Optimal. Leaf size=138 \[ \frac{3}{32 a^2 c^3 \left (a^2 x^2+1\right )}+\frac{1}{32 a^2 c^3 \left (a^2 x^2+1\right )^2}-\frac{\tan ^{-1}(a x)^2}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac{3 x \tan ^{-1}(a x)}{16 a c^3 \left (a^2 x^2+1\right )}+\frac{x \tan ^{-1}(a x)}{8 a c^3 \left (a^2 x^2+1\right )^2}+\frac{3 \tan ^{-1}(a x)^2}{32 a^2 c^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0964198, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4930, 4896, 4892, 261} \[ \frac{3}{32 a^2 c^3 \left (a^2 x^2+1\right )}+\frac{1}{32 a^2 c^3 \left (a^2 x^2+1\right )^2}-\frac{\tan ^{-1}(a x)^2}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac{3 x \tan ^{-1}(a x)}{16 a c^3 \left (a^2 x^2+1\right )}+\frac{x \tan ^{-1}(a x)}{8 a c^3 \left (a^2 x^2+1\right )^2}+\frac{3 \tan ^{-1}(a x)^2}{32 a^2 c^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4930
Rule 4896
Rule 4892
Rule 261
Rubi steps
\begin{align*} \int \frac{x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx &=-\frac{\tan ^{-1}(a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{\int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^3} \, dx}{2 a}\\ &=\frac{1}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{x \tan ^{-1}(a x)}{8 a c^3 \left (1+a^2 x^2\right )^2}-\frac{\tan ^{-1}(a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{8 a c}\\ &=\frac{1}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{x \tan ^{-1}(a x)}{8 a c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)}{16 a c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^2}{32 a^2 c^3}-\frac{\tan ^{-1}(a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 \int \frac{x}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c}\\ &=\frac{1}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{3}{32 a^2 c^3 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)}{8 a c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)}{16 a c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^2}{32 a^2 c^3}-\frac{\tan ^{-1}(a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0385074, size = 71, normalized size = 0.51 \[ \frac{3 a^2 x^2+2 a x \left (3 a^2 x^2+5\right ) \tan ^{-1}(a x)+\left (3 a^4 x^4+6 a^2 x^2-5\right ) \tan ^{-1}(a x)^2+4}{32 c^3 \left (a^3 x^2+a\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.041, size = 127, normalized size = 0.9 \begin{align*}{\frac{1}{32\,{c}^{3}{a}^{2} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{3}{32\,{c}^{3}{a}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{x\arctan \left ( ax \right ) }{8\,a{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{3\,x\arctan \left ( ax \right ) }{16\,a{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}}{32\,{c}^{3}{a}^{2}}}-{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{2}}{4\,{c}^{3}{a}^{2} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.58952, size = 220, normalized size = 1.59 \begin{align*} \frac{{\left (\frac{3 \, a^{2} x^{3} + 5 \, x}{a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}} + \frac{3 \, \arctan \left (a x\right )}{a c^{2}}\right )} \arctan \left (a x\right )}{16 \, a c} + \frac{3 \, a^{2} x^{2} - 3 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 4}{32 \,{\left (a^{6} c^{2} x^{4} + 2 \, a^{4} c^{2} x^{2} + a^{2} c^{2}\right )} c} - \frac{\arctan \left (a x\right )^{2}}{4 \,{\left (a^{2} c x^{2} + c\right )}^{2} a^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.12659, size = 192, normalized size = 1.39 \begin{align*} \frac{3 \, a^{2} x^{2} +{\left (3 \, a^{4} x^{4} + 6 \, a^{2} x^{2} - 5\right )} \arctan \left (a x\right )^{2} + 2 \,{\left (3 \, a^{3} x^{3} + 5 \, a x\right )} \arctan \left (a x\right ) + 4}{32 \,{\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\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*} \frac{\int \frac{x \operatorname{atan}^{2}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \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 \frac{x \arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]