Optimal. Leaf size=374 \[ -\frac{3 i a^2 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{2 c^2}-\frac{3 i a^2 \text{PolyLog}\left (4,-1+\frac{2}{1-i a x}\right )}{2 c^2}+\frac{3 i a^2 \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{c^2}-\frac{3 a^2 \tan ^{-1}(a x) \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )}{c^2}-\frac{3 a^3 x}{8 c^2 \left (a^2 x^2+1\right )}-\frac{a^2 \tan ^{-1}(a x)^3}{2 c^2 \left (a^2 x^2+1\right )}+\frac{3 a^3 x \tan ^{-1}(a x)^2}{4 c^2 \left (a^2 x^2+1\right )}+\frac{3 a^2 \tan ^{-1}(a x)}{4 c^2 \left (a^2 x^2+1\right )}+\frac{i a^2 \tan ^{-1}(a x)^4}{2 c^2}-\frac{a^2 \tan ^{-1}(a x)^3}{4 c^2}-\frac{3 i a^2 \tan ^{-1}(a x)^2}{2 c^2}-\frac{3 a^2 \tan ^{-1}(a x)}{8 c^2}-\frac{2 a^2 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^3}{c^2}+\frac{3 a^2 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)}{c^2}-\frac{\tan ^{-1}(a x)^3}{2 c^2 x^2}-\frac{3 a \tan ^{-1}(a x)^2}{2 c^2 x} \]
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Rubi [A] time = 1.02629, antiderivative size = 374, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 14, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636, Rules used = {4966, 4918, 4852, 4924, 4868, 2447, 4884, 4992, 4996, 6610, 4930, 4892, 199, 205} \[ -\frac{3 i a^2 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{2 c^2}-\frac{3 i a^2 \text{PolyLog}\left (4,-1+\frac{2}{1-i a x}\right )}{2 c^2}+\frac{3 i a^2 \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{c^2}-\frac{3 a^2 \tan ^{-1}(a x) \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )}{c^2}-\frac{3 a^3 x}{8 c^2 \left (a^2 x^2+1\right )}-\frac{a^2 \tan ^{-1}(a x)^3}{2 c^2 \left (a^2 x^2+1\right )}+\frac{3 a^3 x \tan ^{-1}(a x)^2}{4 c^2 \left (a^2 x^2+1\right )}+\frac{3 a^2 \tan ^{-1}(a x)}{4 c^2 \left (a^2 x^2+1\right )}+\frac{i a^2 \tan ^{-1}(a x)^4}{2 c^2}-\frac{a^2 \tan ^{-1}(a x)^3}{4 c^2}-\frac{3 i a^2 \tan ^{-1}(a x)^2}{2 c^2}-\frac{3 a^2 \tan ^{-1}(a x)}{8 c^2}-\frac{2 a^2 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^3}{c^2}+\frac{3 a^2 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)}{c^2}-\frac{\tan ^{-1}(a x)^3}{2 c^2 x^2}-\frac{3 a \tan ^{-1}(a x)^2}{2 c^2 x} \]
Antiderivative was successfully verified.
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Rule 4966
Rule 4918
Rule 4852
Rule 4924
Rule 4868
Rule 2447
Rule 4884
Rule 4992
Rule 4996
Rule 6610
Rule 4930
Rule 4892
Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^3}{x^3 \left (c+a^2 c x^2\right )^2} \, dx &=-\left (a^2 \int \frac{\tan ^{-1}(a x)^3}{x \left (c+a^2 c x^2\right )^2} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)^3}{x^3 \left (c+a^2 c x^2\right )} \, dx}{c}\\ &=a^4 \int \frac{x \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx+\frac{\int \frac{\tan ^{-1}(a x)^3}{x^3} \, dx}{c^2}-2 \frac{a^2 \int \frac{\tan ^{-1}(a x)^3}{x \left (c+a^2 c x^2\right )} \, dx}{c}\\ &=-\frac{\tan ^{-1}(a x)^3}{2 c^2 x^2}-\frac{a^2 \tan ^{-1}(a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac{1}{2} \left (3 a^3\right ) \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx+\frac{(3 a) \int \frac{\tan ^{-1}(a x)^2}{x^2 \left (1+a^2 x^2\right )} \, dx}{2 c^2}-2 \left (-\frac{i a^2 \tan ^{-1}(a x)^4}{4 c^2}+\frac{\left (i a^2\right ) \int \frac{\tan ^{-1}(a x)^3}{x (i+a x)} \, dx}{c^2}\right )\\ &=\frac{3 a^3 x \tan ^{-1}(a x)^2}{4 c^2 \left (1+a^2 x^2\right )}+\frac{a^2 \tan ^{-1}(a x)^3}{4 c^2}-\frac{\tan ^{-1}(a x)^3}{2 c^2 x^2}-\frac{a^2 \tan ^{-1}(a x)^3}{2 c^2 \left (1+a^2 x^2\right )}-\frac{1}{2} \left (3 a^4\right ) \int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx+\frac{(3 a) \int \frac{\tan ^{-1}(a x)^2}{x^2} \, dx}{2 c^2}-\frac{\left (3 a^3\right ) \int \frac{\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{2 c^2}-2 \left (-\frac{i a^2 \tan ^{-1}(a x)^4}{4 c^2}+\frac{a^2 \tan ^{-1}(a x)^3 \log \left (2-\frac{2}{1-i a x}\right )}{c^2}-\frac{\left (3 a^3\right ) \int \frac{\tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^2}\right )\\ &=\frac{3 a^2 \tan ^{-1}(a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac{3 a \tan ^{-1}(a x)^2}{2 c^2 x}+\frac{3 a^3 x \tan ^{-1}(a x)^2}{4 c^2 \left (1+a^2 x^2\right )}-\frac{a^2 \tan ^{-1}(a x)^3}{4 c^2}-\frac{\tan ^{-1}(a x)^3}{2 c^2 x^2}-\frac{a^2 \tan ^{-1}(a x)^3}{2 c^2 \left (1+a^2 x^2\right )}-\frac{1}{4} \left (3 a^3\right ) \int \frac{1}{\left (c+a^2 c x^2\right )^2} \, dx+\frac{\left (3 a^2\right ) \int \frac{\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx}{c^2}-2 \left (-\frac{i a^2 \tan ^{-1}(a x)^4}{4 c^2}+\frac{a^2 \tan ^{-1}(a x)^3 \log \left (2-\frac{2}{1-i a x}\right )}{c^2}-\frac{3 i a^2 \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{2 c^2}+\frac{\left (3 i a^3\right ) \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^2}\right )\\ &=-\frac{3 a^3 x}{8 c^2 \left (1+a^2 x^2\right )}+\frac{3 a^2 \tan ^{-1}(a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac{3 i a^2 \tan ^{-1}(a x)^2}{2 c^2}-\frac{3 a \tan ^{-1}(a x)^2}{2 c^2 x}+\frac{3 a^3 x \tan ^{-1}(a x)^2}{4 c^2 \left (1+a^2 x^2\right )}-\frac{a^2 \tan ^{-1}(a x)^3}{4 c^2}-\frac{\tan ^{-1}(a x)^3}{2 c^2 x^2}-\frac{a^2 \tan ^{-1}(a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac{\left (3 i a^2\right ) \int \frac{\tan ^{-1}(a x)}{x (i+a x)} \, dx}{c^2}-2 \left (-\frac{i a^2 \tan ^{-1}(a x)^4}{4 c^2}+\frac{a^2 \tan ^{-1}(a x)^3 \log \left (2-\frac{2}{1-i a x}\right )}{c^2}-\frac{3 i a^2 \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{2 c^2}+\frac{3 a^2 \tan ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{2 c^2}-\frac{\left (3 a^3\right ) \int \frac{\text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{2 c^2}\right )-\frac{\left (3 a^3\right ) \int \frac{1}{c+a^2 c x^2} \, dx}{8 c}\\ &=-\frac{3 a^3 x}{8 c^2 \left (1+a^2 x^2\right )}-\frac{3 a^2 \tan ^{-1}(a x)}{8 c^2}+\frac{3 a^2 \tan ^{-1}(a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac{3 i a^2 \tan ^{-1}(a x)^2}{2 c^2}-\frac{3 a \tan ^{-1}(a x)^2}{2 c^2 x}+\frac{3 a^3 x \tan ^{-1}(a x)^2}{4 c^2 \left (1+a^2 x^2\right )}-\frac{a^2 \tan ^{-1}(a x)^3}{4 c^2}-\frac{\tan ^{-1}(a x)^3}{2 c^2 x^2}-\frac{a^2 \tan ^{-1}(a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac{3 a^2 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{c^2}-2 \left (-\frac{i a^2 \tan ^{-1}(a x)^4}{4 c^2}+\frac{a^2 \tan ^{-1}(a x)^3 \log \left (2-\frac{2}{1-i a x}\right )}{c^2}-\frac{3 i a^2 \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{2 c^2}+\frac{3 a^2 \tan ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{2 c^2}+\frac{3 i a^2 \text{Li}_4\left (-1+\frac{2}{1-i a x}\right )}{4 c^2}\right )-\frac{\left (3 a^3\right ) \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^2}\\ &=-\frac{3 a^3 x}{8 c^2 \left (1+a^2 x^2\right )}-\frac{3 a^2 \tan ^{-1}(a x)}{8 c^2}+\frac{3 a^2 \tan ^{-1}(a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac{3 i a^2 \tan ^{-1}(a x)^2}{2 c^2}-\frac{3 a \tan ^{-1}(a x)^2}{2 c^2 x}+\frac{3 a^3 x \tan ^{-1}(a x)^2}{4 c^2 \left (1+a^2 x^2\right )}-\frac{a^2 \tan ^{-1}(a x)^3}{4 c^2}-\frac{\tan ^{-1}(a x)^3}{2 c^2 x^2}-\frac{a^2 \tan ^{-1}(a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac{3 a^2 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{c^2}-\frac{3 i a^2 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{2 c^2}-2 \left (-\frac{i a^2 \tan ^{-1}(a x)^4}{4 c^2}+\frac{a^2 \tan ^{-1}(a x)^3 \log \left (2-\frac{2}{1-i a x}\right )}{c^2}-\frac{3 i a^2 \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{2 c^2}+\frac{3 a^2 \tan ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{2 c^2}+\frac{3 i a^2 \text{Li}_4\left (-1+\frac{2}{1-i a x}\right )}{4 c^2}\right )\\ \end{align*}
Mathematica [A] time = 0.7366, size = 243, normalized size = 0.65 \[ \frac{a^2 \left (-96 i \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(a x)}\right )-96 \tan ^{-1}(a x) \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(a x)}\right )-48 i \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(a x)}\right )+48 i \text{PolyLog}\left (4,e^{-2 i \tan ^{-1}(a x)}\right )-\frac{16 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^3}{a^2 x^2}-16 i \tan ^{-1}(a x)^4-\frac{48 \tan ^{-1}(a x)^2}{a x}-48 i \tan ^{-1}(a x)^2-64 \tan ^{-1}(a x)^3 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )+96 \tan ^{-1}(a x) \log \left (1-e^{2 i \tan ^{-1}(a x)}\right )+12 \tan ^{-1}(a x)^2 \sin \left (2 \tan ^{-1}(a x)\right )-6 \sin \left (2 \tan ^{-1}(a x)\right )-8 \tan ^{-1}(a x)^3 \cos \left (2 \tan ^{-1}(a x)\right )+12 \tan ^{-1}(a x) \cos \left (2 \tan ^{-1}(a x)\right )+i \pi ^4\right )}{32 c^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 7.276, size = 815, normalized size = 2.2 \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 )}^{2} x^{3}}\,{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^{4} c^{2} x^{7} + 2 \, a^{2} c^{2} x^{5} + c^{2} x^{3}}, 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^{4} x^{7} + 2 a^{2} x^{5} + x^{3}}\, dx}{c^{2}} \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 )}^{2} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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