Optimal. Leaf size=80 \[ \frac {\tan ^{-1}(a x)^2}{2 a^5 c}-\frac {x \tan ^{-1}(a x)}{a^4 c}-\frac {x^2}{6 a^3 c}+\frac {x^3 \tan ^{-1}(a x)}{3 a^2 c}+\frac {2 \log \left (a^2 x^2+1\right )}{3 a^5 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {4916, 4852, 266, 43, 4846, 260, 4884} \[ -\frac {x^2}{6 a^3 c}+\frac {2 \log \left (a^2 x^2+1\right )}{3 a^5 c}+\frac {x^3 \tan ^{-1}(a x)}{3 a^2 c}-\frac {x \tan ^{-1}(a x)}{a^4 c}+\frac {\tan ^{-1}(a x)^2}{2 a^5 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 260
Rule 266
Rule 4846
Rule 4852
Rule 4884
Rule 4916
Rubi steps
\begin {align*} \int \frac {x^4 \tan ^{-1}(a x)}{c+a^2 c x^2} \, dx &=-\frac {\int \frac {x^2 \tan ^{-1}(a x)}{c+a^2 c x^2} \, dx}{a^2}+\frac {\int x^2 \tan ^{-1}(a x) \, dx}{a^2 c}\\ &=\frac {x^3 \tan ^{-1}(a x)}{3 a^2 c}+\frac {\int \frac {\tan ^{-1}(a x)}{c+a^2 c x^2} \, dx}{a^4}-\frac {\int \tan ^{-1}(a x) \, dx}{a^4 c}-\frac {\int \frac {x^3}{1+a^2 x^2} \, dx}{3 a c}\\ &=-\frac {x \tan ^{-1}(a x)}{a^4 c}+\frac {x^3 \tan ^{-1}(a x)}{3 a^2 c}+\frac {\tan ^{-1}(a x)^2}{2 a^5 c}+\frac {\int \frac {x}{1+a^2 x^2} \, dx}{a^3 c}-\frac {\operatorname {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )}{6 a c}\\ &=-\frac {x \tan ^{-1}(a x)}{a^4 c}+\frac {x^3 \tan ^{-1}(a x)}{3 a^2 c}+\frac {\tan ^{-1}(a x)^2}{2 a^5 c}+\frac {\log \left (1+a^2 x^2\right )}{2 a^5 c}-\frac {\operatorname {Subst}\left (\int \left (\frac {1}{a^2}-\frac {1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )}{6 a c}\\ &=-\frac {x^2}{6 a^3 c}-\frac {x \tan ^{-1}(a x)}{a^4 c}+\frac {x^3 \tan ^{-1}(a x)}{3 a^2 c}+\frac {\tan ^{-1}(a x)^2}{2 a^5 c}+\frac {2 \log \left (1+a^2 x^2\right )}{3 a^5 c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 56, normalized size = 0.70 \[ \frac {-a^2 x^2+4 \log \left (a^2 x^2+1\right )+2 a x \left (a^2 x^2-3\right ) \tan ^{-1}(a x)+3 \tan ^{-1}(a x)^2}{6 a^5 c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 54, normalized size = 0.68 \[ -\frac {a^{2} x^{2} - 2 \, {\left (a^{3} x^{3} - 3 \, a x\right )} \arctan \left (a x\right ) - 3 \, \arctan \left (a x\right )^{2} - 4 \, \log \left (a^{2} x^{2} + 1\right )}{6 \, a^{5} c} \]
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 = 0.04, size = 73, normalized size = 0.91 \[ -\frac {x^{2}}{6 a^{3} c}-\frac {x \arctan \left (a x \right )}{a^{4} c}+\frac {x^{3} \arctan \left (a x \right )}{3 a^{2} c}+\frac {\arctan \left (a x \right )^{2}}{2 a^{5} c}+\frac {2 \ln \left (a^{2} x^{2}+1\right )}{3 a^{5} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.43, size = 74, normalized size = 0.92 \[ \frac {1}{3} \, {\left (\frac {a^{2} x^{3} - 3 \, x}{a^{4} c} + \frac {3 \, \arctan \left (a x\right )}{a^{5} c}\right )} \arctan \left (a x\right ) - \frac {a^{2} x^{2} + 3 \, \arctan \left (a x\right )^{2} - 4 \, \log \left (a^{2} x^{2} + 1\right )}{6 \, a^{5} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.17, size = 73, normalized size = 0.91 \[ \frac {2\,\ln \left (a^2\,x^2+1\right )}{3\,a^5\,c}-a^2\,\mathrm {atan}\left (a\,x\right )\,\left (\frac {x}{a^6\,c}-\frac {x^3}{3\,a^4\,c}\right )-\frac {x^2}{6\,a^3\,c}+\frac {{\mathrm {atan}\left (a\,x\right )}^2}{2\,a^5\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.83, size = 110, normalized size = 1.38 \[ \begin {cases} \frac {x^{3} \operatorname {atan}{\left (a x \right )}}{3 a^{2} c} - \frac {x^{2}}{6 a^{3} c} - \frac {x \operatorname {atan}{\left (a x \right )}}{a^{4} c} + \frac {2 \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{3 a^{5} c} + \frac {\operatorname {atan}^{2}{\left (a x \right )}}{2 a^{5} c} & \text {for}\: c \neq 0 \\\tilde {\infty } \left (\frac {x^{5} \operatorname {atan}{\left (a x \right )}}{5} - \frac {x^{4}}{20 a} + \frac {x^{2}}{10 a^{3}} - \frac {\log {\left (a^{2} x^{2} + 1 \right )}}{10 a^{5}}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________