Optimal. Leaf size=131 \[ \frac{3 c^2 d^2 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right ) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{20 b \sqrt{\sin (2 a+2 b x)}}-\frac{c (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{7/2}}{5 b d}+\frac{c d (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{3/2}}{10 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.175756, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2568, 2569, 2572, 2639} \[ \frac{3 c^2 d^2 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right ) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{20 b \sqrt{\sin (2 a+2 b x)}}-\frac{c (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{7/2}}{5 b d}+\frac{c d (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{3/2}}{10 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2568
Rule 2569
Rule 2572
Rule 2639
Rubi steps
\begin{align*} \int (d \cos (a+b x))^{5/2} (c \sin (a+b x))^{5/2} \, dx &=-\frac{c (d \cos (a+b x))^{7/2} (c \sin (a+b x))^{3/2}}{5 b d}+\frac{1}{10} \left (3 c^2\right ) \int (d \cos (a+b x))^{5/2} \sqrt{c \sin (a+b x)} \, dx\\ &=\frac{c d (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2}}{10 b}-\frac{c (d \cos (a+b x))^{7/2} (c \sin (a+b x))^{3/2}}{5 b d}+\frac{1}{20} \left (3 c^2 d^2\right ) \int \sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)} \, dx\\ &=\frac{c d (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2}}{10 b}-\frac{c (d \cos (a+b x))^{7/2} (c \sin (a+b x))^{3/2}}{5 b d}+\frac{\left (3 c^2 d^2 \sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)}\right ) \int \sqrt{\sin (2 a+2 b x)} \, dx}{20 \sqrt{\sin (2 a+2 b x)}}\\ &=\frac{c d (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2}}{10 b}-\frac{c (d \cos (a+b x))^{7/2} (c \sin (a+b x))^{3/2}}{5 b d}+\frac{3 c^2 d^2 \sqrt{d \cos (a+b x)} E\left (\left .a-\frac{\pi }{4}+b x\right |2\right ) \sqrt{c \sin (a+b x)}}{20 b \sqrt{\sin (2 a+2 b x)}}\\ \end{align*}
Mathematica [C] time = 0.177644, size = 70, normalized size = 0.53 \[ \frac{2 d^2 \sqrt [4]{\cos ^2(a+b x)} \tan (a+b x) (c \sin (a+b x))^{5/2} \sqrt{d \cos (a+b x)} \, _2F_1\left (-\frac{3}{4},\frac{7}{4};\frac{11}{4};\sin ^2(a+b x)\right )}{7 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.08, size = 532, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \cos \left (b x + a\right )\right )^{\frac{5}{2}} \left (c \sin \left (b x + a\right )\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (c^{2} d^{2} \cos \left (b x + a\right )^{4} - c^{2} d^{2} \cos \left (b x + a\right )^{2}\right )} \sqrt{d \cos \left (b x + a\right )} \sqrt{c \sin \left (b x + a\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \cos \left (b x + a\right )\right )^{\frac{5}{2}} \left (c \sin \left (b x + a\right )\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]