Optimal. Leaf size=317 \[ \frac {10 i a^3 \text {Li}_2\left (\frac {2}{1-i a x}-1\right )}{3 c^3}+\frac {35 a^3 \tan ^{-1}(a x)^3}{24 c^3}+\frac {10 i a^3 \tan ^{-1}(a x)^2}{3 c^3}-\frac {205 a^3 \tan ^{-1}(a x)}{192 c^3}-\frac {20 a^3 \log \left (2-\frac {2}{1-i a x}\right ) \tan ^{-1}(a x)}{3 c^3}-\frac {a^2}{3 c^3 x}+\frac {3 a^2 \tan ^{-1}(a x)^2}{c^3 x}-\frac {47 a^4 x}{64 c^3 \left (a^2 x^2+1\right )}-\frac {a^4 x}{32 c^3 \left (a^2 x^2+1\right )^2}+\frac {11 a^4 x \tan ^{-1}(a x)^2}{8 c^3 \left (a^2 x^2+1\right )}+\frac {a^4 x \tan ^{-1}(a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {11 a^3 \tan ^{-1}(a x)}{8 c^3 \left (a^2 x^2+1\right )}+\frac {a^3 \tan ^{-1}(a x)}{8 c^3 \left (a^2 x^2+1\right )^2}-\frac {\tan ^{-1}(a x)^2}{3 c^3 x^3}-\frac {a \tan ^{-1}(a x)}{3 c^3 x^2} \]
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Rubi [A] time = 1.53, antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps used = 48, number of rules used = 14, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {4966, 4918, 4852, 325, 203, 4924, 4868, 2447, 4884, 4892, 4930, 199, 205, 4900} \[ \frac {10 i a^3 \text {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{3 c^3}-\frac {47 a^4 x}{64 c^3 \left (a^2 x^2+1\right )}-\frac {a^4 x}{32 c^3 \left (a^2 x^2+1\right )^2}+\frac {11 a^4 x \tan ^{-1}(a x)^2}{8 c^3 \left (a^2 x^2+1\right )}+\frac {a^4 x \tan ^{-1}(a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {11 a^3 \tan ^{-1}(a x)}{8 c^3 \left (a^2 x^2+1\right )}+\frac {a^3 \tan ^{-1}(a x)}{8 c^3 \left (a^2 x^2+1\right )^2}-\frac {a^2}{3 c^3 x}+\frac {35 a^3 \tan ^{-1}(a x)^3}{24 c^3}+\frac {10 i a^3 \tan ^{-1}(a x)^2}{3 c^3}-\frac {205 a^3 \tan ^{-1}(a x)}{192 c^3}+\frac {3 a^2 \tan ^{-1}(a x)^2}{c^3 x}-\frac {20 a^3 \log \left (2-\frac {2}{1-i a x}\right ) \tan ^{-1}(a x)}{3 c^3}-\frac {a \tan ^{-1}(a x)}{3 c^3 x^2}-\frac {\tan ^{-1}(a x)^2}{3 c^3 x^3} \]
Antiderivative was successfully verified.
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Rule 199
Rule 203
Rule 205
Rule 325
Rule 2447
Rule 4852
Rule 4868
Rule 4884
Rule 4892
Rule 4900
Rule 4918
Rule 4924
Rule 4930
Rule 4966
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)^2}{x^4 \left (c+a^2 c x^2\right )^3} \, dx &=-\left (a^2 \int \frac {\tan ^{-1}(a x)^2}{x^2 \left (c+a^2 c x^2\right )^3} \, dx\right )+\frac {\int \frac {\tan ^{-1}(a x)^2}{x^4 \left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=a^4 \int \frac {\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx+\frac {\int \frac {\tan ^{-1}(a x)^2}{x^4 \left (c+a^2 c x^2\right )} \, dx}{c^2}-2 \frac {a^2 \int \frac {\tan ^{-1}(a x)^2}{x^2 \left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=\frac {a^3 \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac {a^4 x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {1}{8} a^4 \int \frac {1}{\left (c+a^2 c x^2\right )^3} \, dx+\frac {\int \frac {\tan ^{-1}(a x)^2}{x^4} \, dx}{c^3}-\frac {a^2 \int \frac {\tan ^{-1}(a x)^2}{x^2 \left (c+a^2 c x^2\right )} \, dx}{c^2}+\frac {\left (3 a^4\right ) \int \frac {\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}-2 \left (\frac {a^2 \int \frac {\tan ^{-1}(a x)^2}{x^2 \left (c+a^2 c x^2\right )} \, dx}{c^2}-\frac {a^4 \int \frac {\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{c}\right )\\ &=-\frac {a^4 x}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {a^3 \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}-\frac {\tan ^{-1}(a x)^2}{3 c^3 x^3}+\frac {a^4 x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^4 x \tan ^{-1}(a x)^2}{8 c^3 \left (1+a^2 x^2\right )}+\frac {a^3 \tan ^{-1}(a x)^3}{8 c^3}+\frac {(2 a) \int \frac {\tan ^{-1}(a x)}{x^3 \left (1+a^2 x^2\right )} \, dx}{3 c^3}-\frac {a^2 \int \frac {\tan ^{-1}(a x)^2}{x^2} \, dx}{c^3}+\frac {a^4 \int \frac {\tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx}{c^2}-\frac {\left (3 a^4\right ) \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{32 c}-\frac {\left (3 a^5\right ) \int \frac {x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}-2 \left (-\frac {a^4 x \tan ^{-1}(a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac {a^3 \tan ^{-1}(a x)^3}{6 c^3}+\frac {a^2 \int \frac {\tan ^{-1}(a x)^2}{x^2} \, dx}{c^3}-\frac {a^4 \int \frac {\tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx}{c^2}+\frac {a^5 \int \frac {x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{c}\right )\\ &=-\frac {a^4 x}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {3 a^4 x}{64 c^3 \left (1+a^2 x^2\right )}+\frac {a^3 \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^3 \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}-\frac {\tan ^{-1}(a x)^2}{3 c^3 x^3}+\frac {a^2 \tan ^{-1}(a x)^2}{c^3 x}+\frac {a^4 x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^4 x \tan ^{-1}(a x)^2}{8 c^3 \left (1+a^2 x^2\right )}+\frac {11 a^3 \tan ^{-1}(a x)^3}{24 c^3}+\frac {(2 a) \int \frac {\tan ^{-1}(a x)}{x^3} \, dx}{3 c^3}-\frac {\left (2 a^3\right ) \int \frac {\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx}{3 c^3}-\frac {\left (2 a^3\right ) \int \frac {\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx}{c^3}-\frac {\left (3 a^4\right ) \int \frac {1}{c+a^2 c x^2} \, dx}{64 c^2}-\frac {\left (3 a^4\right ) \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{8 c}-2 \left (-\frac {a^3 \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {a^2 \tan ^{-1}(a x)^2}{c^3 x}-\frac {a^4 x \tan ^{-1}(a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac {a^3 \tan ^{-1}(a x)^3}{2 c^3}+\frac {\left (2 a^3\right ) \int \frac {\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx}{c^3}+\frac {a^4 \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{2 c}\right )\\ &=-\frac {a^4 x}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {15 a^4 x}{64 c^3 \left (1+a^2 x^2\right )}-\frac {3 a^3 \tan ^{-1}(a x)}{64 c^3}-\frac {a \tan ^{-1}(a x)}{3 c^3 x^2}+\frac {a^3 \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^3 \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {4 i a^3 \tan ^{-1}(a x)^2}{3 c^3}-\frac {\tan ^{-1}(a x)^2}{3 c^3 x^3}+\frac {a^2 \tan ^{-1}(a x)^2}{c^3 x}+\frac {a^4 x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^4 x \tan ^{-1}(a x)^2}{8 c^3 \left (1+a^2 x^2\right )}+\frac {11 a^3 \tan ^{-1}(a x)^3}{24 c^3}+\frac {a^2 \int \frac {1}{x^2 \left (1+a^2 x^2\right )} \, dx}{3 c^3}-\frac {\left (2 i a^3\right ) \int \frac {\tan ^{-1}(a x)}{x (i+a x)} \, dx}{3 c^3}-\frac {\left (2 i a^3\right ) \int \frac {\tan ^{-1}(a x)}{x (i+a x)} \, dx}{c^3}-\frac {\left (3 a^4\right ) \int \frac {1}{c+a^2 c x^2} \, dx}{16 c^2}-2 \left (\frac {a^4 x}{4 c^3 \left (1+a^2 x^2\right )}-\frac {a^3 \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i a^3 \tan ^{-1}(a x)^2}{c^3}-\frac {a^2 \tan ^{-1}(a x)^2}{c^3 x}-\frac {a^4 x \tan ^{-1}(a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac {a^3 \tan ^{-1}(a x)^3}{2 c^3}+\frac {\left (2 i a^3\right ) \int \frac {\tan ^{-1}(a x)}{x (i+a x)} \, dx}{c^3}+\frac {a^4 \int \frac {1}{c+a^2 c x^2} \, dx}{4 c^2}\right )\\ &=-\frac {a^2}{3 c^3 x}-\frac {a^4 x}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {15 a^4 x}{64 c^3 \left (1+a^2 x^2\right )}-\frac {15 a^3 \tan ^{-1}(a x)}{64 c^3}-\frac {a \tan ^{-1}(a x)}{3 c^3 x^2}+\frac {a^3 \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^3 \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {4 i a^3 \tan ^{-1}(a x)^2}{3 c^3}-\frac {\tan ^{-1}(a x)^2}{3 c^3 x^3}+\frac {a^2 \tan ^{-1}(a x)^2}{c^3 x}+\frac {a^4 x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^4 x \tan ^{-1}(a x)^2}{8 c^3 \left (1+a^2 x^2\right )}+\frac {11 a^3 \tan ^{-1}(a x)^3}{24 c^3}-\frac {8 a^3 \tan ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )}{3 c^3}-\frac {a^4 \int \frac {1}{1+a^2 x^2} \, dx}{3 c^3}+\frac {\left (2 a^4\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{3 c^3}-2 \left (\frac {a^4 x}{4 c^3 \left (1+a^2 x^2\right )}+\frac {a^3 \tan ^{-1}(a x)}{4 c^3}-\frac {a^3 \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i a^3 \tan ^{-1}(a x)^2}{c^3}-\frac {a^2 \tan ^{-1}(a x)^2}{c^3 x}-\frac {a^4 x \tan ^{-1}(a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac {a^3 \tan ^{-1}(a x)^3}{2 c^3}+\frac {2 a^3 \tan ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {\left (2 a^4\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}\right )+\frac {\left (2 a^4\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}\\ &=-\frac {a^2}{3 c^3 x}-\frac {a^4 x}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {15 a^4 x}{64 c^3 \left (1+a^2 x^2\right )}-\frac {109 a^3 \tan ^{-1}(a x)}{192 c^3}-\frac {a \tan ^{-1}(a x)}{3 c^3 x^2}+\frac {a^3 \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^3 \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {4 i a^3 \tan ^{-1}(a x)^2}{3 c^3}-\frac {\tan ^{-1}(a x)^2}{3 c^3 x^3}+\frac {a^2 \tan ^{-1}(a x)^2}{c^3 x}+\frac {a^4 x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^4 x \tan ^{-1}(a x)^2}{8 c^3 \left (1+a^2 x^2\right )}+\frac {11 a^3 \tan ^{-1}(a x)^3}{24 c^3}-\frac {8 a^3 \tan ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )}{3 c^3}+\frac {4 i a^3 \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{3 c^3}-2 \left (\frac {a^4 x}{4 c^3 \left (1+a^2 x^2\right )}+\frac {a^3 \tan ^{-1}(a x)}{4 c^3}-\frac {a^3 \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i a^3 \tan ^{-1}(a x)^2}{c^3}-\frac {a^2 \tan ^{-1}(a x)^2}{c^3 x}-\frac {a^4 x \tan ^{-1}(a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac {a^3 \tan ^{-1}(a x)^3}{2 c^3}+\frac {2 a^3 \tan ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {i a^3 \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{c^3}\right )\\ \end {align*}
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Mathematica [A] time = 0.82, size = 189, normalized size = 0.60 \[ \frac {a^3 \left (-\frac {256 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{a^2 x^2}-\frac {256 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}{a^3 x^3}+2560 i \left (\tan ^{-1}(a x)^2+\text {Li}_2\left (e^{2 i \tan ^{-1}(a x)}\right )\right )+1120 \tan ^{-1}(a x)^3+\frac {256 \left (10 \tan ^{-1}(a x)^2-1\right )}{a x}-5120 \tan ^{-1}(a x) \log \left (1-e^{2 i \tan ^{-1}(a x)}\right )+288 \left (2 \tan ^{-1}(a x)^2-1\right ) \sin \left (2 \tan ^{-1}(a x)\right )+3 \left (8 \tan ^{-1}(a x)^2-1\right ) \sin \left (4 \tan ^{-1}(a x)\right )+576 \tan ^{-1}(a x) \cos \left (2 \tan ^{-1}(a x)\right )+12 \tan ^{-1}(a x) \cos \left (4 \tan ^{-1}(a x)\right )\right )}{768 c^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arctan \left (a x\right )^{2}}{a^{6} c^{3} x^{10} + 3 \, a^{4} c^{3} x^{8} + 3 \, a^{2} c^{3} x^{6} + c^{3} x^{4}}, x\right ) \]
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 = 0.15, size = 517, normalized size = 1.63 \[ \frac {13 a^{4} x \arctan \left (a x \right )^{2}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {11 a^{6} \arctan \left (a x \right )^{2} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {49 a^{4} x}{64 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {a \arctan \left (a x \right )}{3 c^{3} x^{2}}+\frac {a^{3} \arctan \left (a x \right )}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {11 a^{3} \arctan \left (a x \right )}{8 c^{3} \left (a^{2} x^{2}+1\right )}+\frac {3 a^{2} \arctan \left (a x \right )^{2}}{c^{3} x}-\frac {a^{2}}{3 c^{3} x}-\frac {205 a^{3} \arctan \left (a x \right )}{192 c^{3}}-\frac {\arctan \left (a x \right )^{2}}{3 c^{3} x^{3}}+\frac {35 a^{3} \arctan \left (a x \right )^{3}}{24 c^{3}}-\frac {5 i a^{3} \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{3 c^{3}}-\frac {10 i a^{3} \dilog \left (i a x +1\right )}{3 c^{3}}+\frac {5 i a^{3} \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{3 c^{3}}+\frac {5 i a^{3} \ln \left (a x +i\right )^{2}}{6 c^{3}}-\frac {5 i a^{3} \ln \left (a x -i\right )^{2}}{6 c^{3}}+\frac {10 i a^{3} \dilog \left (-i a x +1\right )}{3 c^{3}}-\frac {20 a^{3} \arctan \left (a x \right ) \ln \left (a x \right )}{3 c^{3}}+\frac {10 a^{3} \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{3 c^{3}}-\frac {47 a^{6} x^{3}}{64 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {5 i a^{3} \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{3 c^{3}}+\frac {10 i a^{3} \ln \left (a x \right ) \ln \left (-i a x +1\right )}{3 c^{3}}-\frac {10 i a^{3} \ln \left (a x \right ) \ln \left (i a x +1\right )}{3 c^{3}}-\frac {5 i a^{3} \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{3 c^{3}}-\frac {5 i a^{3} \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{3 c^{3}}+\frac {5 i a^{3} \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{3 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^2}{x^4\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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 \[ \frac {\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{a^{6} x^{10} + 3 a^{4} x^{8} + 3 a^{2} x^{6} + x^{4}}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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