Optimal. Leaf size=16 \[ \frac {1}{4 b \left (a+\frac {b}{x^2}\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267}
\begin {gather*} \frac {1}{4 b \left (a+\frac {b}{x^2}\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 267
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x^3} \, dx &=\frac {1}{4 b \left (a+\frac {b}{x^2}\right )^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 24, normalized size = 1.50 \begin {gather*} -\frac {b+2 a x^2}{4 a^2 \left (b+a x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(30\) vs.
\(2(14)=28\).
time = 0.03, size = 31, normalized size = 1.94
method | result | size |
derivativedivides | \(\frac {1}{4 b \left (\frac {b}{x^{2}}+a \right )^{2}}\) | \(15\) |
gosper | \(-\frac {2 a \,x^{2}+b}{4 \left (a \,x^{2}+b \right )^{2} a^{2}}\) | \(23\) |
risch | \(\frac {-\frac {x^{2}}{2 a}-\frac {b}{4 a^{2}}}{\left (a \,x^{2}+b \right )^{2}}\) | \(26\) |
default | \(\frac {b}{4 a^{2} \left (a \,x^{2}+b \right )^{2}}-\frac {1}{2 a^{2} \left (a \,x^{2}+b \right )}\) | \(31\) |
norman | \(\frac {-\frac {x^{4}}{2 a}-\frac {b \,x^{2}}{4 a^{2}}}{x^{2} \left (a \,x^{2}+b \right )^{2}}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 14, normalized size = 0.88 \begin {gather*} \frac {1}{4 \, {\left (a + \frac {b}{x^{2}}\right )}^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs.
\(2 (14) = 28\).
time = 0.36, size = 36, normalized size = 2.25 \begin {gather*} -\frac {2 \, a x^{2} + b}{4 \, {\left (a^{4} x^{4} + 2 \, a^{3} b x^{2} + a^{2} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs.
\(2 (12) = 24\).
time = 0.12, size = 36, normalized size = 2.25 \begin {gather*} \frac {- 2 a x^{2} - b}{4 a^{4} x^{4} + 8 a^{3} b x^{2} + 4 a^{2} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.82, size = 22, normalized size = 1.38 \begin {gather*} -\frac {2 \, a x^{2} + b}{4 \, {\left (a x^{2} + b\right )}^{2} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.08, size = 37, normalized size = 2.31 \begin {gather*} -\frac {\frac {b}{4\,a^2}+\frac {x^2}{2\,a}}{a^2\,x^4+2\,a\,b\,x^2+b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________