Optimal. Leaf size=80 \[ -\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{c^2 \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c^3 \pi ^{3/2}}+\frac {b \log \left (1+c^2 x^2\right )}{2 c^3 \pi ^{3/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 80, normalized size of antiderivative = 1.00, 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 = {5810, 5783,
266} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 5783
Rule 5810
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 = 0.21, size = 78, normalized size = 0.98 \begin {gather*} \frac {-\frac {2 a c x}{\sqrt {1+c^2 x^2}}+\left (2 a-\frac {2 b c x}{\sqrt {1+c^2 x^2}}\right ) \sinh ^{-1}(c x)+b \sinh ^{-1}(c x)^2+b \log \left (1+c^2 x^2\right )}{2 c^3 \pi ^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(195\) vs.
\(2(70)=140\).
time = 3.33, size = 196, normalized size = 2.45
method | result | size |
default | \(-\frac {a x}{\pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {a \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{\pi \,c^{2} \sqrt {\pi \,c^{2}}}+\frac {b \arcsinh \left (c x \right )^{2}}{2 c^{3} \pi ^{\frac {3}{2}}}-\frac {2 b \arcsinh \left (c x \right )}{c^{3} \pi ^{\frac {3}{2}}}+\frac {b \arcsinh \left (c x \right ) x^{2}}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}+1\right )}-\frac {b \arcsinh \left (c x \right ) x}{\pi ^{\frac {3}{2}} c^{2} \sqrt {c^{2} x^{2}+1}}+\frac {b \arcsinh \left (c x \right )}{\pi ^{\frac {3}{2}} c^{3} \left (c^{2} x^{2}+1\right )}+\frac {b \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c^{3} \pi ^{\frac {3}{2}}}\) | \(196\) |
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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {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 = -0.01 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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