Optimal. Leaf size=42 \[ \frac {c \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}{4 a^2}-\frac {1}{12} a c x^3-\frac {c x}{4 a} \]
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Rubi [A] time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {4930} \[ \frac {c \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}{4 a^2}-\frac {1}{12} a c x^3-\frac {c x}{4 a} \]
Antiderivative was successfully verified.
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Rule 4930
Rubi steps
\begin {align*} \int x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x) \, dx &=\frac {c \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{4 a^2}-\frac {\int \left (c+a^2 c x^2\right ) \, dx}{4 a}\\ &=-\frac {c x}{4 a}-\frac {1}{12} a c x^3+\frac {c \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{4 a^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 58, normalized size = 1.38 \[ \frac {1}{4} a^2 c x^4 \tan ^{-1}(a x)+\frac {c \tan ^{-1}(a x)}{4 a^2}-\frac {1}{12} a c x^3+\frac {1}{2} c x^2 \tan ^{-1}(a x)-\frac {c x}{4 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 44, normalized size = 1.05 \[ -\frac {a^{3} c x^{3} + 3 \, a c x - 3 \, {\left (a^{4} c x^{4} + 2 \, a^{2} c x^{2} + c\right )} \arctan \left (a x\right )}{12 \, a^{2}} \]
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 = 0.02, size = 49, normalized size = 1.17 \[ \frac {a^{2} c \arctan \left (a x \right ) x^{4}}{4}+\frac {c \arctan \left (a x \right ) x^{2}}{2}-\frac {a c \,x^{3}}{12}-\frac {c x}{4 a}+\frac {c \arctan \left (a x \right )}{4 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 50, normalized size = 1.19 \[ \frac {{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )}{4 \, a^{2} c} - \frac {a^{2} c^{2} x^{3} + 3 \, c^{2} x}{12 \, a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.48, size = 48, normalized size = 1.14 \[ \frac {\frac {c\,\mathrm {atan}\left (a\,x\right )}{4}-\frac {a\,c\,x}{4}}{a^2}+\frac {c\,x^2\,\mathrm {atan}\left (a\,x\right )}{2}-\frac {a\,c\,x^3}{12}+\frac {a^2\,c\,x^4\,\mathrm {atan}\left (a\,x\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.80, size = 54, normalized size = 1.29 \[ \begin {cases} \frac {a^{2} c x^{4} \operatorname {atan}{\left (a x \right )}}{4} - \frac {a c x^{3}}{12} + \frac {c x^{2} \operatorname {atan}{\left (a x \right )}}{2} - \frac {c x}{4 a} + \frac {c \operatorname {atan}{\left (a x \right )}}{4 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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