Optimal. Leaf size=37 \[ \frac {2 (c \sin (a+b x))^{7/2}}{7 b c d (d \cos (a+b x))^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2563} \[ \frac {2 (c \sin (a+b x))^{7/2}}{7 b c d (d \cos (a+b x))^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2563
Rubi steps
\begin {align*} \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{9/2}} \, dx &=\frac {2 (c \sin (a+b x))^{7/2}}{7 b c d (d \cos (a+b x))^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 40, normalized size = 1.08 \[ \frac {2 \cot (a+b x) (c \sin (a+b x))^{9/2}}{7 b c^2 (d \cos (a+b x))^{9/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.54, size = 60, normalized size = 1.62 \[ -\frac {2 \, {\left (c^{2} \cos \left (b x + a\right )^{2} - c^{2}\right )} \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )} \sin \left (b x + a\right )}{7 \, b d^{5} \cos \left (b x + a\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.10, size = 38, normalized size = 1.03 \[ \frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right ) \left (c \sin \left (b x +a \right )\right )^{\frac {5}{2}}}{7 b \left (d \cos \left (b x +a \right )\right )^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \sin \left (b x + a\right )\right )^{\frac {5}{2}}}{\left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.78, size = 89, normalized size = 2.41 \[ \frac {2\,c^2\,\sqrt {c\,\sin \left (a+b\,x\right )}\,\left (3\,\sin \left (2\,a+2\,b\,x\right )-\sin \left (6\,a+6\,b\,x\right )\right )}{7\,b\,d^4\,\sqrt {d\,\cos \left (a+b\,x\right )}\,\left (15\,\cos \left (2\,a+2\,b\,x\right )+6\,\cos \left (4\,a+4\,b\,x\right )+\cos \left (6\,a+6\,b\,x\right )+10\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________