Optimal. Leaf size=332 \[ \frac{3 i \text{PolyLog}\left (4,-1+\frac{2}{1-i a x}\right )}{4 c^3}-\frac{3 i \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{2 c^3}+\frac{3 \tan ^{-1}(a x) \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )}{2 c^3}+\frac{141 a x}{256 c^3 \left (a^2 x^2+1\right )}+\frac{3 a x}{128 c^3 \left (a^2 x^2+1\right )^2}+\frac{\tan ^{-1}(a x)^3}{2 c^3 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac{33 a x \tan ^{-1}(a x)^2}{32 c^3 \left (a^2 x^2+1\right )}-\frac{3 a x \tan ^{-1}(a x)^2}{16 c^3 \left (a^2 x^2+1\right )^2}-\frac{33 \tan ^{-1}(a x)}{32 c^3 \left (a^2 x^2+1\right )}-\frac{3 \tan ^{-1}(a x)}{32 c^3 \left (a^2 x^2+1\right )^2}-\frac{i \tan ^{-1}(a x)^4}{4 c^3}-\frac{11 \tan ^{-1}(a x)^3}{32 c^3}+\frac{141 \tan ^{-1}(a x)}{256 c^3}+\frac{\log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^3}{c^3} \]
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Rubi [A] time = 0.709367, antiderivative size = 332, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 12, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {4966, 4924, 4868, 4884, 4992, 4996, 6610, 4930, 4892, 199, 205, 4900} \[ \frac{3 i \text{PolyLog}\left (4,-1+\frac{2}{1-i a x}\right )}{4 c^3}-\frac{3 i \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{2 c^3}+\frac{3 \tan ^{-1}(a x) \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )}{2 c^3}+\frac{141 a x}{256 c^3 \left (a^2 x^2+1\right )}+\frac{3 a x}{128 c^3 \left (a^2 x^2+1\right )^2}+\frac{\tan ^{-1}(a x)^3}{2 c^3 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac{33 a x \tan ^{-1}(a x)^2}{32 c^3 \left (a^2 x^2+1\right )}-\frac{3 a x \tan ^{-1}(a x)^2}{16 c^3 \left (a^2 x^2+1\right )^2}-\frac{33 \tan ^{-1}(a x)}{32 c^3 \left (a^2 x^2+1\right )}-\frac{3 \tan ^{-1}(a x)}{32 c^3 \left (a^2 x^2+1\right )^2}-\frac{i \tan ^{-1}(a x)^4}{4 c^3}-\frac{11 \tan ^{-1}(a x)^3}{32 c^3}+\frac{141 \tan ^{-1}(a x)}{256 c^3}+\frac{\log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^3}{c^3} \]
Antiderivative was successfully verified.
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Rule 4966
Rule 4924
Rule 4868
Rule 4884
Rule 4992
Rule 4996
Rule 6610
Rule 4930
Rule 4892
Rule 199
Rule 205
Rule 4900
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^3}{x \left (c+a^2 c x^2\right )^3} \, dx &=-\left (a^2 \int \frac{x \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)^3}{x \left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=\frac{\tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{1}{4} (3 a) \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx+\frac{\int \frac{\tan ^{-1}(a x)^3}{x \left (c+a^2 c x^2\right )} \, dx}{c^2}-\frac{a^2 \int \frac{x \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=-\frac{3 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 a x \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac{\tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{\tan ^{-1}(a x)^3}{2 c^3 \left (1+a^2 x^2\right )}-\frac{i \tan ^{-1}(a x)^4}{4 c^3}+\frac{1}{32} (3 a) \int \frac{1}{\left (c+a^2 c x^2\right )^3} \, dx+\frac{i \int \frac{\tan ^{-1}(a x)^3}{x (i+a x)} \, dx}{c^3}-\frac{(9 a) \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c}-\frac{(3 a) \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{2 c}\\ &=\frac{3 a x}{128 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 a x \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac{33 a x \tan ^{-1}(a x)^2}{32 c^3 \left (1+a^2 x^2\right )}-\frac{11 \tan ^{-1}(a x)^3}{32 c^3}+\frac{\tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{\tan ^{-1}(a x)^3}{2 c^3 \left (1+a^2 x^2\right )}-\frac{i \tan ^{-1}(a x)^4}{4 c^3}+\frac{\tan ^{-1}(a x)^3 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}-\frac{(3 a) \int \frac{\tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}+\frac{(9 a) \int \frac{1}{\left (c+a^2 c x^2\right )^2} \, dx}{128 c}+\frac{\left (9 a^2\right ) \int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c}+\frac{\left (3 a^2\right ) \int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{2 c}\\ &=\frac{3 a x}{128 c^3 \left (1+a^2 x^2\right )^2}+\frac{9 a x}{256 c^3 \left (1+a^2 x^2\right )}-\frac{3 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac{33 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )}-\frac{3 a x \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac{33 a x \tan ^{-1}(a x)^2}{32 c^3 \left (1+a^2 x^2\right )}-\frac{11 \tan ^{-1}(a x)^3}{32 c^3}+\frac{\tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{\tan ^{-1}(a x)^3}{2 c^3 \left (1+a^2 x^2\right )}-\frac{i \tan ^{-1}(a x)^4}{4 c^3}+\frac{\tan ^{-1}(a x)^3 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}-\frac{3 i \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{2 c^3}+\frac{(3 i a) \int \frac{\tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}+\frac{(9 a) \int \frac{1}{c+a^2 c x^2} \, dx}{256 c^2}+\frac{(9 a) \int \frac{1}{\left (c+a^2 c x^2\right )^2} \, dx}{32 c}+\frac{(3 a) \int \frac{1}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}\\ &=\frac{3 a x}{128 c^3 \left (1+a^2 x^2\right )^2}+\frac{141 a x}{256 c^3 \left (1+a^2 x^2\right )}+\frac{9 \tan ^{-1}(a x)}{256 c^3}-\frac{3 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac{33 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )}-\frac{3 a x \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac{33 a x \tan ^{-1}(a x)^2}{32 c^3 \left (1+a^2 x^2\right )}-\frac{11 \tan ^{-1}(a x)^3}{32 c^3}+\frac{\tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{\tan ^{-1}(a x)^3}{2 c^3 \left (1+a^2 x^2\right )}-\frac{i \tan ^{-1}(a x)^4}{4 c^3}+\frac{\tan ^{-1}(a x)^3 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}-\frac{3 i \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{2 c^3}+\frac{3 \tan ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{2 c^3}-\frac{(3 a) \int \frac{\text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{2 c^3}+\frac{(9 a) \int \frac{1}{c+a^2 c x^2} \, dx}{64 c^2}+\frac{(3 a) \int \frac{1}{c+a^2 c x^2} \, dx}{8 c^2}\\ &=\frac{3 a x}{128 c^3 \left (1+a^2 x^2\right )^2}+\frac{141 a x}{256 c^3 \left (1+a^2 x^2\right )}+\frac{141 \tan ^{-1}(a x)}{256 c^3}-\frac{3 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac{33 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )}-\frac{3 a x \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac{33 a x \tan ^{-1}(a x)^2}{32 c^3 \left (1+a^2 x^2\right )}-\frac{11 \tan ^{-1}(a x)^3}{32 c^3}+\frac{\tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{\tan ^{-1}(a x)^3}{2 c^3 \left (1+a^2 x^2\right )}-\frac{i \tan ^{-1}(a x)^4}{4 c^3}+\frac{\tan ^{-1}(a x)^3 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}-\frac{3 i \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{2 c^3}+\frac{3 \tan ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{2 c^3}+\frac{3 i \text{Li}_4\left (-1+\frac{2}{1-i a x}\right )}{4 c^3}\\ \end{align*}
Mathematica [A] time = 0.313079, size = 208, normalized size = 0.63 \[ \frac{1536 i \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(a x)}\right )+1536 \tan ^{-1}(a x) \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(a x)}\right )-768 i \text{PolyLog}\left (4,e^{-2 i \tan ^{-1}(a x)}\right )+256 i \tan ^{-1}(a x)^4+1024 \tan ^{-1}(a x)^3 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )-576 \tan ^{-1}(a x)^2 \sin \left (2 \tan ^{-1}(a x)\right )-24 \tan ^{-1}(a x)^2 \sin \left (4 \tan ^{-1}(a x)\right )+288 \sin \left (2 \tan ^{-1}(a x)\right )+3 \sin \left (4 \tan ^{-1}(a x)\right )+384 \tan ^{-1}(a x)^3 \cos \left (2 \tan ^{-1}(a x)\right )+32 \tan ^{-1}(a x)^3 \cos \left (4 \tan ^{-1}(a x)\right )-576 \tan ^{-1}(a x) \cos \left (2 \tan ^{-1}(a x)\right )-12 \tan ^{-1}(a x) \cos \left (4 \tan ^{-1}(a x)\right )-16 i \pi ^4}{1024 c^3} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 3.697, size = 2463, normalized size = 7.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\arctan \left (a x\right )^{3}}{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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\operatorname{atan}^{3}{\left (a x \right )}}{a^{6} x^{7} + 3 a^{4} x^{5} + 3 a^{2} x^{3} + x}\, dx}{c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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