Optimal. Leaf size=18 \[ \frac {\sec ^3(c+d x)}{3 a^2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {3175, 2606, 30} \[ \frac {\sec ^3(c+d x)}{3 a^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2606
Rule 3175
Rubi steps
\begin {align*} \int \frac {\sin (c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx &=\frac {\int \sec ^3(c+d x) \tan (c+d x) \, dx}{a^2}\\ &=\frac {\operatorname {Subst}\left (\int x^2 \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=\frac {\sec ^3(c+d x)}{3 a^2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 18, normalized size = 1.00 \[ \frac {\sec ^3(c+d x)}{3 a^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 16, normalized size = 0.89 \[ \frac {1}{3 \, a^{2} d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.14, size = 16, normalized size = 0.89 \[ \frac {1}{3 \, a^{2} d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.15, size = 17, normalized size = 0.94 \[ \frac {1}{3 d \,a^{2} \cos \left (d x +c \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 16, normalized size = 0.89 \[ \frac {1}{3 \, a^{2} d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 13.59, size = 16, normalized size = 0.89 \[ \frac {1}{3\,a^2\,d\,{\cos \left (c+d\,x\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 9.72, size = 156, normalized size = 8.67 \[ \begin {cases} - \frac {6 \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a^{2} d} - \frac {2}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a^{2} d} & \text {for}\: d \neq 0 \\\frac {x \sin {\relax (c )}}{\left (- a \sin ^{2}{\relax (c )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________