Optimal. Leaf size=208 \[ -\frac {45 x}{256 a c^3 \left (a^2 x^2+1\right )}-\frac {3 x}{128 a c^3 \left (a^2 x^2+1\right )^2}-\frac {\tan ^{-1}(a x)^3}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac {9 x \tan ^{-1}(a x)^2}{32 a c^3 \left (a^2 x^2+1\right )}+\frac {3 x \tan ^{-1}(a x)^2}{16 a c^3 \left (a^2 x^2+1\right )^2}+\frac {9 \tan ^{-1}(a x)}{32 a^2 c^3 \left (a^2 x^2+1\right )}+\frac {3 \tan ^{-1}(a x)}{32 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac {3 \tan ^{-1}(a x)^3}{32 a^2 c^3}-\frac {45 \tan ^{-1}(a x)}{256 a^2 c^3} \]
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Rubi [A] time = 0.18, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4930, 4900, 4892, 199, 205} \[ -\frac {45 x}{256 a c^3 \left (a^2 x^2+1\right )}-\frac {3 x}{128 a c^3 \left (a^2 x^2+1\right )^2}-\frac {\tan ^{-1}(a x)^3}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac {9 x \tan ^{-1}(a x)^2}{32 a c^3 \left (a^2 x^2+1\right )}+\frac {3 x \tan ^{-1}(a x)^2}{16 a c^3 \left (a^2 x^2+1\right )^2}+\frac {9 \tan ^{-1}(a x)}{32 a^2 c^3 \left (a^2 x^2+1\right )}+\frac {3 \tan ^{-1}(a x)}{32 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac {3 \tan ^{-1}(a x)^3}{32 a^2 c^3}-\frac {45 \tan ^{-1}(a x)}{256 a^2 c^3} \]
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 4892
Rule 4900
Rule 4930
Rubi steps
\begin {align*} \int \frac {x \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx &=-\frac {\tan ^{-1}(a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \int \frac {\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx}{4 a}\\ &=\frac {3 \tan ^{-1}(a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}-\frac {\tan ^{-1}(a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {3 \int \frac {1}{\left (c+a^2 c x^2\right )^3} \, dx}{32 a}+\frac {9 \int \frac {\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{16 a c}\\ &=-\frac {3 x}{128 a c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \tan ^{-1}(a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 x \tan ^{-1}(a x)^2}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)^3}{32 a^2 c^3}-\frac {\tan ^{-1}(a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {9 \int \frac {x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c}-\frac {9 \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{128 a c}\\ &=-\frac {3 x}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac {9 x}{256 a c^3 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \tan ^{-1}(a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {3 x \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 x \tan ^{-1}(a x)^2}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)^3}{32 a^2 c^3}-\frac {\tan ^{-1}(a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {9 \int \frac {1}{c+a^2 c x^2} \, dx}{256 a c^2}-\frac {9 \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{32 a c}\\ &=-\frac {3 x}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac {45 x}{256 a c^3 \left (1+a^2 x^2\right )}-\frac {9 \tan ^{-1}(a x)}{256 a^2 c^3}+\frac {3 \tan ^{-1}(a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \tan ^{-1}(a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {3 x \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 x \tan ^{-1}(a x)^2}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)^3}{32 a^2 c^3}-\frac {\tan ^{-1}(a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {9 \int \frac {1}{c+a^2 c x^2} \, dx}{64 a c^2}\\ &=-\frac {3 x}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac {45 x}{256 a c^3 \left (1+a^2 x^2\right )}-\frac {45 \tan ^{-1}(a x)}{256 a^2 c^3}+\frac {3 \tan ^{-1}(a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \tan ^{-1}(a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {3 x \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 x \tan ^{-1}(a x)^2}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)^3}{32 a^2 c^3}-\frac {\tan ^{-1}(a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 103, normalized size = 0.50 \[ \frac {-3 a x \left (15 a^2 x^2+17\right )+24 a x \left (3 a^2 x^2+5\right ) \tan ^{-1}(a x)^2+8 \left (3 a^4 x^4+6 a^2 x^2-5\right ) \tan ^{-1}(a x)^3-3 \left (15 a^4 x^4+6 a^2 x^2-17\right ) \tan ^{-1}(a x)}{256 c^3 \left (a^3 x^2+a\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 117, normalized size = 0.56 \[ -\frac {45 \, a^{3} x^{3} - 8 \, {\left (3 \, a^{4} x^{4} + 6 \, a^{2} x^{2} - 5\right )} \arctan \left (a x\right )^{3} - 24 \, {\left (3 \, a^{3} x^{3} + 5 \, a x\right )} \arctan \left (a x\right )^{2} + 51 \, a x + 3 \, {\left (15 \, a^{4} x^{4} + 6 \, a^{2} x^{2} - 17\right )} \arctan \left (a x\right )}{256 \, {\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\right )}} \]
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 = 0.06, size = 191, normalized size = 0.92 \[ -\frac {\arctan \left (a x \right )^{3}}{4 a^{2} c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {3 x \arctan \left (a x \right )^{2}}{16 a \,c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {9 x \arctan \left (a x \right )^{2}}{32 a \,c^{3} \left (a^{2} x^{2}+1\right )}+\frac {3 \arctan \left (a x \right )^{3}}{32 a^{2} c^{3}}+\frac {9 \arctan \left (a x \right )}{32 a^{2} c^{3} \left (a^{2} x^{2}+1\right )}+\frac {3 \arctan \left (a x \right )}{32 a^{2} c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {45 a \,x^{3}}{256 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {51 x}{256 a \,c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {45 \arctan \left (a x \right )}{256 a^{2} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 272, normalized size = 1.31 \[ \frac {3 \, {\left (\frac {3 \, a^{2} x^{3} + 5 \, x}{a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}} + \frac {3 \, \arctan \left (a x\right )}{a c^{2}}\right )} \arctan \left (a x\right )^{2}}{32 \, a c} - \frac {3 \, {\left (\frac {{\left (15 \, a^{3} x^{3} - 8 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} + 17 \, a x + 15 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a^{2}}{a^{7} c^{2} x^{4} + 2 \, a^{5} c^{2} x^{2} + a^{3} c^{2}} - \frac {8 \, {\left (3 \, a^{2} x^{2} - 3 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 4\right )} a \arctan \left (a x\right )}{a^{6} c^{2} x^{4} + 2 \, a^{4} c^{2} x^{2} + a^{2} c^{2}}\right )}}{256 \, a c} - \frac {\arctan \left (a x\right )^{3}}{4 \, {\left (a^{2} c x^{2} + c\right )}^{2} a^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.53, size = 189, normalized size = 0.91 \[ {\mathrm {atan}\left (a\,x\right )}^3\,\left (\frac {3}{32\,a^2\,c^3}-\frac {1}{4\,a^4\,c^3\,\left (\frac {1}{a^2}+2\,x^2+a^2\,x^4\right )}\right )-\frac {\frac {45\,a^2\,x^3}{8}+\frac {51\,x}{8}}{32\,a^5\,c^3\,x^4+64\,a^3\,c^3\,x^2+32\,a\,c^3}+\frac {{\mathrm {atan}\left (a\,x\right )}^2\,\left (\frac {15\,x}{32\,a^3\,c^3}+\frac {9\,x^3}{32\,a\,c^3}\right )}{\frac {1}{a^2}+2\,x^2+a^2\,x^4}-\frac {45\,\mathrm {atan}\left (a\,x\right )}{256\,a^2\,c^3}+\frac {\mathrm {atan}\left (a\,x\right )\,\left (\frac {3}{8\,a^4\,c^3}+\frac {9\,x^2}{32\,a^2\,c^3}\right )}{\frac {1}{a^2}+2\,x^2+a^2\,x^4} \]
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 \[ \frac {\int \frac {x \operatorname {atan}^{3}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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