Optimal. Leaf size=181 \[ \frac {\tan ^{-1}(a x)^3}{24 a^3 c^3}-\frac {\tan ^{-1}(a x)}{64 a^3 c^3}-\frac {x}{64 a^2 c^3 \left (a^2 x^2+1\right )}+\frac {x}{32 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac {x \tan ^{-1}(a x)^2}{8 a^2 c^3 \left (a^2 x^2+1\right )}-\frac {x \tan ^{-1}(a x)^2}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac {\tan ^{-1}(a x)}{8 a^3 c^3 \left (a^2 x^2+1\right )}-\frac {\tan ^{-1}(a x)}{8 a^3 c^3 \left (a^2 x^2+1\right )^2} \]
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Rubi [A] time = 0.27, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4964, 4892, 4930, 199, 205, 4900} \[ -\frac {x}{64 a^2 c^3 \left (a^2 x^2+1\right )}+\frac {x}{32 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac {x \tan ^{-1}(a x)^2}{8 a^2 c^3 \left (a^2 x^2+1\right )}-\frac {x \tan ^{-1}(a x)^2}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac {\tan ^{-1}(a x)}{8 a^3 c^3 \left (a^2 x^2+1\right )}-\frac {\tan ^{-1}(a x)}{8 a^3 c^3 \left (a^2 x^2+1\right )^2}+\frac {\tan ^{-1}(a x)^3}{24 a^3 c^3}-\frac {\tan ^{-1}(a x)}{64 a^3 c^3} \]
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 4892
Rule 4900
Rule 4930
Rule 4964
Rubi steps
\begin {align*} \int \frac {x^2 \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx &=-\frac {\int \frac {\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx}{a^2}+\frac {\int \frac {\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{a^2 c}\\ &=-\frac {\tan ^{-1}(a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )^2}-\frac {x \tan ^{-1}(a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \tan ^{-1}(a x)^2}{2 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^3}{6 a^3 c^3}+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^3} \, dx}{8 a^2}-\frac {3 \int \frac {\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a^2 c}-\frac {\int \frac {x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{a c}\\ &=\frac {x}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {\tan ^{-1}(a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {\tan ^{-1}(a x)}{2 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \tan ^{-1}(a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \tan ^{-1}(a x)^2}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^3}{24 a^3 c^3}+\frac {3 \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{32 a^2 c}-\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{2 a^2 c}+\frac {3 \int \frac {x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a c}\\ &=\frac {x}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {13 x}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac {\tan ^{-1}(a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {\tan ^{-1}(a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \tan ^{-1}(a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \tan ^{-1}(a x)^2}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^3}{24 a^3 c^3}+\frac {3 \int \frac {1}{c+a^2 c x^2} \, dx}{64 a^2 c^2}-\frac {\int \frac {1}{c+a^2 c x^2} \, dx}{4 a^2 c^2}+\frac {3 \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{8 a^2 c}\\ &=\frac {x}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {x}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac {13 \tan ^{-1}(a x)}{64 a^3 c^3}-\frac {\tan ^{-1}(a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {\tan ^{-1}(a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \tan ^{-1}(a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \tan ^{-1}(a x)^2}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^3}{24 a^3 c^3}+\frac {3 \int \frac {1}{c+a^2 c x^2} \, dx}{16 a^2 c^2}\\ &=\frac {x}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {x}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac {\tan ^{-1}(a x)}{64 a^3 c^3}-\frac {\tan ^{-1}(a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {\tan ^{-1}(a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \tan ^{-1}(a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \tan ^{-1}(a x)^2}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^3}{24 a^3 c^3}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 95, normalized size = 0.52 \[ \frac {-3 a^3 x^3+24 a x \left (a^2 x^2-1\right ) \tan ^{-1}(a x)^2+8 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^3-3 \left (a^4 x^4-6 a^2 x^2+1\right ) \tan ^{-1}(a x)+3 a x}{192 a^3 c^3 \left (a^2 x^2+1\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 114, normalized size = 0.63 \[ -\frac {3 \, a^{3} x^{3} - 8 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} - 24 \, {\left (a^{3} x^{3} - a x\right )} \arctan \left (a x\right )^{2} - 3 \, a x + 3 \, {\left (a^{4} x^{4} - 6 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )}{192 \, {\left (a^{7} c^{3} x^{4} + 2 \, a^{5} c^{3} x^{2} + a^{3} c^{3}\right )}} \]
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 = 164, normalized size = 0.91 \[ \frac {\arctan \left (a x \right )^{2} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {x \arctan \left (a x \right )^{2}}{8 a^{2} c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )^{3}}{24 a^{3} c^{3}}-\frac {\arctan \left (a x \right )}{8 a^{3} c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )}{8 a^{3} c^{3} \left (a^{2} x^{2}+1\right )}-\frac {x^{3}}{64 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {x}{64 a^{2} c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {\arctan \left (a x \right )}{64 a^{3} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 232, normalized size = 1.28 \[ \frac {1}{8} \, {\left (\frac {a^{2} x^{3} - x}{a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}} + \frac {\arctan \left (a x\right )}{a^{3} c^{3}}\right )} \arctan \left (a x\right )^{2} - \frac {{\left (3 \, a^{3} x^{3} - 8 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} - 3 \, a x + 3 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a^{2}}{192 \, {\left (a^{9} c^{3} x^{4} + 2 \, a^{7} c^{3} x^{2} + a^{5} c^{3}\right )}} + \frac {{\left (a^{2} x^{2} - {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2}\right )} a \arctan \left (a x\right )}{8 \, {\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.49, size = 150, normalized size = 0.83 \[ \frac {\frac {x}{8\,a^2}-\frac {x^3}{8}}{8\,a^4\,c^3\,x^4+16\,a^2\,c^3\,x^2+8\,c^3}-\frac {{\mathrm {atan}\left (a\,x\right )}^2\,\left (\frac {x}{8\,a^4\,c^3}-\frac {x^3}{8\,a^2\,c^3}\right )}{\frac {1}{a^2}+2\,x^2+a^2\,x^4}-\frac {\mathrm {atan}\left (a\,x\right )}{64\,a^3\,c^3}+\frac {{\mathrm {atan}\left (a\,x\right )}^3}{24\,a^3\,c^3}+\frac {x^2\,\mathrm {atan}\left (a\,x\right )}{8\,a^3\,c^3\,\left (\frac {1}{a^2}+2\,x^2+a^2\,x^4\right )} \]
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 \[ \frac {\int \frac {x^{2} \operatorname {atan}^{2}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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