Optimal. Leaf size=322 \[ -\frac{3 a^2 \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )}{2 c^3}+\frac{3 i a^2 \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{c^3}+\frac{19 a^2}{32 c^3 \left (a^2 x^2+1\right )}+\frac{a^2}{32 c^3 \left (a^2 x^2+1\right )^2}-\frac{a^2 \log \left (a^2 x^2+1\right )}{2 c^3}+\frac{19 a^3 x \tan ^{-1}(a x)}{16 c^3 \left (a^2 x^2+1\right )}+\frac{a^3 x \tan ^{-1}(a x)}{8 c^3 \left (a^2 x^2+1\right )^2}-\frac{a^2 \tan ^{-1}(a x)^2}{c^3 \left (a^2 x^2+1\right )}-\frac{a^2 \tan ^{-1}(a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac{a^2 \log (x)}{c^3}+\frac{i a^2 \tan ^{-1}(a x)^3}{c^3}+\frac{3 a^2 \tan ^{-1}(a x)^2}{32 c^3}-\frac{3 a^2 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^2}{c^3}-\frac{\tan ^{-1}(a x)^2}{2 c^3 x^2}-\frac{a \tan ^{-1}(a x)}{c^3 x} \]
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Rubi [A] time = 1.33344, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 36, number of rules used = 16, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727, Rules used = {4966, 4918, 4852, 266, 36, 29, 31, 4884, 4924, 4868, 4992, 6610, 4930, 4892, 261, 4896} \[ -\frac{3 a^2 \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )}{2 c^3}+\frac{3 i a^2 \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{c^3}+\frac{19 a^2}{32 c^3 \left (a^2 x^2+1\right )}+\frac{a^2}{32 c^3 \left (a^2 x^2+1\right )^2}-\frac{a^2 \log \left (a^2 x^2+1\right )}{2 c^3}+\frac{19 a^3 x \tan ^{-1}(a x)}{16 c^3 \left (a^2 x^2+1\right )}+\frac{a^3 x \tan ^{-1}(a x)}{8 c^3 \left (a^2 x^2+1\right )^2}-\frac{a^2 \tan ^{-1}(a x)^2}{c^3 \left (a^2 x^2+1\right )}-\frac{a^2 \tan ^{-1}(a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac{a^2 \log (x)}{c^3}+\frac{i a^2 \tan ^{-1}(a x)^3}{c^3}+\frac{3 a^2 \tan ^{-1}(a x)^2}{32 c^3}-\frac{3 a^2 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^2}{c^3}-\frac{\tan ^{-1}(a x)^2}{2 c^3 x^2}-\frac{a \tan ^{-1}(a x)}{c^3 x} \]
Antiderivative was successfully verified.
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Rule 4966
Rule 4918
Rule 4852
Rule 266
Rule 36
Rule 29
Rule 31
Rule 4884
Rule 4924
Rule 4868
Rule 4992
Rule 6610
Rule 4930
Rule 4892
Rule 261
Rule 4896
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^2}{x^3 \left (c+a^2 c x^2\right )^3} \, dx &=-\left (a^2 \int \frac{\tan ^{-1}(a x)^2}{x \left (c+a^2 c x^2\right )^3} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)^2}{x^3 \left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=a^4 \int \frac{x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx+\frac{\int \frac{\tan ^{-1}(a x)^2}{x^3 \left (c+a^2 c x^2\right )} \, dx}{c^2}-2 \frac{a^2 \int \frac{\tan ^{-1}(a x)^2}{x \left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=-\frac{a^2 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{1}{2} a^3 \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^3} \, dx}{c^3}-\frac{a^2 \int \frac{\tan ^{-1}(a x)^2}{x \left (c+a^2 c x^2\right )} \, dx}{c^2}-2 \left (\frac{a^2 \int \frac{\tan ^{-1}(a x)^2}{x \left (c+a^2 c x^2\right )} \, dx}{c^2}-\frac{a^4 \int \frac{x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{c}\right )\\ &=\frac{a^2}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{a^3 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}-\frac{\tan ^{-1}(a x)^2}{2 c^3 x^2}-\frac{a^2 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{i a^2 \tan ^{-1}(a x)^3}{3 c^3}+\frac{a \int \frac{\tan ^{-1}(a x)}{x^2 \left (1+a^2 x^2\right )} \, dx}{c^3}-\frac{\left (i a^2\right ) \int \frac{\tan ^{-1}(a x)^2}{x (i+a x)} \, dx}{c^3}+\frac{\left (3 a^3\right ) \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{8 c}-2 \left (\frac{a^2 \tan ^{-1}(a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac{i a^2 \tan ^{-1}(a x)^3}{3 c^3}+\frac{\left (i a^2\right ) \int \frac{\tan ^{-1}(a x)^2}{x (i+a x)} \, dx}{c^3}-\frac{a^3 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{c}\right )\\ &=\frac{a^2}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{a^3 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^3 x \tan ^{-1}(a x)}{16 c^3 \left (1+a^2 x^2\right )}+\frac{3 a^2 \tan ^{-1}(a x)^2}{32 c^3}-\frac{\tan ^{-1}(a x)^2}{2 c^3 x^2}-\frac{a^2 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{i a^2 \tan ^{-1}(a x)^3}{3 c^3}-\frac{a^2 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}+\frac{a \int \frac{\tan ^{-1}(a x)}{x^2} \, dx}{c^3}-\frac{a^3 \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{c^3}+\frac{\left (2 a^3\right ) \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^4\right ) \int \frac{x}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c}-2 \left (-\frac{a^3 x \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac{a^2 \tan ^{-1}(a x)^2}{4 c^3}+\frac{a^2 \tan ^{-1}(a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac{i a^2 \tan ^{-1}(a x)^3}{3 c^3}+\frac{a^2 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}-\frac{\left (2 a^3\right ) \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{a^4 \int \frac{x}{\left (c+a^2 c x^2\right )^2} \, dx}{2 c}\right )\\ &=\frac{a^2}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^2}{32 c^3 \left (1+a^2 x^2\right )}-\frac{a \tan ^{-1}(a x)}{c^3 x}+\frac{a^3 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^3 x \tan ^{-1}(a x)}{16 c^3 \left (1+a^2 x^2\right )}-\frac{13 a^2 \tan ^{-1}(a x)^2}{32 c^3}-\frac{\tan ^{-1}(a x)^2}{2 c^3 x^2}-\frac{a^2 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{i a^2 \tan ^{-1}(a x)^3}{3 c^3}-\frac{a^2 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}+\frac{i a^2 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^3}+\frac{a^2 \int \frac{1}{x \left (1+a^2 x^2\right )} \, dx}{c^3}-\frac{\left (i a^3\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}-2 \left (-\frac{a^2}{4 c^3 \left (1+a^2 x^2\right )}-\frac{a^3 x \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac{a^2 \tan ^{-1}(a x)^2}{4 c^3}+\frac{a^2 \tan ^{-1}(a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac{i a^2 \tan ^{-1}(a x)^3}{3 c^3}+\frac{a^2 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}-\frac{i a^2 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^3}+\frac{\left (i a^3\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}\right )\\ &=\frac{a^2}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^2}{32 c^3 \left (1+a^2 x^2\right )}-\frac{a \tan ^{-1}(a x)}{c^3 x}+\frac{a^3 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^3 x \tan ^{-1}(a x)}{16 c^3 \left (1+a^2 x^2\right )}-\frac{13 a^2 \tan ^{-1}(a x)^2}{32 c^3}-\frac{\tan ^{-1}(a x)^2}{2 c^3 x^2}-\frac{a^2 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{i a^2 \tan ^{-1}(a x)^3}{3 c^3}-\frac{a^2 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}+\frac{i a^2 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^3}-\frac{a^2 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{2 c^3}-2 \left (-\frac{a^2}{4 c^3 \left (1+a^2 x^2\right )}-\frac{a^3 x \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac{a^2 \tan ^{-1}(a x)^2}{4 c^3}+\frac{a^2 \tan ^{-1}(a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac{i a^2 \tan ^{-1}(a x)^3}{3 c^3}+\frac{a^2 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}-\frac{i a^2 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^3}+\frac{a^2 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{2 c^3}\right )+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )}{2 c^3}\\ &=\frac{a^2}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^2}{32 c^3 \left (1+a^2 x^2\right )}-\frac{a \tan ^{-1}(a x)}{c^3 x}+\frac{a^3 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^3 x \tan ^{-1}(a x)}{16 c^3 \left (1+a^2 x^2\right )}-\frac{13 a^2 \tan ^{-1}(a x)^2}{32 c^3}-\frac{\tan ^{-1}(a x)^2}{2 c^3 x^2}-\frac{a^2 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{i a^2 \tan ^{-1}(a x)^3}{3 c^3}-\frac{a^2 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}+\frac{i a^2 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^3}-\frac{a^2 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{2 c^3}-2 \left (-\frac{a^2}{4 c^3 \left (1+a^2 x^2\right )}-\frac{a^3 x \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac{a^2 \tan ^{-1}(a x)^2}{4 c^3}+\frac{a^2 \tan ^{-1}(a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac{i a^2 \tan ^{-1}(a x)^3}{3 c^3}+\frac{a^2 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}-\frac{i a^2 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^3}+\frac{a^2 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{2 c^3}\right )+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )}{2 c^3}-\frac{a^4 \operatorname{Subst}\left (\int \frac{1}{1+a^2 x} \, dx,x,x^2\right )}{2 c^3}\\ &=\frac{a^2}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^2}{32 c^3 \left (1+a^2 x^2\right )}-\frac{a \tan ^{-1}(a x)}{c^3 x}+\frac{a^3 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^3 x \tan ^{-1}(a x)}{16 c^3 \left (1+a^2 x^2\right )}-\frac{13 a^2 \tan ^{-1}(a x)^2}{32 c^3}-\frac{\tan ^{-1}(a x)^2}{2 c^3 x^2}-\frac{a^2 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{i a^2 \tan ^{-1}(a x)^3}{3 c^3}+\frac{a^2 \log (x)}{c^3}-\frac{a^2 \log \left (1+a^2 x^2\right )}{2 c^3}-\frac{a^2 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}+\frac{i a^2 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^3}-\frac{a^2 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{2 c^3}-2 \left (-\frac{a^2}{4 c^3 \left (1+a^2 x^2\right )}-\frac{a^3 x \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac{a^2 \tan ^{-1}(a x)^2}{4 c^3}+\frac{a^2 \tan ^{-1}(a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac{i a^2 \tan ^{-1}(a x)^3}{3 c^3}+\frac{a^2 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}-\frac{i a^2 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^3}+\frac{a^2 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{2 c^3}\right )\\ \end{align*}
Mathematica [A] time = 0.771168, size = 226, normalized size = 0.7 \[ \frac{a^2 \left (-3 i \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(a x)}\right )-\frac{3}{2} \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(a x)}\right )+\log \left (\frac{a x}{\sqrt{a^2 x^2+1}}\right )-\frac{\left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}{2 a^2 x^2}-i \tan ^{-1}(a x)^3-\frac{\tan ^{-1}(a x)}{a x}-3 \tan ^{-1}(a x)^2 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )+\frac{5}{8} \tan ^{-1}(a x) \sin \left (2 \tan ^{-1}(a x)\right )+\frac{1}{64} \tan ^{-1}(a x) \sin \left (4 \tan ^{-1}(a x)\right )-\frac{5}{8} \tan ^{-1}(a x)^2 \cos \left (2 \tan ^{-1}(a x)\right )-\frac{1}{32} \tan ^{-1}(a x)^2 \cos \left (4 \tan ^{-1}(a x)\right )+\frac{5}{16} \cos \left (2 \tan ^{-1}(a x)\right )+\frac{1}{256} \cos \left (4 \tan ^{-1}(a x)\right )+\frac{i \pi ^3}{8}\right )}{c^3} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 2.632, size = 2421, normalized size = 7.5 \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 )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{3} 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 )^{2}}{a^{6} c^{3} x^{9} + 3 \, a^{4} c^{3} x^{7} + 3 \, a^{2} c^{3} x^{5} + c^{3} 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}^{2}{\left (a x \right )}}{a^{6} x^{9} + 3 a^{4} x^{7} + 3 a^{2} x^{5} + x^{3}}\, 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 )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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