Optimal. Leaf size=166 \[ \frac {4 i a^3 \text {Li}_2\left (\frac {2}{1-i a x}-1\right )}{3 c}+\frac {a^3 \tan ^{-1}(a x)^3}{3 c}+\frac {4 i a^3 \tan ^{-1}(a x)^2}{3 c}-\frac {a^3 \tan ^{-1}(a x)}{3 c}-\frac {8 a^3 \log \left (2-\frac {2}{1-i a x}\right ) \tan ^{-1}(a x)}{3 c}-\frac {a^2}{3 c x}+\frac {a^2 \tan ^{-1}(a x)^2}{c x}-\frac {\tan ^{-1}(a x)^2}{3 c x^3}-\frac {a \tan ^{-1}(a x)}{3 c x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.44, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4918, 4852, 325, 203, 4924, 4868, 2447, 4884} \[ \frac {4 i a^3 \text {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{3 c}-\frac {a^2}{3 c x}+\frac {a^3 \tan ^{-1}(a x)^3}{3 c}+\frac {4 i a^3 \tan ^{-1}(a x)^2}{3 c}-\frac {a^3 \tan ^{-1}(a x)}{3 c}+\frac {a^2 \tan ^{-1}(a x)^2}{c x}-\frac {8 a^3 \log \left (2-\frac {2}{1-i a x}\right ) \tan ^{-1}(a x)}{3 c}-\frac {a \tan ^{-1}(a x)}{3 c x^2}-\frac {\tan ^{-1}(a x)^2}{3 c x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 325
Rule 2447
Rule 4852
Rule 4868
Rule 4884
Rule 4918
Rule 4924
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)^2}{x^4 \left (c+a^2 c x^2\right )} \, dx &=-\left (a^2 \int \frac {\tan ^{-1}(a x)^2}{x^2 \left (c+a^2 c x^2\right )} \, dx\right )+\frac {\int \frac {\tan ^{-1}(a x)^2}{x^4} \, dx}{c}\\ &=-\frac {\tan ^{-1}(a x)^2}{3 c x^3}+a^4 \int \frac {\tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx+\frac {(2 a) \int \frac {\tan ^{-1}(a x)}{x^3 \left (1+a^2 x^2\right )} \, dx}{3 c}-\frac {a^2 \int \frac {\tan ^{-1}(a x)^2}{x^2} \, dx}{c}\\ &=-\frac {\tan ^{-1}(a x)^2}{3 c x^3}+\frac {a^2 \tan ^{-1}(a x)^2}{c x}+\frac {a^3 \tan ^{-1}(a x)^3}{3 c}+\frac {(2 a) \int \frac {\tan ^{-1}(a x)}{x^3} \, dx}{3 c}-\frac {\left (2 a^3\right ) \int \frac {\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx}{3 c}-\frac {\left (2 a^3\right ) \int \frac {\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx}{c}\\ &=-\frac {a \tan ^{-1}(a x)}{3 c x^2}+\frac {4 i a^3 \tan ^{-1}(a x)^2}{3 c}-\frac {\tan ^{-1}(a x)^2}{3 c x^3}+\frac {a^2 \tan ^{-1}(a x)^2}{c x}+\frac {a^3 \tan ^{-1}(a x)^3}{3 c}+\frac {a^2 \int \frac {1}{x^2 \left (1+a^2 x^2\right )} \, dx}{3 c}-\frac {\left (2 i a^3\right ) \int \frac {\tan ^{-1}(a x)}{x (i+a x)} \, dx}{3 c}-\frac {\left (2 i a^3\right ) \int \frac {\tan ^{-1}(a x)}{x (i+a x)} \, dx}{c}\\ &=-\frac {a^2}{3 c x}-\frac {a \tan ^{-1}(a x)}{3 c x^2}+\frac {4 i a^3 \tan ^{-1}(a x)^2}{3 c}-\frac {\tan ^{-1}(a x)^2}{3 c x^3}+\frac {a^2 \tan ^{-1}(a x)^2}{c x}+\frac {a^3 \tan ^{-1}(a x)^3}{3 c}-\frac {8 a^3 \tan ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )}{3 c}-\frac {a^4 \int \frac {1}{1+a^2 x^2} \, dx}{3 c}+\frac {\left (2 a^4\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{3 c}+\frac {\left (2 a^4\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c}\\ &=-\frac {a^2}{3 c x}-\frac {a^3 \tan ^{-1}(a x)}{3 c}-\frac {a \tan ^{-1}(a x)}{3 c x^2}+\frac {4 i a^3 \tan ^{-1}(a x)^2}{3 c}-\frac {\tan ^{-1}(a x)^2}{3 c x^3}+\frac {a^2 \tan ^{-1}(a x)^2}{c x}+\frac {a^3 \tan ^{-1}(a x)^3}{3 c}-\frac {8 a^3 \tan ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )}{3 c}+\frac {4 i a^3 \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{3 c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.38, size = 120, normalized size = 0.72 \[ \frac {a^3 \left (-\frac {\frac {\left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}{a^2 x^2}-4 \tan ^{-1}(a x)^2+1}{a x}+\tan ^{-1}(a x) \left (-\frac {a^2 x^2+1}{a^2 x^2}+\tan ^{-1}(a x) \left (\tan ^{-1}(a x)+4 i\right )-8 \log \left (1-e^{2 i \tan ^{-1}(a x)}\right )\right )+4 i \text {Li}_2\left (e^{2 i \tan ^{-1}(a x)}\right )\right )}{3 c} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arctan \left (a x\right )^{2}}{a^{2} c x^{6} + c x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.12, size = 374, normalized size = 2.25 \[ -\frac {\arctan \left (a x \right )^{2}}{3 c \,x^{3}}+\frac {a^{2} \arctan \left (a x \right )^{2}}{c x}+\frac {a^{3} \arctan \left (a x \right )^{3}}{3 c}-\frac {a \arctan \left (a x \right )}{3 c \,x^{2}}-\frac {8 a^{3} \arctan \left (a x \right ) \ln \left (a x \right )}{3 c}+\frac {4 a^{3} \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{3 c}-\frac {a^{2}}{3 c x}-\frac {a^{3} \arctan \left (a x \right )}{3 c}-\frac {i a^{3} \ln \left (a x -i\right )^{2}}{3 c}+\frac {4 i a^{3} \ln \left (a x \right ) \ln \left (-i a x +1\right )}{3 c}+\frac {4 i a^{3} \dilog \left (-i a x +1\right )}{3 c}-\frac {2 i a^{3} \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{3 c}+\frac {2 i a^{3} \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{3 c}-\frac {2 i a^{3} \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{3 c}+\frac {2 i a^{3} \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{3 c}+\frac {2 i a^{3} \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{3 c}-\frac {4 i a^{3} \ln \left (a x \right ) \ln \left (i a x +1\right )}{3 c}+\frac {i a^{3} \ln \left (a x +i\right )^{2}}{3 c}-\frac {2 i a^{3} \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{3 c}-\frac {4 i a^{3} \dilog \left (i a x +1\right )}{3 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^2}{x^4\,\left (c\,a^2\,x^2+c\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{a^{2} x^{6} + x^{4}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________