Optimal. Leaf size=97 \[ -\frac {a}{4 c^2 \left (a^2 x^2+1\right )}-\frac {a \log \left (a^2 x^2+1\right )}{2 c^2}-\frac {a^2 x \tan ^{-1}(a x)}{2 c^2 \left (a^2 x^2+1\right )}+\frac {a \log (x)}{c^2}-\frac {3 a \tan ^{-1}(a x)^2}{4 c^2}-\frac {\tan ^{-1}(a x)}{c^2 x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4966, 4918, 4852, 266, 36, 29, 31, 4884, 4892, 261} \[ -\frac {a}{4 c^2 \left (a^2 x^2+1\right )}-\frac {a \log \left (a^2 x^2+1\right )}{2 c^2}-\frac {a^2 x \tan ^{-1}(a x)}{2 c^2 \left (a^2 x^2+1\right )}+\frac {a \log (x)}{c^2}-\frac {3 a \tan ^{-1}(a x)^2}{4 c^2}-\frac {\tan ^{-1}(a x)}{c^2 x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 261
Rule 266
Rule 4852
Rule 4884
Rule 4892
Rule 4918
Rule 4966
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)}{x^2 \left (c+a^2 c x^2\right )^2} \, dx &=-\left (a^2 \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx\right )+\frac {\int \frac {\tan ^{-1}(a x)}{x^2 \left (c+a^2 c x^2\right )} \, dx}{c}\\ &=-\frac {a^2 x \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac {a \tan ^{-1}(a x)^2}{4 c^2}+\frac {1}{2} a^3 \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx+\frac {\int \frac {\tan ^{-1}(a x)}{x^2} \, dx}{c^2}-\frac {a^2 \int \frac {\tan ^{-1}(a x)}{c+a^2 c x^2} \, dx}{c}\\ &=-\frac {a}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\tan ^{-1}(a x)}{c^2 x}-\frac {a^2 x \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac {3 a \tan ^{-1}(a x)^2}{4 c^2}+\frac {a \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx}{c^2}\\ &=-\frac {a}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\tan ^{-1}(a x)}{c^2 x}-\frac {a^2 x \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac {3 a \tan ^{-1}(a x)^2}{4 c^2}+\frac {a \operatorname {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )}{2 c^2}\\ &=-\frac {a}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\tan ^{-1}(a x)}{c^2 x}-\frac {a^2 x \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac {3 a \tan ^{-1}(a x)^2}{4 c^2}+\frac {a \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 c^2}-\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right )}{2 c^2}\\ &=-\frac {a}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\tan ^{-1}(a x)}{c^2 x}-\frac {a^2 x \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac {3 a \tan ^{-1}(a x)^2}{4 c^2}+\frac {a \log (x)}{c^2}-\frac {a \log \left (1+a^2 x^2\right )}{2 c^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 94, normalized size = 0.97 \[ -\frac {a}{4 c^2 \left (a^2 x^2+1\right )}-\frac {a \log \left (a^2 x^2+1\right )}{2 c^2}-\frac {\left (3 a^2 x^2+2\right ) \tan ^{-1}(a x)}{2 c^2 x \left (a^2 x^2+1\right )}+\frac {a \log (x)}{c^2}-\frac {3 a \tan ^{-1}(a x)^2}{4 c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.49, size = 97, normalized size = 1.00 \[ -\frac {3 \, {\left (a^{3} x^{3} + a x\right )} \arctan \left (a x\right )^{2} + a x + 2 \, {\left (3 \, a^{2} x^{2} + 2\right )} \arctan \left (a x\right ) + 2 \, {\left (a^{3} x^{3} + a x\right )} \log \left (a^{2} x^{2} + 1\right ) - 4 \, {\left (a^{3} x^{3} + a x\right )} \log \relax (x)}{4 \, {\left (a^{2} c^{2} x^{3} + c^{2} x\right )}} \]
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 [A] time = 0.05, size = 92, normalized size = 0.95 \[ -\frac {\arctan \left (a x \right )}{c^{2} x}-\frac {a^{2} x \arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}-\frac {3 a \arctan \left (a x \right )^{2}}{4 c^{2}}+\frac {a \ln \left (a x \right )}{c^{2}}-\frac {a \ln \left (a^{2} x^{2}+1\right )}{2 c^{2}}-\frac {a}{4 c^{2} \left (a^{2} x^{2}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.44, size = 119, normalized size = 1.23 \[ -\frac {1}{2} \, {\left (\frac {3 \, a^{2} x^{2} + 2}{a^{2} c^{2} x^{3} + c^{2} x} + \frac {3 \, a \arctan \left (a x\right )}{c^{2}}\right )} \arctan \left (a x\right ) + \frac {{\left (3 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} - 2 \, {\left (a^{2} x^{2} + 1\right )} \log \left (a^{2} x^{2} + 1\right ) + 4 \, {\left (a^{2} x^{2} + 1\right )} \log \relax (x) - 1\right )} a}{4 \, {\left (a^{2} c^{2} x^{2} + c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.48, size = 91, normalized size = 0.94 \[ \frac {a\,\ln \relax (x)}{c^2}-\frac {a\,\ln \left (a^2\,x^2+1\right )}{2\,c^2}-\frac {\mathrm {atan}\left (a\,x\right )\,\left (\frac {1}{a^2\,c^2}+\frac {3\,x^2}{2\,c^2}\right )}{\frac {x}{a^2}+x^3}-\frac {a}{2\,\left (2\,a^2\,c^2\,x^2+2\,c^2\right )}-\frac {3\,a\,{\mathrm {atan}\left (a\,x\right )}^2}{4\,c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 1.42, size = 272, normalized size = 2.80 \[ \frac {4 a^{3} x^{3} \log {\relax (x )}}{4 a^{2} c^{2} x^{3} + 4 c^{2} x} - \frac {2 a^{3} x^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{4 a^{2} c^{2} x^{3} + 4 c^{2} x} - \frac {3 a^{3} x^{3} \operatorname {atan}^{2}{\left (a x \right )}}{4 a^{2} c^{2} x^{3} + 4 c^{2} x} - \frac {6 a^{2} x^{2} \operatorname {atan}{\left (a x \right )}}{4 a^{2} c^{2} x^{3} + 4 c^{2} x} + \frac {4 a x \log {\relax (x )}}{4 a^{2} c^{2} x^{3} + 4 c^{2} x} - \frac {2 a x \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{4 a^{2} c^{2} x^{3} + 4 c^{2} x} - \frac {3 a x \operatorname {atan}^{2}{\left (a x \right )}}{4 a^{2} c^{2} x^{3} + 4 c^{2} x} - \frac {a x}{4 a^{2} c^{2} x^{3} + 4 c^{2} x} - \frac {4 \operatorname {atan}{\left (a x \right )}}{4 a^{2} c^{2} x^{3} + 4 c^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________