Optimal. Leaf size=250 \[ -\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 a^3 \sqrt{a^2 c x^2+c}}+\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 a^3 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c}}{2 a^3 c}+\frac{i \sqrt{a^2 x^2+1} \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right ) \tan ^{-1}(a x)}{a^3 \sqrt{a^2 c x^2+c}}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{2 a^2 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.146525, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4952, 261, 4890, 4886} \[ -\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 a^3 \sqrt{a^2 c x^2+c}}+\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 a^3 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c}}{2 a^3 c}+\frac{i \sqrt{a^2 x^2+1} \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right ) \tan ^{-1}(a x)}{a^3 \sqrt{a^2 c x^2+c}}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{2 a^2 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4952
Rule 261
Rule 4890
Rule 4886
Rubi steps
\begin{align*} \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx &=\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{2 a^2 c}-\frac{\int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{2 a^2}-\frac{\int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{2 a}\\ &=-\frac{\sqrt{c+a^2 c x^2}}{2 a^3 c}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{2 a^2 c}-\frac{\sqrt{1+a^2 x^2} \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{2 a^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{\sqrt{c+a^2 c x^2}}{2 a^3 c}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{2 a^2 c}+\frac{i \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{i \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 a^3 \sqrt{c+a^2 c x^2}}+\frac{i \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 a^3 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.5769, size = 158, normalized size = 0.63 \[ -\frac{\sqrt{c \left (a^2 x^2+1\right )} \left (i \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )-i \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )+\sqrt{a^2 x^2+1}-a x \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+\tan ^{-1}(a x) \log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\tan ^{-1}(a x) \log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right )}{2 a^3 c \sqrt{a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.895, size = 184, normalized size = 0.7 \begin{align*}{\frac{\arctan \left ( ax \right ) xa-1}{2\,c{a}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{{\frac{i}{2}}}{c{a}^{3}} \left ( i\arctan \left ( ax \right ) \ln \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -i\arctan \left ( ax \right ) \ln \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +{\it dilog} \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -{\it dilog} \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) \right ) \sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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 (\frac{x^{2} \arctan \left (a x\right )}{\sqrt{a^{2} c x^{2} + c}}, 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*} \int \frac{x^{2} \operatorname{atan}{\left (a x \right )}}{\sqrt{c \left (a^{2} x^{2} + 1\right )}}\, dx \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^{2} \arctan \left (a x\right )}{\sqrt{a^{2} c x^{2} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]