Optimal. Leaf size=242 \[ \frac{7 i a^3 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{3 c^2}-\frac{a^4 x}{4 c^2 \left (a^2 x^2+1\right )}+\frac{a^4 x \tan ^{-1}(a x)^2}{2 c^2 \left (a^2 x^2+1\right )}+\frac{a^3 \tan ^{-1}(a x)}{2 c^2 \left (a^2 x^2+1\right )}-\frac{a^2}{3 c^2 x}+\frac{5 a^3 \tan ^{-1}(a x)^3}{6 c^2}+\frac{7 i a^3 \tan ^{-1}(a x)^2}{3 c^2}-\frac{7 a^3 \tan ^{-1}(a x)}{12 c^2}+\frac{2 a^2 \tan ^{-1}(a x)^2}{c^2 x}-\frac{14 a^3 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)}{3 c^2}-\frac{a \tan ^{-1}(a x)}{3 c^2 x^2}-\frac{\tan ^{-1}(a x)^2}{3 c^2 x^3} \]
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Rubi [A] time = 0.87263, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 13, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.591, Rules used = {4966, 4918, 4852, 325, 203, 4924, 4868, 2447, 4884, 4892, 4930, 199, 205} \[ \frac{7 i a^3 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{3 c^2}-\frac{a^4 x}{4 c^2 \left (a^2 x^2+1\right )}+\frac{a^4 x \tan ^{-1}(a x)^2}{2 c^2 \left (a^2 x^2+1\right )}+\frac{a^3 \tan ^{-1}(a x)}{2 c^2 \left (a^2 x^2+1\right )}-\frac{a^2}{3 c^2 x}+\frac{5 a^3 \tan ^{-1}(a x)^3}{6 c^2}+\frac{7 i a^3 \tan ^{-1}(a x)^2}{3 c^2}-\frac{7 a^3 \tan ^{-1}(a x)}{12 c^2}+\frac{2 a^2 \tan ^{-1}(a x)^2}{c^2 x}-\frac{14 a^3 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)}{3 c^2}-\frac{a \tan ^{-1}(a x)}{3 c^2 x^2}-\frac{\tan ^{-1}(a x)^2}{3 c^2 x^3} \]
Antiderivative was successfully verified.
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Rule 4966
Rule 4918
Rule 4852
Rule 325
Rule 203
Rule 4924
Rule 4868
Rule 2447
Rule 4884
Rule 4892
Rule 4930
Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^2}{x^4 \left (c+a^2 c x^2\right )^2} \, dx &=-\left (a^2 \int \frac{\tan ^{-1}(a x)^2}{x^2 \left (c+a^2 c x^2\right )^2} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)^2}{x^4 \left (c+a^2 c x^2\right )} \, dx}{c}\\ &=a^4 \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx+\frac{\int \frac{\tan ^{-1}(a x)^2}{x^4} \, dx}{c^2}-2 \frac{a^2 \int \frac{\tan ^{-1}(a x)^2}{x^2 \left (c+a^2 c x^2\right )} \, dx}{c}\\ &=-\frac{\tan ^{-1}(a x)^2}{3 c^2 x^3}+\frac{a^4 x \tan ^{-1}(a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac{a^3 \tan ^{-1}(a x)^3}{6 c^2}-a^5 \int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx+\frac{(2 a) \int \frac{\tan ^{-1}(a x)}{x^3 \left (1+a^2 x^2\right )} \, dx}{3 c^2}-2 \left (\frac{a^2 \int \frac{\tan ^{-1}(a x)^2}{x^2} \, dx}{c^2}-\frac{a^4 \int \frac{\tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx}{c}\right )\\ &=\frac{a^3 \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac{\tan ^{-1}(a x)^2}{3 c^2 x^3}+\frac{a^4 x \tan ^{-1}(a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac{a^3 \tan ^{-1}(a x)^3}{6 c^2}-\frac{1}{2} a^4 \int \frac{1}{\left (c+a^2 c x^2\right )^2} \, dx+\frac{(2 a) \int \frac{\tan ^{-1}(a x)}{x^3} \, dx}{3 c^2}-\frac{\left (2 a^3\right ) \int \frac{\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx}{3 c^2}-2 \left (-\frac{a^2 \tan ^{-1}(a x)^2}{c^2 x}-\frac{a^3 \tan ^{-1}(a x)^3}{3 c^2}+\frac{\left (2 a^3\right ) \int \frac{\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx}{c^2}\right )\\ &=-\frac{a^4 x}{4 c^2 \left (1+a^2 x^2\right )}-\frac{a \tan ^{-1}(a x)}{3 c^2 x^2}+\frac{a^3 \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac{i a^3 \tan ^{-1}(a x)^2}{3 c^2}-\frac{\tan ^{-1}(a x)^2}{3 c^2 x^3}+\frac{a^4 x \tan ^{-1}(a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac{a^3 \tan ^{-1}(a x)^3}{6 c^2}+\frac{a^2 \int \frac{1}{x^2 \left (1+a^2 x^2\right )} \, dx}{3 c^2}-\frac{\left (2 i a^3\right ) \int \frac{\tan ^{-1}(a x)}{x (i+a x)} \, dx}{3 c^2}-2 \left (-\frac{i a^3 \tan ^{-1}(a x)^2}{c^2}-\frac{a^2 \tan ^{-1}(a x)^2}{c^2 x}-\frac{a^3 \tan ^{-1}(a x)^3}{3 c^2}+\frac{\left (2 i a^3\right ) \int \frac{\tan ^{-1}(a x)}{x (i+a x)} \, dx}{c^2}\right )-\frac{a^4 \int \frac{1}{c+a^2 c x^2} \, dx}{4 c}\\ &=-\frac{a^2}{3 c^2 x}-\frac{a^4 x}{4 c^2 \left (1+a^2 x^2\right )}-\frac{a^3 \tan ^{-1}(a x)}{4 c^2}-\frac{a \tan ^{-1}(a x)}{3 c^2 x^2}+\frac{a^3 \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac{i a^3 \tan ^{-1}(a x)^2}{3 c^2}-\frac{\tan ^{-1}(a x)^2}{3 c^2 x^3}+\frac{a^4 x \tan ^{-1}(a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac{a^3 \tan ^{-1}(a x)^3}{6 c^2}-\frac{2 a^3 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{3 c^2}-\frac{a^4 \int \frac{1}{1+a^2 x^2} \, dx}{3 c^2}+\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^2}-2 \left (-\frac{i a^3 \tan ^{-1}(a x)^2}{c^2}-\frac{a^2 \tan ^{-1}(a x)^2}{c^2 x}-\frac{a^3 \tan ^{-1}(a x)^3}{3 c^2}+\frac{2 a^3 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{c^2}-\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^2}\right )\\ &=-\frac{a^2}{3 c^2 x}-\frac{a^4 x}{4 c^2 \left (1+a^2 x^2\right )}-\frac{7 a^3 \tan ^{-1}(a x)}{12 c^2}-\frac{a \tan ^{-1}(a x)}{3 c^2 x^2}+\frac{a^3 \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac{i a^3 \tan ^{-1}(a x)^2}{3 c^2}-\frac{\tan ^{-1}(a x)^2}{3 c^2 x^3}+\frac{a^4 x \tan ^{-1}(a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac{a^3 \tan ^{-1}(a x)^3}{6 c^2}-\frac{2 a^3 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{3 c^2}+\frac{i a^3 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{3 c^2}-2 \left (-\frac{i a^3 \tan ^{-1}(a x)^2}{c^2}-\frac{a^2 \tan ^{-1}(a x)^2}{c^2 x}-\frac{a^3 \tan ^{-1}(a x)^3}{3 c^2}+\frac{2 a^3 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{c^2}-\frac{i a^3 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^2}\right )\\ \end{align*}
Mathematica [A] time = 0.419227, size = 166, normalized size = 0.69 \[ \frac{56 i a^3 x^3 \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(a x)}\right )+20 a^3 x^3 \tan ^{-1}(a x)^3-a^2 x^2 \left (3 a x \sin \left (2 \tan ^{-1}(a x)\right )+8\right )+\tan ^{-1}(a x)^2 \left (56 i a^3 x^3+48 a^2 x^2+6 a^3 x^3 \sin \left (2 \tan ^{-1}(a x)\right )-8\right )+2 a x \tan ^{-1}(a x) \left (-4 a^2 x^2-56 a^2 x^2 \log \left (1-e^{2 i \tan ^{-1}(a x)}\right )+3 a^2 x^2 \cos \left (2 \tan ^{-1}(a x)\right )-4\right )}{24 c^2 x^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.12, size = 444, normalized size = 1.8 \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{4} c^{2} x^{8} + 2 \, a^{2} c^{2} x^{6} + c^{2} x^{4}}, 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^{4} x^{8} + 2 a^{2} x^{6} + x^{4}}\, 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 )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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