Optimal. Leaf size=74 \[ -\frac {1}{56} a^5 c^3 x^7-\frac {3}{40} a^3 c^3 x^5+\frac {c^3 \left (a^2 x^2+1\right )^4 \tan ^{-1}(a x)}{8 a^2}-\frac {1}{8} a c^3 x^3-\frac {c^3 x}{8 a} \]
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Rubi [A] time = 0.05, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4930, 194} \[ -\frac {1}{56} a^5 c^3 x^7-\frac {3}{40} a^3 c^3 x^5+\frac {c^3 \left (a^2 x^2+1\right )^4 \tan ^{-1}(a x)}{8 a^2}-\frac {1}{8} a c^3 x^3-\frac {c^3 x}{8 a} \]
Antiderivative was successfully verified.
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Rule 194
Rule 4930
Rubi steps
\begin {align*} \int x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x) \, dx &=\frac {c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)}{8 a^2}-\frac {\int \left (c+a^2 c x^2\right )^3 \, dx}{8 a}\\ &=\frac {c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)}{8 a^2}-\frac {\int \left (c^3+3 a^2 c^3 x^2+3 a^4 c^3 x^4+a^6 c^3 x^6\right ) \, dx}{8 a}\\ &=-\frac {c^3 x}{8 a}-\frac {1}{8} a c^3 x^3-\frac {3}{40} a^3 c^3 x^5-\frac {1}{56} a^5 c^3 x^7+\frac {c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)}{8 a^2}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 128, normalized size = 1.73 \[ \frac {1}{8} a^6 c^3 x^8 \tan ^{-1}(a x)-\frac {1}{56} a^5 c^3 x^7+\frac {1}{2} a^4 c^3 x^6 \tan ^{-1}(a x)-\frac {3}{40} a^3 c^3 x^5+\frac {3}{4} a^2 c^3 x^4 \tan ^{-1}(a x)+\frac {c^3 \tan ^{-1}(a x)}{8 a^2}-\frac {1}{8} a c^3 x^3+\frac {1}{2} c^3 x^2 \tan ^{-1}(a x)-\frac {c^3 x}{8 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 99, normalized size = 1.34 \[ -\frac {5 \, a^{7} c^{3} x^{7} + 21 \, a^{5} c^{3} x^{5} + 35 \, a^{3} c^{3} x^{3} + 35 \, a c^{3} x - 35 \, {\left (a^{8} c^{3} x^{8} + 4 \, a^{6} c^{3} x^{6} + 6 \, a^{4} c^{3} x^{4} + 4 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )}{280 \, a^{2}} \]
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 = 0.03, size = 111, normalized size = 1.50 \[ \frac {a^{6} c^{3} \arctan \left (a x \right ) x^{8}}{8}+\frac {a^{4} c^{3} \arctan \left (a x \right ) x^{6}}{2}+\frac {3 a^{2} c^{3} \arctan \left (a x \right ) x^{4}}{4}+\frac {c^{3} \arctan \left (a x \right ) x^{2}}{2}-\frac {a^{5} c^{3} x^{7}}{56}-\frac {3 a^{3} c^{3} x^{5}}{40}-\frac {a \,c^{3} x^{3}}{8}-\frac {c^{3} x}{8 a}+\frac {c^{3} \arctan \left (a x \right )}{8 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 73, normalized size = 0.99 \[ \frac {{\left (a^{2} c x^{2} + c\right )}^{4} \arctan \left (a x\right )}{8 \, a^{2} c} - \frac {5 \, a^{6} c^{4} x^{7} + 21 \, a^{4} c^{4} x^{5} + 35 \, a^{2} c^{4} x^{3} + 35 \, c^{4} x}{280 \, a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 100, normalized size = 1.35 \[ \mathrm {atan}\left (a\,x\right )\,\left (\frac {a^6\,c^3\,x^8}{8}+\frac {a^4\,c^3\,x^6}{2}+\frac {3\,a^2\,c^3\,x^4}{4}+\frac {c^3\,x^2}{2}\right )-\frac {c^3\,x}{8\,a}-\frac {a\,c^3\,x^3}{8}+\frac {c^3\,\mathrm {atan}\left (a\,x\right )}{8\,a^2}-\frac {3\,a^3\,c^3\,x^5}{40}-\frac {a^5\,c^3\,x^7}{56} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.43, size = 124, normalized size = 1.68 \[ \begin {cases} \frac {a^{6} c^{3} x^{8} \operatorname {atan}{\left (a x \right )}}{8} - \frac {a^{5} c^{3} x^{7}}{56} + \frac {a^{4} c^{3} x^{6} \operatorname {atan}{\left (a x \right )}}{2} - \frac {3 a^{3} c^{3} x^{5}}{40} + \frac {3 a^{2} c^{3} x^{4} \operatorname {atan}{\left (a x \right )}}{4} - \frac {a c^{3} x^{3}}{8} + \frac {c^{3} x^{2} \operatorname {atan}{\left (a x \right )}}{2} - \frac {c^{3} x}{8 a} + \frac {c^{3} \operatorname {atan}{\left (a x \right )}}{8 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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