Optimal. Leaf size=80 \[ -\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\pi c^2 \sqrt{\pi c^2 x^2+\pi }}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 \pi ^{3/2} b c^3}+\frac{b \log \left (c^2 x^2+1\right )}{2 \pi ^{3/2} c^3} \]
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Rubi [A] time = 0.139521, antiderivative size = 105, normalized size of antiderivative = 1.31, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {5751, 5675, 260} \[ -\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\pi c^2 \sqrt{\pi c^2 x^2+\pi }}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 \pi ^{3/2} b c^3}+\frac{b \sqrt{c^2 x^2+1} \log \left (c^2 x^2+1\right )}{2 \pi c^3 \sqrt{\pi c^2 x^2+\pi }} \]
Antiderivative was successfully verified.
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Rule 5751
Rule 5675
Rule 260
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx &=-\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{c^2 \pi \sqrt{\pi +c^2 \pi x^2}}+\frac{\int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{\pi +c^2 \pi x^2}} \, dx}{c^2 \pi }+\frac{\left (b \sqrt{1+c^2 x^2}\right ) \int \frac{x}{1+c^2 x^2} \, dx}{c \pi \sqrt{\pi +c^2 \pi x^2}}\\ &=-\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{c^2 \pi \sqrt{\pi +c^2 \pi x^2}}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c^3 \pi ^{3/2}}+\frac{b \sqrt{1+c^2 x^2} \log \left (1+c^2 x^2\right )}{2 c^3 \pi \sqrt{\pi +c^2 \pi x^2}}\\ \end{align*}
Mathematica [A] time = 0.279468, size = 78, normalized size = 0.98 \[ \frac{\sinh ^{-1}(c x) \left (2 a-\frac{2 b c x}{\sqrt{c^2 x^2+1}}\right )-\frac{2 a c x}{\sqrt{c^2 x^2+1}}+b \log \left (c^2 x^2+1\right )+b \sinh ^{-1}(c x)^2}{2 \pi ^{3/2} c^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.105, size = 196, normalized size = 2.5 \begin{align*} -{\frac{ax}{\pi \,{c}^{2}}{\frac{1}{\sqrt{\pi \,{c}^{2}{x}^{2}+\pi }}}}+{\frac{a}{\pi \,{c}^{2}}\ln \left ({\pi \,{c}^{2}x{\frac{1}{\sqrt{\pi \,{c}^{2}}}}}+\sqrt{\pi \,{c}^{2}{x}^{2}+\pi } \right ){\frac{1}{\sqrt{\pi \,{c}^{2}}}}}+{\frac{b \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{2\,{c}^{3}{\pi }^{3/2}}}-2\,{\frac{b{\it Arcsinh} \left ( cx \right ) }{{c}^{3}{\pi }^{3/2}}}+{\frac{b{\it Arcsinh} \left ( cx \right ){x}^{2}}{{\pi }^{{\frac{3}{2}}}c \left ({c}^{2}{x}^{2}+1 \right ) }}-{\frac{b{\it Arcsinh} \left ( cx \right ) x}{{\pi }^{{\frac{3}{2}}}{c}^{2}}{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}+{\frac{b{\it Arcsinh} \left ( cx \right ) }{{c}^{3}{\pi }^{{\frac{3}{2}}} \left ({c}^{2}{x}^{2}+1 \right ) }}+{\frac{b}{{c}^{3}{\pi }^{{\frac{3}{2}}}}\ln \left ( 1+ \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) ^{2} \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{x}{\pi \sqrt{\pi + \pi c^{2} x^{2}} c^{2}} - \frac{\operatorname{arsinh}\left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\pi \sqrt{\pi c^{2}} c^{2}}\right )} + b \int \frac{x^{2} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac{3}{2}}}\,{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 x^{2} \operatorname{arsinh}\left (c x\right ) + a x^{2}\right )}}{\pi ^{2} c^{4} x^{4} + 2 \, \pi ^{2} c^{2} x^{2} + \pi ^{2}}, 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^{2}}{c^{2} x^{2} \sqrt{c^{2} x^{2} + 1} + \sqrt{c^{2} x^{2} + 1}}\, dx + \int \frac{b x^{2} \operatorname{asinh}{\left (c x \right )}}{c^{2} x^{2} \sqrt{c^{2} x^{2} + 1} + \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{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )} x^{2}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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