Optimal. Leaf size=69 \[ -\frac {c \tan ^{-1}(a x)}{12 a^4}+\frac {c x}{12 a^3}+\frac {1}{6} a^2 c x^6 \tan ^{-1}(a x)-\frac {1}{30} a c x^5+\frac {1}{4} c x^4 \tan ^{-1}(a x)-\frac {c x^3}{36 a} \]
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Rubi [A] time = 0.09, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4950, 4852, 302, 203} \[ \frac {1}{6} a^2 c x^6 \tan ^{-1}(a x)+\frac {c x}{12 a^3}-\frac {c \tan ^{-1}(a x)}{12 a^4}-\frac {1}{30} a c x^5-\frac {c x^3}{36 a}+\frac {1}{4} c x^4 \tan ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 302
Rule 4852
Rule 4950
Rubi steps
\begin {align*} \int x^3 \left (c+a^2 c x^2\right ) \tan ^{-1}(a x) \, dx &=c \int x^3 \tan ^{-1}(a x) \, dx+\left (a^2 c\right ) \int x^5 \tan ^{-1}(a x) \, dx\\ &=\frac {1}{4} c x^4 \tan ^{-1}(a x)+\frac {1}{6} a^2 c x^6 \tan ^{-1}(a x)-\frac {1}{4} (a c) \int \frac {x^4}{1+a^2 x^2} \, dx-\frac {1}{6} \left (a^3 c\right ) \int \frac {x^6}{1+a^2 x^2} \, dx\\ &=\frac {1}{4} c x^4 \tan ^{-1}(a x)+\frac {1}{6} a^2 c x^6 \tan ^{-1}(a x)-\frac {1}{4} (a c) \int \left (-\frac {1}{a^4}+\frac {x^2}{a^2}+\frac {1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx-\frac {1}{6} \left (a^3 c\right ) \int \left (\frac {1}{a^6}-\frac {x^2}{a^4}+\frac {x^4}{a^2}-\frac {1}{a^6 \left (1+a^2 x^2\right )}\right ) \, dx\\ &=\frac {c x}{12 a^3}-\frac {c x^3}{36 a}-\frac {1}{30} a c x^5+\frac {1}{4} c x^4 \tan ^{-1}(a x)+\frac {1}{6} a^2 c x^6 \tan ^{-1}(a x)+\frac {c \int \frac {1}{1+a^2 x^2} \, dx}{6 a^3}-\frac {c \int \frac {1}{1+a^2 x^2} \, dx}{4 a^3}\\ &=\frac {c x}{12 a^3}-\frac {c x^3}{36 a}-\frac {1}{30} a c x^5-\frac {c \tan ^{-1}(a x)}{12 a^4}+\frac {1}{4} c x^4 \tan ^{-1}(a x)+\frac {1}{6} a^2 c x^6 \tan ^{-1}(a x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 69, normalized size = 1.00 \[ -\frac {c \tan ^{-1}(a x)}{12 a^4}+\frac {c x}{12 a^3}+\frac {1}{6} a^2 c x^6 \tan ^{-1}(a x)-\frac {1}{30} a c x^5+\frac {1}{4} c x^4 \tan ^{-1}(a x)-\frac {c x^3}{36 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 57, normalized size = 0.83 \[ -\frac {6 \, a^{5} c x^{5} + 5 \, a^{3} c x^{3} - 15 \, a c x - 15 \, {\left (2 \, a^{6} c x^{6} + 3 \, a^{4} c x^{4} - c\right )} \arctan \left (a x\right )}{180 \, a^{4}} \]
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 = 58, normalized size = 0.84 \[ \frac {c x}{12 a^{3}}-\frac {c \,x^{3}}{36 a}-\frac {a c \,x^{5}}{30}-\frac {c \arctan \left (a x \right )}{12 a^{4}}+\frac {c \,x^{4} \arctan \left (a x \right )}{4}+\frac {a^{2} c \,x^{6} \arctan \left (a x \right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 64, normalized size = 0.93 \[ -\frac {1}{180} \, a {\left (\frac {6 \, a^{4} c x^{5} + 5 \, a^{2} c x^{3} - 15 \, c x}{a^{4}} + \frac {15 \, c \arctan \left (a x\right )}{a^{5}}\right )} + \frac {1}{12} \, {\left (2 \, a^{2} c x^{6} + 3 \, c x^{4}\right )} \arctan \left (a x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 57, normalized size = 0.83 \[ -\frac {c\,\left (15\,\mathrm {atan}\left (a\,x\right )-15\,a\,x+5\,a^3\,x^3+6\,a^5\,x^5-45\,a^4\,x^4\,\mathrm {atan}\left (a\,x\right )-30\,a^6\,x^6\,\mathrm {atan}\left (a\,x\right )\right )}{180\,a^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.46, size = 65, normalized size = 0.94 \[ \begin {cases} \frac {a^{2} c x^{6} \operatorname {atan}{\left (a x \right )}}{6} - \frac {a c x^{5}}{30} + \frac {c x^{4} \operatorname {atan}{\left (a x \right )}}{4} - \frac {c x^{3}}{36 a} + \frac {c x}{12 a^{3}} - \frac {c \operatorname {atan}{\left (a x \right )}}{12 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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