Optimal. Leaf size=89 \[ -\sqrt{\pi } b \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )+\sqrt{\pi } b \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )+\sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )-2 \sqrt{\pi } \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )+\sqrt{\pi } (-b) c x \]
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Rubi [A] time = 0.19175, antiderivative size = 177, normalized size of antiderivative = 1.99, 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 = {5742, 5760, 4182, 2279, 2391, 8} \[ -\frac{b \sqrt{\pi c^2 x^2+\pi } \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}+\frac{b \sqrt{\pi c^2 x^2+\pi } \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}+\sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )-\frac{2 \sqrt{\pi c^2 x^2+\pi } \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}}-\frac{b c x \sqrt{\pi c^2 x^2+\pi }}{\sqrt{c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 5742
Rule 5760
Rule 4182
Rule 2279
Rule 2391
Rule 8
Rubi steps
\begin{align*} \int \frac{\sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx &=\sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{\sqrt{\pi +c^2 \pi x^2} \int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{1+c^2 x^2}} \, dx}{\sqrt{1+c^2 x^2}}-\frac{\left (b c \sqrt{\pi +c^2 \pi x^2}\right ) \int 1 \, dx}{\sqrt{1+c^2 x^2}}\\ &=-\frac{b c x \sqrt{\pi +c^2 \pi x^2}}{\sqrt{1+c^2 x^2}}+\sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{\sqrt{\pi +c^2 \pi x^2} \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}\\ &=-\frac{b c x \sqrt{\pi +c^2 \pi x^2}}{\sqrt{1+c^2 x^2}}+\sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2 \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{\left (b \sqrt{\pi +c^2 \pi x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}+\frac{\left (b \sqrt{\pi +c^2 \pi x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}\\ &=-\frac{b c x \sqrt{\pi +c^2 \pi x^2}}{\sqrt{1+c^2 x^2}}+\sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2 \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{\left (b \sqrt{\pi +c^2 \pi x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}+\frac{\left (b \sqrt{\pi +c^2 \pi x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}\\ &=-\frac{b c x \sqrt{\pi +c^2 \pi x^2}}{\sqrt{1+c^2 x^2}}+\sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2 \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{b \sqrt{\pi +c^2 \pi x^2} \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}+\frac{b \sqrt{\pi +c^2 \pi x^2} \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.188321, size = 131, normalized size = 1.47 \[ \sqrt{\pi } \left (b \left (\text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-\text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-c x+\sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-\sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )\right )+a \sqrt{c^2 x^2+1}-a \log \left (\pi \left (\sqrt{c^2 x^2+1}+1\right )\right )+a \log (x)\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.148, size = 171, normalized size = 1.9 \begin{align*} -{\it Artanh} \left ({\sqrt{\pi }{\frac{1}{\sqrt{\pi \,{c}^{2}{x}^{2}+\pi }}}} \right ) \sqrt{\pi }a+a\sqrt{\pi \,{c}^{2}{x}^{2}+\pi }+{\it Arcsinh} \left ( cx \right ) \sqrt{\pi }\ln \left ( 1-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) b-{\it Arcsinh} \left ( cx \right ) \sqrt{\pi }\ln \left ( 1+cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) b+{\it Arcsinh} \left ( cx \right ) \sqrt{\pi }\sqrt{{c}^{2}{x}^{2}+1}b-bcx\sqrt{\pi }+b{\it polylog} \left ( 2,cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) \sqrt{\pi }-b{\it polylog} \left ( 2,-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) \sqrt{\pi } \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*} -{\left (\sqrt{\pi } \operatorname{arsinh}\left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right ) - \sqrt{\pi + \pi c^{2} x^{2}}\right )} a + b \int \frac{\sqrt{\pi + \pi c^{2} x^{2}} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{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 )}}{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*} \sqrt{\pi } \left (\int \frac{a \sqrt{c^{2} x^{2} + 1}}{x}\, dx + \int \frac{b \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{x}\, dx\right ) \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{\sqrt{\pi + \pi c^{2} x^{2}}{\left (b \operatorname{arsinh}\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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