\left (\tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )}{\pi ^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.153, size = 156, normalized size = 1.7 \begin{align*}{\frac{a}{\pi }{\frac{1}{\sqrt{\pi \,{c}^{2}{x}^{2}+\pi }}}}-{\frac{a}{{\pi }^{{\frac{3}{2}}}}{\it Artanh} \left ({\sqrt{\pi }{\frac{1}{\sqrt{\pi \,{c}^{2}{x}^{2}+\pi }}}} \right ) }+{\frac{b{\it Arcsinh} \left ( cx \right ) }{{\pi }^{{\frac{3}{2}}}}{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}-2\,{\frac{b\arctan \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) }{{\pi }^{3/2}}}-{\frac{b}{{\pi }^{{\frac{3}{2}}}}{\it dilog} \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) }-{\frac{b}{{\pi }^{{\frac{3}{2}}}}{\it dilog} \left ( 1+cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) }-{\frac{b{\it Arcsinh} \left ( cx \right ) }{{\pi }^{{\frac{3}{2}}}}\ln \left ( 1+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*} -a{\left (\frac{\operatorname{arsinh}\left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right )}{\pi ^{\frac{3}{2}}} - \frac{1}{\pi \sqrt{\pi + \pi c^{2} x^{2}}}\right )} + b \int \frac{\log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac{3}{2}} x}\,{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 ^{2} c^{4} x^{5} + 2 \, \pi ^{2} c^{2} x^{3} + \pi ^{2} 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*} \frac{\int \frac{a}{c^{2} x^{3} \sqrt{c^{2} x^{2} + 1} + x \sqrt{c^{2} x^{2} + 1}}\, dx + \int \frac{b \operatorname{asinh}{\left (c x \right )}}{c^{2} x^{3} \sqrt{c^{2} x^{2} + 1} + x \sqrt{c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac{3}{2}}} \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}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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