Optimal. Leaf size=6 \[ \frac {\sin (x)}{a^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3254, 2717}
\begin {gather*} \frac {\sin (x)}{a^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2717
Rule 3254
Rubi steps
\begin {align*} \int \frac {\cos ^5(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx &=\frac {\int \cos (x) \, dx}{a^2}\\ &=\frac {\sin (x)}{a^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 6, normalized size = 1.00 \begin {gather*} \frac {\sin (x)}{a^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.16, size = 7, normalized size = 1.17
method | result | size |
derivativedivides | \(\frac {\sin \left (x \right )}{a^{2}}\) | \(7\) |
default | \(\frac {\sin \left (x \right )}{a^{2}}\) | \(7\) |
risch | \(\frac {\sin \left (x \right )}{a^{2}}\) | \(7\) |
norman | \(\frac {-\frac {2 \tan \left (\frac {x}{2}\right )}{a}-\frac {2 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}+\frac {6 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}+\frac {6 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{a}-\frac {6 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{a}-\frac {6 \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{a}+\frac {2 \left (\tan ^{13}\left (\frac {x}{2}\right )\right )}{a}+\frac {2 \left (\tan ^{15}\left (\frac {x}{2}\right )\right )}{a}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{5} a \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )^{3}}\) | \(112\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 6, normalized size = 1.00 \begin {gather*} \frac {\sin \left (x\right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 6, normalized size = 1.00 \begin {gather*} \frac {\sin \left (x\right )}{a^{2}} \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. 19 vs.
\(2 (5) = 10\).
time = 6.60, size = 19, normalized size = 3.17 \begin {gather*} \frac {2 \tan {\left (\frac {x}{2} \right )}}{a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.43, size = 6, normalized size = 1.00 \begin {gather*} \frac {\sin \left (x\right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.02, size = 6, normalized size = 1.00 \begin {gather*} \frac {\sin \left (x\right )}{a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________