Optimal. Leaf size=45 \[ -\frac {a+b \sinh ^{-1}(c x)}{c^2 \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {b \text {ArcTan}(c x)}{c^2 \pi ^{3/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {5798, 209}
\begin {gather*} \frac {b \text {ArcTan}(c x)}{\pi ^{3/2} c^2}-\frac {a+b \sinh ^{-1}(c x)}{\pi c^2 \sqrt {\pi c^2 x^2+\pi }} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 5798
Rubi steps
\begin {align*} \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx &=-\frac {a+b \sinh ^{-1}(c x)}{c^2 \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int \frac {1}{1+c^2 x^2} \, dx}{c \pi \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{c^2 \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {b \sqrt {1+c^2 x^2} \tan ^{-1}(c x)}{c^2 \pi \sqrt {\pi +c^2 \pi x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 52, normalized size = 1.16 \begin {gather*} \frac {-a-b \sinh ^{-1}(c x)+b \sqrt {1+c^2 x^2} \text {ArcTan}(c x)}{c^2 \pi ^{3/2} \sqrt {1+c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 2.38, size = 103, normalized size = 2.29
method | result | size |
default | \(-\frac {a}{\pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {b \arcsinh \left (c x \right )}{\pi ^{\frac {3}{2}} \sqrt {c^{2} x^{2}+1}\, c^{2}}+\frac {i b \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right )}{c^{2} \pi ^{\frac {3}{2}}}-\frac {i b \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right )}{c^{2} \pi ^{\frac {3}{2}}}\) | \(103\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 127 vs.
\(2 (41) = 82\).
time = 0.46, size = 127, normalized size = 2.82 \begin {gather*} -\frac {\sqrt {\pi } {\left (b c^{2} x^{2} + b\right )} \arctan \left (-\frac {2 \, \sqrt {\pi } \sqrt {\pi + \pi c^{2} x^{2}} \sqrt {c^{2} x^{2} + 1} c x}{\pi - \pi c^{4} x^{4}}\right ) + 2 \, \sqrt {\pi + \pi c^{2} x^{2}} b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, \sqrt {\pi + \pi c^{2} x^{2}} a}{2 \, {\left (\pi ^{2} c^{4} x^{2} + \pi ^{2} c^{2}\right )}} \end {gather*}
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 \begin {gather*} \frac {\int \frac {a x}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b x \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 {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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