Optimal. Leaf size=80 \[ \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 \pi ^{3/2} b c^3}-\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\pi c^2 \sqrt {\pi c^2 x^2+\pi }}+\frac {b \log \left (c^2 x^2+1\right )}{2 \pi ^{3/2} c^3} \]
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Rubi [A] time = 0.14, 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 260
Rule 5675
Rule 5751
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*}
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Mathematica [A] time = 0.29, 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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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\pi + \pi c^{2} x^{2}} {\left (b x^{2} \operatorname {arsinh}\left (c x\right ) + a x^{2}\right )}}{\pi ^{2} c^{4} x^{4} + 2 \, \pi ^{2} c^{2} x^{2} + \pi ^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.16, size = 196, normalized size = 2.45 \[ -\frac {a x}{\pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {a \ln \left (\frac {\pi x \,c^{2}}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{\pi \,c^{2} \sqrt {\pi \,c^{2}}}+\frac {b \arcsinh \left (c x \right )^{2}}{2 c^{3} \pi ^{\frac {3}{2}}}-\frac {2 b \arcsinh \left (c x \right )}{c^{3} \pi ^{\frac {3}{2}}}+\frac {b \arcsinh \left (c x \right ) x^{2}}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}+1\right )}-\frac {b \arcsinh \left (c x \right ) x}{\pi ^{\frac {3}{2}} c^{2} \sqrt {c^{2} x^{2}+1}}+\frac {b \arcsinh \left (c x \right )}{\pi ^{\frac {3}{2}} c^{3} \left (c^{2} x^{2}+1\right )}+\frac {b \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c^{3} \pi ^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -a {\left (\frac {x}{\pi \sqrt {\pi + \pi c^{2} x^{2}} c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{\pi ^{\frac {3}{2}} c^{3}}\right )} + b \int \frac {x^{2} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a x^{2}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b x^{2} \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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