Optimal. Leaf size=225 \[ -\frac {45}{128 a c^3 \left (a^2 x^2+1\right )}-\frac {3}{128 a c^3 \left (a^2 x^2+1\right )^2}+\frac {3 x \tan ^{-1}(a x)^3}{8 c^3 \left (a^2 x^2+1\right )}+\frac {x \tan ^{-1}(a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {9 \tan ^{-1}(a x)^2}{16 a c^3 \left (a^2 x^2+1\right )}+\frac {3 \tan ^{-1}(a x)^2}{16 a c^3 \left (a^2 x^2+1\right )^2}-\frac {45 x \tan ^{-1}(a x)}{64 c^3 \left (a^2 x^2+1\right )}-\frac {3 x \tan ^{-1}(a x)}{32 c^3 \left (a^2 x^2+1\right )^2}+\frac {3 \tan ^{-1}(a x)^4}{32 a c^3}-\frac {45 \tan ^{-1}(a x)^2}{128 a c^3} \]
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Rubi [A] time = 0.20, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {4900, 4892, 4930, 261, 4896} \[ -\frac {45}{128 a c^3 \left (a^2 x^2+1\right )}-\frac {3}{128 a c^3 \left (a^2 x^2+1\right )^2}+\frac {3 x \tan ^{-1}(a x)^3}{8 c^3 \left (a^2 x^2+1\right )}+\frac {x \tan ^{-1}(a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {9 \tan ^{-1}(a x)^2}{16 a c^3 \left (a^2 x^2+1\right )}+\frac {3 \tan ^{-1}(a x)^2}{16 a c^3 \left (a^2 x^2+1\right )^2}-\frac {45 x \tan ^{-1}(a x)}{64 c^3 \left (a^2 x^2+1\right )}-\frac {3 x \tan ^{-1}(a x)}{32 c^3 \left (a^2 x^2+1\right )^2}+\frac {3 \tan ^{-1}(a x)^4}{32 a c^3}-\frac {45 \tan ^{-1}(a x)^2}{128 a c^3} \]
Antiderivative was successfully verified.
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Rule 261
Rule 4892
Rule 4896
Rule 4900
Rule 4930
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx &=\frac {3 \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {x \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {3}{8} \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^3} \, dx+\frac {3 \int \frac {\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}\\ &=-\frac {3}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac {3 x \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {x \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \tan ^{-1}(a x)^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)^4}{32 a c^3}-\frac {9 \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{32 c}-\frac {(9 a) \int \frac {x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{8 c}\\ &=-\frac {3}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac {3 x \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {9 x \tan ^{-1}(a x)}{64 c^3 \left (1+a^2 x^2\right )}-\frac {9 \tan ^{-1}(a x)^2}{128 a c^3}+\frac {3 \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )}+\frac {x \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \tan ^{-1}(a x)^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)^4}{32 a c^3}-\frac {9 \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{8 c}+\frac {(9 a) \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx}{64 c}\\ &=-\frac {3}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac {9}{128 a c^3 \left (1+a^2 x^2\right )}-\frac {3 x \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {45 x \tan ^{-1}(a x)}{64 c^3 \left (1+a^2 x^2\right )}-\frac {45 \tan ^{-1}(a x)^2}{128 a c^3}+\frac {3 \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )}+\frac {x \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \tan ^{-1}(a x)^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)^4}{32 a c^3}+\frac {(9 a) \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c}\\ &=-\frac {3}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac {45}{128 a c^3 \left (1+a^2 x^2\right )}-\frac {3 x \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {45 x \tan ^{-1}(a x)}{64 c^3 \left (1+a^2 x^2\right )}-\frac {45 \tan ^{-1}(a x)^2}{128 a c^3}+\frac {3 \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )}+\frac {x \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \tan ^{-1}(a x)^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)^4}{32 a c^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 114, normalized size = 0.51 \[ -\frac {45 a^2 x^2-12 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^4-16 a x \left (3 a^2 x^2+5\right ) \tan ^{-1}(a x)^3+6 a x \left (15 a^2 x^2+17\right ) \tan ^{-1}(a x)+3 \left (15 a^4 x^4+6 a^2 x^2-17\right ) \tan ^{-1}(a x)^2+48}{128 a c^3 \left (a^2 x^2+1\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 132, normalized size = 0.59 \[ \frac {12 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{4} - 45 \, a^{2} x^{2} + 16 \, {\left (3 \, a^{3} x^{3} + 5 \, a x\right )} \arctan \left (a x\right )^{3} - 3 \, {\left (15 \, a^{4} x^{4} + 6 \, a^{2} x^{2} - 17\right )} \arctan \left (a x\right )^{2} - 6 \, {\left (15 \, a^{3} x^{3} + 17 \, a x\right )} \arctan \left (a x\right ) - 48}{128 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \]
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 = 211, normalized size = 0.94 \[ \frac {x \arctan \left (a x \right )^{3}}{4 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {3 x \arctan \left (a x \right )^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )}+\frac {3 \arctan \left (a x \right )^{4}}{32 a \,c^{3}}+\frac {9 \arctan \left (a x \right )^{2}}{16 a \,c^{3} \left (a^{2} x^{2}+1\right )}+\frac {3 \arctan \left (a x \right )^{2}}{16 a \,c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {45 a^{2} \arctan \left (a x \right ) x^{3}}{64 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {51 x \arctan \left (a x \right )}{64 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {45 \arctan \left (a x \right )^{2}}{128 a \,c^{3}}-\frac {45}{128 a \,c^{3} \left (a^{2} x^{2}+1\right )}-\frac {3}{128 a \,c^{3} \left (a^{2} x^{2}+1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 335, normalized size = 1.49 \[ \frac {1}{8} \, {\left (\frac {3 \, a^{2} x^{3} + 5 \, x}{a^{4} c^{3} x^{4} + 2 \, a^{2} c^{3} x^{2} + c^{3}} + \frac {3 \, \arctan \left (a x\right )}{a c^{3}}\right )} \arctan \left (a x\right )^{3} + \frac {3 \, {\left (3 \, a^{2} x^{2} - 3 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 4\right )} a \arctan \left (a x\right )^{2}}{16 \, {\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\right )}} - \frac {3}{128} \, {\left (\frac {{\left (4 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{4} + 15 \, a^{2} x^{2} - 15 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 16\right )} a^{2}}{a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}} + \frac {2 \, {\left (15 \, a^{3} x^{3} - 8 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} + 17 \, a x + 15 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a \arctan \left (a x\right )}{a^{7} c^{3} x^{4} + 2 \, a^{5} c^{3} x^{2} + a^{3} c^{3}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.54, size = 199, normalized size = 0.88 \[ {\mathrm {atan}\left (a\,x\right )}^2\,\left (\frac {\frac {3}{4\,a^3\,c^3}+\frac {9\,x^2}{16\,a\,c^3}}{\frac {1}{a^2}+2\,x^2+a^2\,x^4}-\frac {45}{128\,a\,c^3}\right )-\frac {\frac {45\,a\,x^2}{2}+\frac {24}{a}}{64\,a^4\,c^3\,x^4+128\,a^2\,c^3\,x^2+64\,c^3}-\frac {\mathrm {atan}\left (a\,x\right )\,\left (\frac {45\,x^3}{64\,c^3}+\frac {51\,x}{64\,a^2\,c^3}\right )}{\frac {1}{a^2}+2\,x^2+a^2\,x^4}+\frac {{\mathrm {atan}\left (a\,x\right )}^3\,\left (\frac {3\,x^3}{8\,c^3}+\frac {5\,x}{8\,a^2\,c^3}\right )}{\frac {1}{a^2}+2\,x^2+a^2\,x^4}+\frac {3\,{\mathrm {atan}\left (a\,x\right )}^4}{32\,a\,c^3} \]
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 \[ \frac {\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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