Optimal. Leaf size=216 \[ -\frac{5}{3} i a^3 c^2 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )+i a^3 c^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )-\frac{a^2 c^2}{3 x}+a^4 c^2 x \tan ^{-1}(a x)^2-\frac{2}{3} i a^3 c^2 \tan ^{-1}(a x)^2-\frac{1}{3} a^3 c^2 \tan ^{-1}(a x)-\frac{2 a^2 c^2 \tan ^{-1}(a x)^2}{x}+2 a^3 c^2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)+\frac{10}{3} a^3 c^2 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)-\frac{a c^2 \tan ^{-1}(a x)}{3 x^2}-\frac{c^2 \tan ^{-1}(a x)^2}{3 x^3} \]
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Rubi [A] time = 0.439726, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 13, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.591, Rules used = {4948, 4846, 4920, 4854, 2402, 2315, 4852, 4918, 325, 203, 4924, 4868, 2447} \[ -\frac{5}{3} i a^3 c^2 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )+i a^3 c^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )-\frac{a^2 c^2}{3 x}+a^4 c^2 x \tan ^{-1}(a x)^2-\frac{2}{3} i a^3 c^2 \tan ^{-1}(a x)^2-\frac{1}{3} a^3 c^2 \tan ^{-1}(a x)-\frac{2 a^2 c^2 \tan ^{-1}(a x)^2}{x}+2 a^3 c^2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)+\frac{10}{3} a^3 c^2 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)-\frac{a c^2 \tan ^{-1}(a x)}{3 x^2}-\frac{c^2 \tan ^{-1}(a x)^2}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 4948
Rule 4846
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rule 4852
Rule 4918
Rule 325
Rule 203
Rule 4924
Rule 4868
Rule 2447
Rubi steps
\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2}{x^4} \, dx &=\int \left (a^4 c^2 \tan ^{-1}(a x)^2+\frac{c^2 \tan ^{-1}(a x)^2}{x^4}+\frac{2 a^2 c^2 \tan ^{-1}(a x)^2}{x^2}\right ) \, dx\\ &=c^2 \int \frac{\tan ^{-1}(a x)^2}{x^4} \, dx+\left (2 a^2 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{x^2} \, dx+\left (a^4 c^2\right ) \int \tan ^{-1}(a x)^2 \, dx\\ &=-\frac{c^2 \tan ^{-1}(a x)^2}{3 x^3}-\frac{2 a^2 c^2 \tan ^{-1}(a x)^2}{x}+a^4 c^2 x \tan ^{-1}(a x)^2+\frac{1}{3} \left (2 a c^2\right ) \int \frac{\tan ^{-1}(a x)}{x^3 \left (1+a^2 x^2\right )} \, dx+\left (4 a^3 c^2\right ) \int \frac{\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx-\left (2 a^5 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-i a^3 c^2 \tan ^{-1}(a x)^2-\frac{c^2 \tan ^{-1}(a x)^2}{3 x^3}-\frac{2 a^2 c^2 \tan ^{-1}(a x)^2}{x}+a^4 c^2 x \tan ^{-1}(a x)^2+\frac{1}{3} \left (2 a c^2\right ) \int \frac{\tan ^{-1}(a x)}{x^3} \, dx+\left (4 i a^3 c^2\right ) \int \frac{\tan ^{-1}(a x)}{x (i+a x)} \, dx-\frac{1}{3} \left (2 a^3 c^2\right ) \int \frac{\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx+\left (2 a^4 c^2\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx\\ &=-\frac{a c^2 \tan ^{-1}(a x)}{3 x^2}-\frac{2}{3} i a^3 c^2 \tan ^{-1}(a x)^2-\frac{c^2 \tan ^{-1}(a x)^2}{3 x^3}-\frac{2 a^2 c^2 \tan ^{-1}(a x)^2}{x}+a^4 c^2 x \tan ^{-1}(a x)^2+2 a^3 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )+4 a^3 c^2 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )+\frac{1}{3} \left (a^2 c^2\right ) \int \frac{1}{x^2 \left (1+a^2 x^2\right )} \, dx-\frac{1}{3} \left (2 i a^3 c^2\right ) \int \frac{\tan ^{-1}(a x)}{x (i+a x)} \, dx-\left (2 a^4 c^2\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (4 a^4 c^2\right ) \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac{a^2 c^2}{3 x}-\frac{a c^2 \tan ^{-1}(a x)}{3 x^2}-\frac{2}{3} i a^3 c^2 \tan ^{-1}(a x)^2-\frac{c^2 \tan ^{-1}(a x)^2}{3 x^3}-\frac{2 a^2 c^2 \tan ^{-1}(a x)^2}{x}+a^4 c^2 x \tan ^{-1}(a x)^2+2 a^3 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )+\frac{10}{3} a^3 c^2 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-2 i a^3 c^2 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+\left (2 i a^3 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )-\frac{1}{3} \left (a^4 c^2\right ) \int \frac{1}{1+a^2 x^2} \, dx+\frac{1}{3} \left (2 a^4 c^2\right ) \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac{a^2 c^2}{3 x}-\frac{1}{3} a^3 c^2 \tan ^{-1}(a x)-\frac{a c^2 \tan ^{-1}(a x)}{3 x^2}-\frac{2}{3} i a^3 c^2 \tan ^{-1}(a x)^2-\frac{c^2 \tan ^{-1}(a x)^2}{3 x^3}-\frac{2 a^2 c^2 \tan ^{-1}(a x)^2}{x}+a^4 c^2 x \tan ^{-1}(a x)^2+2 a^3 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )+\frac{10}{3} a^3 c^2 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-\frac{5}{3} i a^3 c^2 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+i a^3 c^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )\\ \end{align*}
Mathematica [A] time = 0.381343, size = 189, normalized size = 0.88 \[ \frac{c^2 \left (-3 i a^3 x^3 \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )-5 i a^3 x^3 \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(a x)}\right )-a^2 x^2+3 a^4 x^4 \tan ^{-1}(a x)^2-8 i a^3 x^3 \tan ^{-1}(a x)^2-a^3 x^3 \tan ^{-1}(a x)-6 a^2 x^2 \tan ^{-1}(a x)^2+10 a^3 x^3 \tan ^{-1}(a x) \log \left (1-e^{2 i \tan ^{-1}(a x)}\right )+6 a^3 x^3 \tan ^{-1}(a x) \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-a x \tan ^{-1}(a x)-\tan ^{-1}(a x)^2\right )}{3 x^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.102, size = 375, normalized size = 1.7 \begin{align*}{a}^{4}{c}^{2}x \left ( \arctan \left ( ax \right ) \right ) ^{2}-2\,{\frac{{a}^{2}{c}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{x}}-{\frac{{c}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{3\,{x}^{3}}}-{\frac{8\,{a}^{3}{c}^{2}\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{3}}-{\frac{a{c}^{2}\arctan \left ( ax \right ) }{3\,{x}^{2}}}+{\frac{10\,{a}^{3}{c}^{2}\arctan \left ( ax \right ) \ln \left ( ax \right ) }{3}}-{\frac{{a}^{3}{c}^{2}\arctan \left ( ax \right ) }{3}}-{\frac{{a}^{2}{c}^{2}}{3\,x}}-{\frac{4\,i}{3}}{a}^{3}{c}^{2}{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) +{\frac{4\,i}{3}}{a}^{3}{c}^{2}\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) -{\frac{5\,i}{3}}{a}^{3}{c}^{2}\ln \left ( ax \right ) \ln \left ( 1-iax \right ) -{\frac{4\,i}{3}}{a}^{3}{c}^{2}\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) -{\frac{4\,i}{3}}{a}^{3}{c}^{2}\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax-i \right ) -{\frac{2\,i}{3}}{a}^{3}{c}^{2} \left ( \ln \left ( ax+i \right ) \right ) ^{2}+{\frac{4\,i}{3}}{a}^{3}{c}^{2}{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) +{\frac{5\,i}{3}}{a}^{3}{c}^{2}\ln \left ( ax \right ) \ln \left ( 1+iax \right ) +{\frac{4\,i}{3}}{a}^{3}{c}^{2}\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax+i \right ) +{\frac{5\,i}{3}}{a}^{3}{c}^{2}{\it dilog} \left ( 1+iax \right ) -{\frac{5\,i}{3}}{a}^{3}{c}^{2}{\it dilog} \left ( 1-iax \right ) +{\frac{2\,i}{3}}{a}^{3}{c}^{2} \left ( \ln \left ( ax-i \right ) \right ) ^{2} \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{{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \arctan \left (a x\right )^{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*} c^{2} \left (\int a^{4} \operatorname{atan}^{2}{\left (a x \right )}\, dx + \int \frac{\operatorname{atan}^{2}{\left (a x \right )}}{x^{4}}\, dx + \int \frac{2 a^{2} \operatorname{atan}^{2}{\left (a x \right )}}{x^{2}}\, dx\right ) \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{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{2}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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