Optimal. Leaf size=51 \[ \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {b \log \left (c^2 x^2+1\right )}{2 \pi ^{3/2} c} \]
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Rubi [A] time = 0.04, antiderivative size = 76, normalized size of antiderivative = 1.49, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {5687, 260} \[ \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {b \sqrt {c^2 x^2+1} \log \left (c^2 x^2+1\right )}{2 \pi c \sqrt {\pi c^2 x^2+\pi }} \]
Antiderivative was successfully verified.
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Rule 260
Rule 5687
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx &=\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\pi \sqrt {\pi +c^2 \pi x^2}}-\frac {\left (b c \sqrt {1+c^2 x^2}\right ) \int \frac {x}{1+c^2 x^2} \, dx}{\pi \sqrt {\pi +c^2 \pi x^2}}\\ &=\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\pi \sqrt {\pi +c^2 \pi x^2}}-\frac {b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{2 c \pi \sqrt {\pi +c^2 \pi x^2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 66, normalized size = 1.29 \[ \frac {2 a c x-b \sqrt {c^2 x^2+1} \log \left (c^2 x^2+1\right )+2 b c x \sinh ^{-1}(c x)}{2 \pi ^{3/2} c \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\pi + \pi c^{2} x^{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{\pi ^{2} c^{4} x^{4} + 2 \, \pi ^{2} c^{2} x^{2} + \pi ^{2}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 132, normalized size = 2.59 \[ \frac {a x}{\pi \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {2 b \arcsinh \left (c x \right )}{c \,\pi ^{\frac {3}{2}}}-\frac {b \arcsinh \left (c x \right ) c \,x^{2}}{\pi ^{\frac {3}{2}} \left (c^{2} x^{2}+1\right )}+\frac {b \arcsinh \left (c x \right ) x}{\pi ^{\frac {3}{2}} \sqrt {c^{2} x^{2}+1}}-\frac {b \arcsinh \left (c x \right )}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}+1\right )}-\frac {b \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c \,\pi ^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 58, normalized size = 1.14 \[ \frac {b x \operatorname {arsinh}\left (c x\right )}{\pi \sqrt {\pi + \pi c^{2} x^{2}}} + \frac {a x}{\pi \sqrt {\pi + \pi c^{2} x^{2}}} - \frac {b \log \left (x^{2} + \frac {1}{c^{2}}\right )}{2 \, \pi ^{\frac {3}{2}} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}} \,d x \]
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 \[ \frac {\int \frac {a}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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