Optimal. Leaf size=311 \[ -\frac{5 i a^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}}+\frac{5 i a^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}}-\frac{a^2 \sqrt{a^2 c x^2+c}}{3 c x}+\frac{2 a^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 c x}-\frac{a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 c x^2}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 c x^3}+\frac{10 a^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.627991, antiderivative size = 311, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4962, 264, 4958, 4954, 4944} \[ -\frac{5 i a^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}}+\frac{5 i a^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}}-\frac{a^2 \sqrt{a^2 c x^2+c}}{3 c x}+\frac{2 a^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 c x}-\frac{a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 c x^2}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 c x^3}+\frac{10 a^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4962
Rule 264
Rule 4958
Rule 4954
Rule 4944
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^2}{x^4 \sqrt{c+a^2 c x^2}} \, dx &=-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 c x^3}+\frac{1}{3} (2 a) \int \frac{\tan ^{-1}(a x)}{x^3 \sqrt{c+a^2 c x^2}} \, dx-\frac{1}{3} \left (2 a^2\right ) \int \frac{\tan ^{-1}(a x)^2}{x^2 \sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{a \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 c x^2}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 c x^3}+\frac{2 a^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 c x}+\frac{1}{3} a^2 \int \frac{1}{x^2 \sqrt{c+a^2 c x^2}} \, dx-\frac{1}{3} a^3 \int \frac{\tan ^{-1}(a x)}{x \sqrt{c+a^2 c x^2}} \, dx-\frac{1}{3} \left (4 a^3\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{a^2 \sqrt{c+a^2 c x^2}}{3 c x}-\frac{a \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 c x^2}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 c x^3}+\frac{2 a^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 c x}-\frac{\left (a^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{1+a^2 x^2}} \, dx}{3 \sqrt{c+a^2 c x^2}}-\frac{\left (4 a^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{1+a^2 x^2}} \, dx}{3 \sqrt{c+a^2 c x^2}}\\ &=-\frac{a^2 \sqrt{c+a^2 c x^2}}{3 c x}-\frac{a \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 c x^2}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 c x^3}+\frac{2 a^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 c x}+\frac{10 a^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{c+a^2 c x^2}}-\frac{5 i a^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{c+a^2 c x^2}}+\frac{5 i a^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 2.60985, size = 228, normalized size = 0.73 \[ \frac{a^3 \sqrt{a^2 c x^2+c} \left (\frac{\left (a^2 x^2+1\right )^{3/2} \left (\frac{20 i a^3 x^3 \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{3/2}}+\tan ^{-1}(a x) \left (-2 \sin \left (2 \tan ^{-1}(a x)\right )+\frac{5 \left (\log \left (1-e^{i \tan ^{-1}(a x)}\right )-\log \left (1+e^{i \tan ^{-1}(a x)}\right )\right ) \left (\sqrt{a^2 x^2+1} \sin \left (3 \tan ^{-1}(a x)\right )-3 a x\right )}{\sqrt{a^2 x^2+1}}\right )+\tan ^{-1}(a x)^2 \left (2-6 \cos \left (2 \tan ^{-1}(a x)\right )\right )+2 \left (\cos \left (2 \tan ^{-1}(a x)\right )-1\right )\right )}{a^3 x^3}-20 i \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )\right )}{12 c \sqrt{a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.798, size = 206, normalized size = 0.7 \begin{align*}{\frac{2\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{2}{a}^{2}-{a}^{2}{x}^{2}-\arctan \left ( ax \right ) xa- \left ( \arctan \left ( ax \right ) \right ) ^{2}}{3\,c{x}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{{\frac{5\,i}{3}}{a}^{3}}{c} \left ( i\arctan \left ( ax \right ) \ln \left ( 1-{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -i\arctan \left ( ax \right ) \ln \left ( 1+{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +{\it polylog} \left ( 2,{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -{\it polylog} \left ( 2,-{(1+iax){\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{\sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right )^{2}}{a^{2} c x^{6} + c x^{4}}, 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{\operatorname{atan}^{2}{\left (a x \right )}}{x^{4} \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{\arctan \left (a x\right )^{2}}{\sqrt{a^{2} c x^{2} + c} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]