Optimal. Leaf size=36 \[ \frac {1}{2} x^2 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac {b \log \left (c^2 x^4+1\right )}{4 c} \]
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Rubi [A] time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5033, 260} \[ \frac {1}{2} x^2 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac {b \log \left (c^2 x^4+1\right )}{4 c} \]
Antiderivative was successfully verified.
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Rule 260
Rule 5033
Rubi steps
\begin {align*} \int x \left (a+b \tan ^{-1}\left (c x^2\right )\right ) \, dx &=\frac {1}{2} x^2 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-(b c) \int \frac {x^3}{1+c^2 x^4} \, dx\\ &=\frac {1}{2} x^2 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac {b \log \left (1+c^2 x^4\right )}{4 c}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 41, normalized size = 1.14 \[ \frac {a x^2}{2}-\frac {b \log \left (c^2 x^4+1\right )}{4 c}+\frac {1}{2} b x^2 \tan ^{-1}\left (c x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 39, normalized size = 1.08 \[ \frac {2 \, b c x^{2} \arctan \left (c x^{2}\right ) + 2 \, a c x^{2} - b \log \left (c^{2} x^{4} + 1\right )}{4 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 40, normalized size = 1.11 \[ \frac {2 \, a c x^{2} + {\left (2 \, c x^{2} \arctan \left (c x^{2}\right ) - \log \left (c^{2} x^{4} + 1\right )\right )} b}{4 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 36, normalized size = 1.00 \[ \frac {a \,x^{2}}{2}+\frac {b \,x^{2} \arctan \left (c \,x^{2}\right )}{2}-\frac {b \ln \left (c^{2} x^{4}+1\right )}{4 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 38, normalized size = 1.06 \[ \frac {1}{2} \, a x^{2} + \frac {{\left (2 \, c x^{2} \arctan \left (c x^{2}\right ) - \log \left (c^{2} x^{4} + 1\right )\right )} b}{4 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 35, normalized size = 0.97 \[ \frac {a\,x^2}{2}-\frac {b\,\ln \left (c^2\,x^4+1\right )}{4\,c}+\frac {b\,x^2\,\mathrm {atan}\left (c\,x^2\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.50, size = 66, normalized size = 1.83 \[ \begin {cases} \frac {a x^{2}}{2} + \frac {b x^{2} \operatorname {atan}{\left (c x^{2} \right )}}{2} - \frac {i b \sqrt {\frac {1}{c^{2}}} \operatorname {atan}{\left (c x^{2} \right )}}{2} - \frac {b \log {\left (x^{2} + i \sqrt {\frac {1}{c^{2}}} \right )}}{2 c} & \text {for}\: c \neq 0 \\\frac {a x^{2}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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