Optimal. Leaf size=96 \[ \frac {c \left (a^2 x^2+1\right )}{12 a^2}+\frac {c \log \left (a^2 x^2+1\right )}{6 a^2}+\frac {c \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}{4 a^2}-\frac {c x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{6 a}-\frac {c x \tan ^{-1}(a x)}{3 a} \]
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Rubi [A] time = 0.05, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4930, 4878, 4846, 260} \[ \frac {c \left (a^2 x^2+1\right )}{12 a^2}+\frac {c \log \left (a^2 x^2+1\right )}{6 a^2}+\frac {c \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}{4 a^2}-\frac {c x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{6 a}-\frac {c x \tan ^{-1}(a x)}{3 a} \]
Antiderivative was successfully verified.
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Rule 260
Rule 4846
Rule 4878
Rule 4930
Rubi steps
\begin {align*} \int x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2 \, dx &=\frac {c \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{4 a^2}-\frac {\int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x) \, dx}{2 a}\\ &=\frac {c \left (1+a^2 x^2\right )}{12 a^2}-\frac {c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{6 a}+\frac {c \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{4 a^2}-\frac {c \int \tan ^{-1}(a x) \, dx}{3 a}\\ &=\frac {c \left (1+a^2 x^2\right )}{12 a^2}-\frac {c x \tan ^{-1}(a x)}{3 a}-\frac {c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{6 a}+\frac {c \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{4 a^2}+\frac {1}{3} c \int \frac {x}{1+a^2 x^2} \, dx\\ &=\frac {c \left (1+a^2 x^2\right )}{12 a^2}-\frac {c x \tan ^{-1}(a x)}{3 a}-\frac {c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{6 a}+\frac {c \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{4 a^2}+\frac {c \log \left (1+a^2 x^2\right )}{6 a^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 64, normalized size = 0.67 \[ \frac {c \left (a^2 x^2+2 \log \left (a^2 x^2+1\right )-2 a x \left (a^2 x^2+3\right ) \tan ^{-1}(a x)+3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2\right )}{12 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 74, normalized size = 0.77 \[ \frac {a^{2} c x^{2} + 3 \, {\left (a^{4} c x^{4} + 2 \, a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{2} - 2 \, {\left (a^{3} c x^{3} + 3 \, a c x\right )} \arctan \left (a x\right ) + 2 \, c \log \left (a^{2} x^{2} + 1\right )}{12 \, a^{2}} \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 85, normalized size = 0.89 \[ \frac {a^{2} c \arctan \left (a x \right )^{2} x^{4}}{4}+\frac {c \arctan \left (a x \right )^{2} x^{2}}{2}-\frac {a c \arctan \left (a x \right ) x^{3}}{6}-\frac {c x \arctan \left (a x \right )}{2 a}+\frac {c \arctan \left (a x \right )^{2}}{4 a^{2}}+\frac {c \,x^{2}}{12}+\frac {c \ln \left (a^{2} x^{2}+1\right )}{6 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 87, normalized size = 0.91 \[ \frac {{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{2}}{4 \, a^{2} c} + \frac {{\left (c^{2} x^{2} + \frac {2 \, c^{2} \log \left (a^{2} x^{2} + 1\right )}{a^{2}}\right )} a - 2 \, {\left (a^{2} c^{2} x^{3} + 3 \, c^{2} x\right )} \arctan \left (a x\right )}{12 \, a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.51, size = 83, normalized size = 0.86 \[ \frac {c\,\left (6\,x^2\,{\mathrm {atan}\left (a\,x\right )}^2+x^2\right )}{12}+\frac {\frac {c\,\left (3\,{\mathrm {atan}\left (a\,x\right )}^2+2\,\ln \left (a^2\,x^2+1\right )\right )}{12}-\frac {a\,c\,x\,\mathrm {atan}\left (a\,x\right )}{2}}{a^2}+\frac {a^2\,c\,x^4\,{\mathrm {atan}\left (a\,x\right )}^2}{4}-\frac {a\,c\,x^3\,\mathrm {atan}\left (a\,x\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.13, size = 94, normalized size = 0.98 \[ \begin {cases} \frac {a^{2} c x^{4} \operatorname {atan}^{2}{\left (a x \right )}}{4} - \frac {a c x^{3} \operatorname {atan}{\left (a x \right )}}{6} + \frac {c x^{2} \operatorname {atan}^{2}{\left (a x \right )}}{2} + \frac {c x^{2}}{12} - \frac {c x \operatorname {atan}{\left (a x \right )}}{2 a} + \frac {c \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{6 a^{2}} + \frac {c \operatorname {atan}^{2}{\left (a x \right )}}{4 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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