Optimal. Leaf size=37 \[ -\frac {3 x}{8 \left (x^2+9\right )}-\frac {x^3}{4 \left (x^2+9\right )^2}+\frac {1}{8} \tan ^{-1}\left (\frac {x}{3}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {288, 203} \[ -\frac {x^3}{4 \left (x^2+9\right )^2}-\frac {3 x}{8 \left (x^2+9\right )}+\frac {1}{8} \tan ^{-1}\left (\frac {x}{3}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 288
Rubi steps
\begin {align*} \int \frac {x^4}{\left (9+x^2\right )^3} \, dx &=-\frac {x^3}{4 \left (9+x^2\right )^2}+\frac {3}{4} \int \frac {x^2}{\left (9+x^2\right )^2} \, dx\\ &=-\frac {x^3}{4 \left (9+x^2\right )^2}-\frac {3 x}{8 \left (9+x^2\right )}+\frac {3}{8} \int \frac {1}{9+x^2} \, dx\\ &=-\frac {x^3}{4 \left (9+x^2\right )^2}-\frac {3 x}{8 \left (9+x^2\right )}+\frac {1}{8} \tan ^{-1}\left (\frac {x}{3}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 28, normalized size = 0.76 \[ \frac {1}{8} \left (\tan ^{-1}\left (\frac {x}{3}\right )-\frac {x \left (5 x^2+27\right )}{\left (x^2+9\right )^2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.39, size = 39, normalized size = 1.05 \[ -\frac {5 \, x^{3} - {\left (x^{4} + 18 \, x^{2} + 81\right )} \arctan \left (\frac {1}{3} \, x\right ) + 27 \, x}{8 \, {\left (x^{4} + 18 \, x^{2} + 81\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.98, size = 25, normalized size = 0.68 \[ -\frac {5 \, x^{3} + 27 \, x}{8 \, {\left (x^{2} + 9\right )}^{2}} + \frac {1}{8} \, \arctan \left (\frac {1}{3} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 25, normalized size = 0.68 \[ \frac {\arctan \left (\frac {x}{3}\right )}{8}+\frac {-\frac {5}{8} x^{3}-\frac {27}{8} x}{\left (x^{2}+9\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.14, size = 30, normalized size = 0.81 \[ -\frac {5 \, x^{3} + 27 \, x}{8 \, {\left (x^{4} + 18 \, x^{2} + 81\right )}} + \frac {1}{8} \, \arctan \left (\frac {1}{3} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.16, size = 30, normalized size = 0.81 \[ \frac {\mathrm {atan}\left (\frac {x}{3}\right )}{8}-\frac {\frac {5\,x^3}{8}+\frac {27\,x}{8}}{x^4+18\,x^2+81} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.14, size = 27, normalized size = 0.73 \[ \frac {- 5 x^{3} - 27 x}{8 x^{4} + 144 x^{2} + 648} + \frac {\operatorname {atan}{\left (\frac {x}{3} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________