Optimal. Leaf size=78 \[ \frac{2}{a^2 c \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)^2}{a^2 c \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)}{a c \sqrt{a^2 c x^2+c}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.110577, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {4930, 4894} \[ \frac{2}{a^2 c \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)^2}{a^2 c \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)}{a c \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4930
Rule 4894
Rubi steps
\begin{align*} \int \frac{x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\frac{\tan ^{-1}(a x)^2}{a^2 c \sqrt{c+a^2 c x^2}}+\frac{2 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a}\\ &=\frac{2}{a^2 c \sqrt{c+a^2 c x^2}}+\frac{2 x \tan ^{-1}(a x)}{a c \sqrt{c+a^2 c x^2}}-\frac{\tan ^{-1}(a x)^2}{a^2 c \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0748785, size = 50, normalized size = 0.64 \[ \frac{\sqrt{a^2 c x^2+c} \left (-\tan ^{-1}(a x)^2+2 a x \tan ^{-1}(a x)+2\right )}{a^2 c^2 \left (a^2 x^2+1\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.265, size = 116, normalized size = 1.5 \begin{align*} -{\frac{ \left ( \left ( \arctan \left ( ax \right ) \right ) ^{2}-2+2\,i\arctan \left ( ax \right ) \right ) \left ( 1+iax \right ) }{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( -1+iax \right ) \left ( \left ( \arctan \left ( ax \right ) \right ) ^{2}-2-2\,i\arctan \left ( ax \right ) \right ) }{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 2.53735, size = 99, normalized size = 1.27 \begin{align*} \sqrt{c}{\left (\frac{2 \, x \arctan \left (a x\right )}{\sqrt{a^{2} x^{2} + 1} a c^{2}} - \frac{\arctan \left (a x\right )^{2}}{\sqrt{a^{2} x^{2} + 1} a^{2} c^{2}} + \frac{2}{\sqrt{a^{2} x^{2} + 1} a^{2} c^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.24516, size = 115, normalized size = 1.47 \begin{align*} \frac{\sqrt{a^{2} c x^{2} + c}{\left (2 \, a x \arctan \left (a x\right ) - \arctan \left (a x\right )^{2} + 2\right )}}{a^{4} c^{2} x^{2} + a^{2} c^{2}} \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*} \int \frac{x \operatorname{atan}^{2}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.22588, size = 97, normalized size = 1.24 \begin{align*} \frac{2 \, x \arctan \left (a x\right )}{\sqrt{a^{2} c x^{2} + c} a c} - \frac{\arctan \left (a x\right )^{2}}{\sqrt{a^{2} c x^{2} + c} a^{2} c} + \frac{2}{\sqrt{a^{2} c x^{2} + c} a^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]