Optimal. Leaf size=140 \[ -\frac {3 \tan ^{-1}(a x)^2}{32 a^4 c^3}-\frac {x^4}{32 c^3 \left (a^2 x^2+1\right )^2}+\frac {x^4 \tan ^{-1}(a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {x^3 \tan ^{-1}(a x)}{8 a c^3 \left (a^2 x^2+1\right )^2}+\frac {3}{32 a^4 c^3 \left (a^2 x^2+1\right )}+\frac {3 x \tan ^{-1}(a x)}{16 a^3 c^3 \left (a^2 x^2+1\right )} \]
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Rubi [A] time = 0.19, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4944, 4938, 4934, 4884} \[ -\frac {x^4}{32 c^3 \left (a^2 x^2+1\right )^2}+\frac {3}{32 a^4 c^3 \left (a^2 x^2+1\right )}+\frac {x^4 \tan ^{-1}(a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {x^3 \tan ^{-1}(a x)}{8 a c^3 \left (a^2 x^2+1\right )^2}+\frac {3 x \tan ^{-1}(a x)}{16 a^3 c^3 \left (a^2 x^2+1\right )}-\frac {3 \tan ^{-1}(a x)^2}{32 a^4 c^3} \]
Antiderivative was successfully verified.
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Rule 4884
Rule 4934
Rule 4938
Rule 4944
Rubi steps
\begin {align*} \int \frac {x^3 \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx &=\frac {x^4 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {1}{2} a \int \frac {x^4 \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^3} \, dx\\ &=-\frac {x^4}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {x^3 \tan ^{-1}(a x)}{8 a c^3 \left (1+a^2 x^2\right )^2}+\frac {x^4 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {3 \int \frac {x^2 \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{8 a c}\\ &=-\frac {x^4}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {3}{32 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {x^3 \tan ^{-1}(a x)}{8 a c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \tan ^{-1}(a x)}{16 a^3 c^3 \left (1+a^2 x^2\right )}+\frac {x^4 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {3 \int \frac {\tan ^{-1}(a x)}{c+a^2 c x^2} \, dx}{16 a^3 c^2}\\ &=-\frac {x^4}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {3}{32 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {x^3 \tan ^{-1}(a x)}{8 a c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \tan ^{-1}(a x)}{16 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \tan ^{-1}(a x)^2}{32 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 74, normalized size = 0.53 \[ \frac {5 a^2 x^2+2 a x \left (5 a^2 x^2+3\right ) \tan ^{-1}(a x)+\left (5 a^4 x^4-6 a^2 x^2-3\right ) \tan ^{-1}(a x)^2+4}{32 a^4 c^3 \left (a^2 x^2+1\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 87, normalized size = 0.62 \[ \frac {5 \, a^{2} x^{2} + {\left (5 \, a^{4} x^{4} - 6 \, a^{2} x^{2} - 3\right )} \arctan \left (a x\right )^{2} + 2 \, {\left (5 \, a^{3} x^{3} + 3 \, a x\right )} \arctan \left (a x\right ) + 4}{32 \, {\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )}} \]
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 = 0.07, size = 154, normalized size = 1.10 \[ \frac {\arctan \left (a x \right )^{2}}{4 a^{4} c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {\arctan \left (a x \right )^{2}}{2 a^{4} c^{3} \left (a^{2} x^{2}+1\right )}+\frac {5 x^{3} \arctan \left (a x \right )}{16 a \,c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {3 \arctan \left (a x \right ) x}{16 a^{3} c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {5 \arctan \left (a x \right )^{2}}{32 a^{4} c^{3}}-\frac {1}{32 a^{4} c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {5}{32 a^{4} c^{3} \left (a^{2} x^{2}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 185, normalized size = 1.32 \[ \frac {1}{16} \, a {\left (\frac {5 \, a^{2} x^{3} + 3 \, x}{a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}} + \frac {5 \, \arctan \left (a x\right )}{a^{5} c^{3}}\right )} \arctan \left (a x\right ) + \frac {{\left (5 \, a^{2} x^{2} - 5 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 4\right )} a^{2}}{32 \, {\left (a^{10} c^{3} x^{4} + 2 \, a^{8} c^{3} x^{2} + a^{6} c^{3}\right )}} - \frac {{\left (2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2}}{4 \, {\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.57, size = 85, normalized size = 0.61 \[ \frac {5\,a^4\,x^4\,{\mathrm {atan}\left (a\,x\right )}^2+10\,a^3\,x^3\,\mathrm {atan}\left (a\,x\right )-6\,a^2\,x^2\,{\mathrm {atan}\left (a\,x\right )}^2+5\,a^2\,x^2+6\,a\,x\,\mathrm {atan}\left (a\,x\right )-3\,{\mathrm {atan}\left (a\,x\right )}^2+4}{32\,a^4\,c^3\,{\left (a^2\,x^2+1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {x^{3} \operatorname {atan}^{2}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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