Optimal. Leaf size=236 \[ -\frac {11}{32 c^3 \left (a^2 x^2+1\right )}-\frac {1}{32 c^3 \left (a^2 x^2+1\right )^2}+\frac {\tan ^{-1}(a x)^2}{2 c^3 \left (a^2 x^2+1\right )}+\frac {\tan ^{-1}(a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {11 a x \tan ^{-1}(a x)}{16 c^3 \left (a^2 x^2+1\right )}-\frac {a x \tan ^{-1}(a x)}{8 c^3 \left (a^2 x^2+1\right )^2}+\frac {\text {Li}_3\left (\frac {2}{1-i a x}-1\right )}{2 c^3}-\frac {i \text {Li}_2\left (\frac {2}{1-i a x}-1\right ) \tan ^{-1}(a x)}{c^3}-\frac {i \tan ^{-1}(a x)^3}{3 c^3}-\frac {11 \tan ^{-1}(a x)^2}{32 c^3}+\frac {\log \left (2-\frac {2}{1-i a x}\right ) \tan ^{-1}(a x)^2}{c^3} \]
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Rubi [A] time = 0.48, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {4966, 4924, 4868, 4884, 4992, 6610, 4930, 4892, 261, 4896} \[ \frac {\text {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c^3}-\frac {i \tan ^{-1}(a x) \text {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c^3}-\frac {11}{32 c^3 \left (a^2 x^2+1\right )}-\frac {1}{32 c^3 \left (a^2 x^2+1\right )^2}+\frac {\tan ^{-1}(a x)^2}{2 c^3 \left (a^2 x^2+1\right )}+\frac {\tan ^{-1}(a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {11 a x \tan ^{-1}(a x)}{16 c^3 \left (a^2 x^2+1\right )}-\frac {a x \tan ^{-1}(a x)}{8 c^3 \left (a^2 x^2+1\right )^2}-\frac {i \tan ^{-1}(a x)^3}{3 c^3}-\frac {11 \tan ^{-1}(a x)^2}{32 c^3}+\frac {\log \left (2-\frac {2}{1-i a x}\right ) \tan ^{-1}(a x)^2}{c^3} \]
Antiderivative was successfully verified.
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Rule 261
Rule 4868
Rule 4884
Rule 4892
Rule 4896
Rule 4924
Rule 4930
Rule 4966
Rule 4992
Rule 6610
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)^2}{x \left (c+a^2 c x^2\right )^3} \, dx &=-\left (a^2 \int \frac {x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx\right )+\frac {\int \frac {\tan ^{-1}(a x)^2}{x \left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=\frac {\tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {1}{2} a \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^3} \, dx+\frac {\int \frac {\tan ^{-1}(a x)^2}{x \left (c+a^2 c x^2\right )} \, dx}{c^2}-\frac {a^2 \int \frac {x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=-\frac {1}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {a x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac {\tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {\tan ^{-1}(a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i \tan ^{-1}(a x)^3}{3 c^3}+\frac {i \int \frac {\tan ^{-1}(a x)^2}{x (i+a x)} \, dx}{c^3}-\frac {(3 a) \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{8 c}-\frac {a \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=-\frac {1}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {a x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}-\frac {11 a x \tan ^{-1}(a x)}{16 c^3 \left (1+a^2 x^2\right )}-\frac {11 \tan ^{-1}(a x)^2}{32 c^3}+\frac {\tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {\tan ^{-1}(a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i \tan ^{-1}(a x)^3}{3 c^3}+\frac {\tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {(2 a) \int \frac {\tan ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}+\frac {\left (3 a^2\right ) \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c}+\frac {a^2 \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx}{2 c}\\ &=-\frac {1}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {11}{32 c^3 \left (1+a^2 x^2\right )}-\frac {a x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}-\frac {11 a x \tan ^{-1}(a x)}{16 c^3 \left (1+a^2 x^2\right )}-\frac {11 \tan ^{-1}(a x)^2}{32 c^3}+\frac {\tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {\tan ^{-1}(a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i \tan ^{-1}(a x)^3}{3 c^3}+\frac {\tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {i \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{c^3}+\frac {(i a) \int \frac {\text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}\\ &=-\frac {1}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {11}{32 c^3 \left (1+a^2 x^2\right )}-\frac {a x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}-\frac {11 a x \tan ^{-1}(a x)}{16 c^3 \left (1+a^2 x^2\right )}-\frac {11 \tan ^{-1}(a x)^2}{32 c^3}+\frac {\tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {\tan ^{-1}(a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i \tan ^{-1}(a x)^3}{3 c^3}+\frac {\tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {i \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{c^3}+\frac {\text {Li}_3\left (-1+\frac {2}{1-i a x}\right )}{2 c^3}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 156, normalized size = 0.66 \[ \frac {768 i \tan ^{-1}(a x) \text {Li}_2\left (e^{-2 i \tan ^{-1}(a x)}\right )+384 \text {Li}_3\left (e^{-2 i \tan ^{-1}(a x)}\right )+256 i \tan ^{-1}(a x)^3+768 \tan ^{-1}(a x)^2 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )-288 \tan ^{-1}(a x) \sin \left (2 \tan ^{-1}(a x)\right )-12 \tan ^{-1}(a x) \sin \left (4 \tan ^{-1}(a x)\right )+288 \tan ^{-1}(a x)^2 \cos \left (2 \tan ^{-1}(a x)\right )+24 \tan ^{-1}(a x)^2 \cos \left (4 \tan ^{-1}(a x)\right )-144 \cos \left (2 \tan ^{-1}(a x)\right )-3 \cos \left (4 \tan ^{-1}(a x)\right )-32 i \pi ^3}{768 c^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arctan \left (a x\right )^{2}}{a^{6} c^{3} x^{7} + 3 \, a^{4} c^{3} x^{5} + 3 \, a^{2} c^{3} x^{3} + c^{3} x}, x\right ) \]
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 [C] time = 1.63, size = 1986, normalized size = 8.42 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{3} x}\,{d x} \]
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 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^2}{x\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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 \[ \frac {\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{a^{6} x^{7} + 3 a^{4} x^{5} + 3 a^{2} x^{3} + x}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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