Optimal. Leaf size=156 \[ -\frac{2 i c \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{15 a^3}+\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)^2+\frac{c x}{30 a^2}-\frac{2 i c \tan ^{-1}(a x)^2}{15 a^3}-\frac{c \tan ^{-1}(a x)}{30 a^3}-\frac{4 c \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{15 a^3}-\frac{1}{10} a c x^4 \tan ^{-1}(a x)+\frac{1}{3} c x^3 \tan ^{-1}(a x)^2-\frac{2 c x^2 \tan ^{-1}(a x)}{15 a}+\frac{c x^3}{30} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.40942, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4950, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 302} \[ -\frac{2 i c \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{15 a^3}+\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)^2+\frac{c x}{30 a^2}-\frac{2 i c \tan ^{-1}(a x)^2}{15 a^3}-\frac{c \tan ^{-1}(a x)}{30 a^3}-\frac{4 c \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{15 a^3}-\frac{1}{10} a c x^4 \tan ^{-1}(a x)+\frac{1}{3} c x^3 \tan ^{-1}(a x)^2-\frac{2 c x^2 \tan ^{-1}(a x)}{15 a}+\frac{c x^3}{30} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4950
Rule 4852
Rule 4916
Rule 321
Rule 203
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rule 302
Rubi steps
\begin{align*} \int x^2 \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2 \, dx &=c \int x^2 \tan ^{-1}(a x)^2 \, dx+\left (a^2 c\right ) \int x^4 \tan ^{-1}(a x)^2 \, dx\\ &=\frac{1}{3} c x^3 \tan ^{-1}(a x)^2+\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)^2-\frac{1}{3} (2 a c) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{5} \left (2 a^3 c\right ) \int \frac{x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{1}{3} c x^3 \tan ^{-1}(a x)^2+\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)^2-\frac{(2 c) \int x \tan ^{-1}(a x) \, dx}{3 a}+\frac{(2 c) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a}-\frac{1}{5} (2 a c) \int x^3 \tan ^{-1}(a x) \, dx+\frac{1}{5} (2 a c) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac{c x^2 \tan ^{-1}(a x)}{3 a}-\frac{1}{10} a c x^4 \tan ^{-1}(a x)-\frac{i c \tan ^{-1}(a x)^2}{3 a^3}+\frac{1}{3} c x^3 \tan ^{-1}(a x)^2+\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)^2+\frac{1}{3} c \int \frac{x^2}{1+a^2 x^2} \, dx-\frac{(2 c) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{3 a^2}+\frac{(2 c) \int x \tan ^{-1}(a x) \, dx}{5 a}-\frac{(2 c) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a}+\frac{1}{10} \left (a^2 c\right ) \int \frac{x^4}{1+a^2 x^2} \, dx\\ &=\frac{c x}{3 a^2}-\frac{2 c x^2 \tan ^{-1}(a x)}{15 a}-\frac{1}{10} a c x^4 \tan ^{-1}(a x)-\frac{2 i c \tan ^{-1}(a x)^2}{15 a^3}+\frac{1}{3} c x^3 \tan ^{-1}(a x)^2+\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)^2-\frac{2 c \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{3 a^3}-\frac{1}{5} c \int \frac{x^2}{1+a^2 x^2} \, dx-\frac{c \int \frac{1}{1+a^2 x^2} \, dx}{3 a^2}+\frac{(2 c) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{5 a^2}+\frac{(2 c) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{3 a^2}+\frac{1}{10} \left (a^2 c\right ) \int \left (-\frac{1}{a^4}+\frac{x^2}{a^2}+\frac{1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx\\ &=\frac{c x}{30 a^2}+\frac{c x^3}{30}-\frac{c \tan ^{-1}(a x)}{3 a^3}-\frac{2 c x^2 \tan ^{-1}(a x)}{15 a}-\frac{1}{10} a c x^4 \tan ^{-1}(a x)-\frac{2 i c \tan ^{-1}(a x)^2}{15 a^3}+\frac{1}{3} c x^3 \tan ^{-1}(a x)^2+\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)^2-\frac{4 c \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{15 a^3}-\frac{(2 i c) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{3 a^3}+\frac{c \int \frac{1}{1+a^2 x^2} \, dx}{10 a^2}+\frac{c \int \frac{1}{1+a^2 x^2} \, dx}{5 a^2}-\frac{(2 c) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^2}\\ &=\frac{c x}{30 a^2}+\frac{c x^3}{30}-\frac{c \tan ^{-1}(a x)}{30 a^3}-\frac{2 c x^2 \tan ^{-1}(a x)}{15 a}-\frac{1}{10} a c x^4 \tan ^{-1}(a x)-\frac{2 i c \tan ^{-1}(a x)^2}{15 a^3}+\frac{1}{3} c x^3 \tan ^{-1}(a x)^2+\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)^2-\frac{4 c \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{15 a^3}-\frac{i c \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{3 a^3}+\frac{(2 i c) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{5 a^3}\\ &=\frac{c x}{30 a^2}+\frac{c x^3}{30}-\frac{c \tan ^{-1}(a x)}{30 a^3}-\frac{2 c x^2 \tan ^{-1}(a x)}{15 a}-\frac{1}{10} a c x^4 \tan ^{-1}(a x)-\frac{2 i c \tan ^{-1}(a x)^2}{15 a^3}+\frac{1}{3} c x^3 \tan ^{-1}(a x)^2+\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)^2-\frac{4 c \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{15 a^3}-\frac{2 i c \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{15 a^3}\\ \end{align*}
Mathematica [A] time = 0.607208, size = 104, normalized size = 0.67 \[ \frac{c \left (4 i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )+a^3 x^3+2 \left (3 a^5 x^5+5 a^3 x^3+2 i\right ) \tan ^{-1}(a x)^2-\tan ^{-1}(a x) \left (3 a^4 x^4+4 a^2 x^2+8 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )+1\right )+a x\right )}{30 a^3} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.095, size = 258, normalized size = 1.7 \begin{align*}{\frac{{a}^{2}c{x}^{5} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{5}}+{\frac{c{x}^{3} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{3}}-{\frac{ac{x}^{4}\arctan \left ( ax \right ) }{10}}-{\frac{2\,c{x}^{2}\arctan \left ( ax \right ) }{15\,a}}+{\frac{2\,c\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{15\,{a}^{3}}}+{\frac{c{x}^{3}}{30}}+{\frac{cx}{30\,{a}^{2}}}-{\frac{c\arctan \left ( ax \right ) }{30\,{a}^{3}}}+{\frac{{\frac{i}{15}}c\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{3}}}+{\frac{{\frac{i}{15}}c{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{3}}}-{\frac{{\frac{i}{15}}c\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{3}}}-{\frac{{\frac{i}{15}}c{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{3}}}+{\frac{{\frac{i}{30}}c \left ( \ln \left ( ax+i \right ) \right ) ^{2}}{{a}^{3}}}+{\frac{{\frac{i}{15}}c\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax-i \right ) }{{a}^{3}}}-{\frac{{\frac{i}{30}}c \left ( \ln \left ( ax-i \right ) \right ) ^{2}}{{a}^{3}}}-{\frac{{\frac{i}{15}}c\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax+i \right ) }{{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{60} \,{\left (3 \, a^{2} c x^{5} + 5 \, c x^{3}\right )} \arctan \left (a x\right )^{2} - \frac{1}{240} \,{\left (3 \, a^{2} c x^{5} + 5 \, c x^{3}\right )} \log \left (a^{2} x^{2} + 1\right )^{2} + \int \frac{180 \,{\left (a^{4} c x^{6} + 2 \, a^{2} c x^{4} + c x^{2}\right )} \arctan \left (a x\right )^{2} + 15 \,{\left (a^{4} c x^{6} + 2 \, a^{2} c x^{4} + c x^{2}\right )} \log \left (a^{2} x^{2} + 1\right )^{2} - 8 \,{\left (3 \, a^{3} c x^{5} + 5 \, a c x^{3}\right )} \arctan \left (a x\right ) + 4 \,{\left (3 \, a^{4} c x^{6} + 5 \, a^{2} c x^{4}\right )} \log \left (a^{2} x^{2} + 1\right )}{240 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{2} c x^{4} + c x^{2}\right )} \arctan \left (a x\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} c \left (\int x^{2} \operatorname{atan}^{2}{\left (a x \right )}\, dx + \int a^{2} x^{4} \operatorname{atan}^{2}{\left (a x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} c x^{2} + c\right )} x^{2} \arctan \left (a x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]