Optimal. Leaf size=166 \[ -\frac {a^2}{3 c x}-\frac {a^3 \text {ArcTan}(a x)}{3 c}-\frac {a \text {ArcTan}(a x)}{3 c x^2}+\frac {4 i a^3 \text {ArcTan}(a x)^2}{3 c}-\frac {\text {ArcTan}(a x)^2}{3 c x^3}+\frac {a^2 \text {ArcTan}(a x)^2}{c x}+\frac {a^3 \text {ArcTan}(a x)^3}{3 c}-\frac {8 a^3 \text {ArcTan}(a x) \log \left (2-\frac {2}{1-i a x}\right )}{3 c}+\frac {4 i a^3 \text {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{3 c} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.32, 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 = {5038, 4946,
331, 209, 5044, 4988, 2497, 5004} \begin {gather*} \frac {a^3 \text {ArcTan}(a x)^3}{3 c}+\frac {4 i a^3 \text {ArcTan}(a x)^2}{3 c}-\frac {a^3 \text {ArcTan}(a x)}{3 c}-\frac {8 a^3 \text {ArcTan}(a x) \log \left (2-\frac {2}{1-i a x}\right )}{3 c}+\frac {4 i a^3 \text {Li}_2\left (\frac {2}{1-i a x}-1\right )}{3 c}+\frac {a^2 \text {ArcTan}(a x)^2}{c x}-\frac {a^2}{3 c x}-\frac {\text {ArcTan}(a x)^2}{3 c x^3}-\frac {a \text {ArcTan}(a x)}{3 c x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 209
Rule 331
Rule 2497
Rule 4946
Rule 4988
Rule 5004
Rule 5038
Rule 5044
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 = 0.28, size = 120, normalized size = 0.72 \begin {gather*} \frac {a^3 \left (-\frac {1-4 \text {ArcTan}(a x)^2+\frac {\left (1+a^2 x^2\right ) \text {ArcTan}(a x)^2}{a^2 x^2}}{a x}+\text {ArcTan}(a x) \left (-\frac {1+a^2 x^2}{a^2 x^2}+\text {ArcTan}(a x) (4 i+\text {ArcTan}(a x))-8 \log \left (1-e^{2 i \text {ArcTan}(a x)}\right )\right )+4 i \text {PolyLog}\left (2,e^{2 i \text {ArcTan}(a x)}\right )\right )}{3 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 294 vs. \(2 (146 ) = 292\).
time = 0.42, size = 295, normalized size = 1.78
method | result | size |
derivativedivides | \(a^{3} \left (\frac {\arctan \left (a x \right )^{3}}{c}-\frac {\arctan \left (a x \right )^{2}}{3 c \,a^{3} x^{3}}+\frac {\arctan \left (a x \right )^{2}}{c a x}-\frac {2 \left (-2 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+\frac {\arctan \left (a x \right )}{2 a^{2} x^{2}}+4 \arctan \left (a x \right ) \ln \left (a x \right )+\frac {1}{2 a x}+\frac {\arctan \left (a x \right )}{2}-i \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-2 i \dilog \left (-i a x +1\right )+\frac {i \ln \left (a x -i\right )^{2}}{2}-i \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )+i \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )+2 i \ln \left (a x \right ) \ln \left (i a x +1\right )+i \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )+i \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-i \dilog \left (\frac {i \left (a x -i\right )}{2}\right )-2 i \ln \left (a x \right ) \ln \left (-i a x +1\right )-\frac {i \ln \left (a x +i\right )^{2}}{2}+2 i \dilog \left (i a x +1\right )+\arctan \left (a x \right )^{3}\right )}{3 c}\right )\) | \(295\) |
default | \(a^{3} \left (\frac {\arctan \left (a x \right )^{3}}{c}-\frac {\arctan \left (a x \right )^{2}}{3 c \,a^{3} x^{3}}+\frac {\arctan \left (a x \right )^{2}}{c a x}-\frac {2 \left (-2 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+\frac {\arctan \left (a x \right )}{2 a^{2} x^{2}}+4 \arctan \left (a x \right ) \ln \left (a x \right )+\frac {1}{2 a x}+\frac {\arctan \left (a x \right )}{2}-i \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-2 i \dilog \left (-i a x +1\right )+\frac {i \ln \left (a x -i\right )^{2}}{2}-i \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )+i \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )+2 i \ln \left (a x \right ) \ln \left (i a x +1\right )+i \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )+i \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-i \dilog \left (\frac {i \left (a x -i\right )}{2}\right )-2 i \ln \left (a x \right ) \ln \left (-i a x +1\right )-\frac {i \ln \left (a x +i\right )^{2}}{2}+2 i \dilog \left (i a x +1\right )+\arctan \left (a x \right )^{3}\right )}{3 c}\right )\) | \(295\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{a^{2} x^{6} + x^{4}}\, dx}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {atan}\left (a\,x\right )}^2}{x^4\,\left (c\,a^2\,x^2+c\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________