Optimal. Leaf size=205 \[ -i a c^2 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )+\frac{5}{3} i a c^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^2-\frac{1}{3} a^3 c^2 x^2 \tan ^{-1}(a x)+\frac{1}{3} a^2 c^2 x+2 a^2 c^2 x \tan ^{-1}(a x)^2+\frac{2}{3} i a c^2 \tan ^{-1}(a x)^2-\frac{1}{3} a c^2 \tan ^{-1}(a x)-\frac{c^2 \tan ^{-1}(a x)^2}{x}+\frac{10}{3} a c^2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)+2 a c^2 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x) \]
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Rubi [A] time = 0.4231, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 20, 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, 4924, 4868, 2447, 4916, 321, 203} \[ -i a c^2 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )+\frac{5}{3} i a c^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^2-\frac{1}{3} a^3 c^2 x^2 \tan ^{-1}(a x)+\frac{1}{3} a^2 c^2 x+2 a^2 c^2 x \tan ^{-1}(a x)^2+\frac{2}{3} i a c^2 \tan ^{-1}(a x)^2-\frac{1}{3} a c^2 \tan ^{-1}(a x)-\frac{c^2 \tan ^{-1}(a x)^2}{x}+\frac{10}{3} a c^2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)+2 a c^2 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 4948
Rule 4846
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rule 4852
Rule 4924
Rule 4868
Rule 2447
Rule 4916
Rule 321
Rule 203
Rubi steps
\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2}{x^2} \, dx &=\int \left (2 a^2 c^2 \tan ^{-1}(a x)^2+\frac{c^2 \tan ^{-1}(a x)^2}{x^2}+a^4 c^2 x^2 \tan ^{-1}(a x)^2\right ) \, dx\\ &=c^2 \int \frac{\tan ^{-1}(a x)^2}{x^2} \, dx+\left (2 a^2 c^2\right ) \int \tan ^{-1}(a x)^2 \, dx+\left (a^4 c^2\right ) \int x^2 \tan ^{-1}(a x)^2 \, dx\\ &=-\frac{c^2 \tan ^{-1}(a x)^2}{x}+2 a^2 c^2 x \tan ^{-1}(a x)^2+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^2+\left (2 a c^2\right ) \int \frac{\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx-\left (4 a^3 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{3} \left (2 a^5 c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=i a c^2 \tan ^{-1}(a x)^2-\frac{c^2 \tan ^{-1}(a x)^2}{x}+2 a^2 c^2 x \tan ^{-1}(a x)^2+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^2+\left (2 i a c^2\right ) \int \frac{\tan ^{-1}(a x)}{x (i+a x)} \, dx+\left (4 a^2 c^2\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx-\frac{1}{3} \left (2 a^3 c^2\right ) \int x \tan ^{-1}(a x) \, dx+\frac{1}{3} \left (2 a^3 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac{1}{3} a^3 c^2 x^2 \tan ^{-1}(a x)+\frac{2}{3} i a c^2 \tan ^{-1}(a x)^2-\frac{c^2 \tan ^{-1}(a x)^2}{x}+2 a^2 c^2 x \tan ^{-1}(a x)^2+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^2+4 a c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )+2 a c^2 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-\frac{1}{3} \left (2 a^2 c^2\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx-\left (2 a^2 c^2\right ) \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx-\left (4 a^2 c^2\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac{1}{3} \left (a^4 c^2\right ) \int \frac{x^2}{1+a^2 x^2} \, dx\\ &=\frac{1}{3} a^2 c^2 x-\frac{1}{3} a^3 c^2 x^2 \tan ^{-1}(a x)+\frac{2}{3} i a c^2 \tan ^{-1}(a x)^2-\frac{c^2 \tan ^{-1}(a x)^2}{x}+2 a^2 c^2 x \tan ^{-1}(a x)^2+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^2+\frac{10}{3} a c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )+2 a c^2 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-i a c^2 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+\left (4 i a 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^2 c^2\right ) \int \frac{1}{1+a^2 x^2} \, dx+\frac{1}{3} \left (2 a^2 c^2\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=\frac{1}{3} a^2 c^2 x-\frac{1}{3} a c^2 \tan ^{-1}(a x)-\frac{1}{3} a^3 c^2 x^2 \tan ^{-1}(a x)+\frac{2}{3} i a c^2 \tan ^{-1}(a x)^2-\frac{c^2 \tan ^{-1}(a x)^2}{x}+2 a^2 c^2 x \tan ^{-1}(a x)^2+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^2+\frac{10}{3} a c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )+2 a c^2 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-i a c^2 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+2 i a c^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )-\frac{1}{3} \left (2 i a 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} a^2 c^2 x-\frac{1}{3} a c^2 \tan ^{-1}(a x)-\frac{1}{3} a^3 c^2 x^2 \tan ^{-1}(a x)+\frac{2}{3} i a c^2 \tan ^{-1}(a x)^2-\frac{c^2 \tan ^{-1}(a x)^2}{x}+2 a^2 c^2 x \tan ^{-1}(a x)^2+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^2+\frac{10}{3} a c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )+2 a c^2 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-i a c^2 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+\frac{5}{3} i a c^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )\\ \end{align*}
Mathematica [A] time = 0.353405, size = 167, normalized size = 0.81 \[ \frac{c^2 \left (-5 i a x \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )-3 i a x \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(a x)}\right )+a^2 x^2+a^4 x^4 \tan ^{-1}(a x)^2-a^3 x^3 \tan ^{-1}(a x)+6 a^2 x^2 \tan ^{-1}(a x)^2-8 i a x \tan ^{-1}(a x)^2-a x \tan ^{-1}(a x)-3 \tan ^{-1}(a x)^2+6 a x \tan ^{-1}(a x) \log \left (1-e^{2 i \tan ^{-1}(a x)}\right )+10 a x \tan ^{-1}(a x) \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )\right )}{3 x} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.096, size = 346, normalized size = 1.7 \begin{align*}{\frac{{a}^{4}{c}^{2}{x}^{3} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{3}}+2\,{a}^{2}{c}^{2}x \left ( \arctan \left ( ax \right ) \right ) ^{2}-{\frac{{c}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{x}}-{\frac{{a}^{3}{c}^{2}{x}^{2}\arctan \left ( ax \right ) }{3}}-{\frac{8\,a{c}^{2}\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{3}}+2\,a{c}^{2}\arctan \left ( ax \right ) \ln \left ( ax \right ) +{\frac{{a}^{2}{c}^{2}x}{3}}-{\frac{a{c}^{2}\arctan \left ( ax \right ) }{3}}+ia{c}^{2}\ln \left ( ax \right ) \ln \left ( 1+iax \right ) +{\frac{4\,i}{3}}a{c}^{2}\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) -ia{c}^{2}\ln \left ( ax \right ) \ln \left ( 1-iax \right ) -ia{c}^{2}{\it dilog} \left ( 1-iax \right ) +ia{c}^{2}{\it dilog} \left ( 1+iax \right ) +{\frac{4\,i}{3}}a{c}^{2}\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax+i \right ) -{\frac{2\,i}{3}}a{c}^{2} \left ( \ln \left ( ax+i \right ) \right ) ^{2}-{\frac{4\,i}{3}}a{c}^{2}\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax-i \right ) +{\frac{2\,i}{3}}a{c}^{2} \left ( \ln \left ( ax-i \right ) \right ) ^{2}-{\frac{4\,i}{3}}a{c}^{2}{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) +{\frac{4\,i}{3}}a{c}^{2}{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) -{\frac{4\,i}{3}}a{c}^{2}\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) \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^{2}}, 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 2 a^{2} \operatorname{atan}^{2}{\left (a x \right )}\, dx + \int \frac{\operatorname{atan}^{2}{\left (a x \right )}}{x^{2}}\, dx + \int a^{4} x^{2} \operatorname{atan}^{2}{\left (a x \right )}\, 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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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