Optimal. Leaf size=106 \[ -\frac{8 c \sqrt{c \sin (a+b x)}}{45 b d^5 \sqrt{d \cos (a+b x)}}-\frac{2 c \sqrt{c \sin (a+b x)}}{45 b d^3 (d \cos (a+b x))^{5/2}}+\frac{2 c \sqrt{c \sin (a+b x)}}{9 b d (d \cos (a+b x))^{9/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.18419, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2566, 2571, 2563} \[ -\frac{8 c \sqrt{c \sin (a+b x)}}{45 b d^5 \sqrt{d \cos (a+b x)}}-\frac{2 c \sqrt{c \sin (a+b x)}}{45 b d^3 (d \cos (a+b x))^{5/2}}+\frac{2 c \sqrt{c \sin (a+b x)}}{9 b d (d \cos (a+b x))^{9/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2566
Rule 2571
Rule 2563
Rubi steps
\begin{align*} \int \frac{(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{11/2}} \, dx &=\frac{2 c \sqrt{c \sin (a+b x)}}{9 b d (d \cos (a+b x))^{9/2}}-\frac{c^2 \int \frac{1}{(d \cos (a+b x))^{7/2} \sqrt{c \sin (a+b x)}} \, dx}{9 d^2}\\ &=\frac{2 c \sqrt{c \sin (a+b x)}}{9 b d (d \cos (a+b x))^{9/2}}-\frac{2 c \sqrt{c \sin (a+b x)}}{45 b d^3 (d \cos (a+b x))^{5/2}}-\frac{\left (4 c^2\right ) \int \frac{1}{(d \cos (a+b x))^{3/2} \sqrt{c \sin (a+b x)}} \, dx}{45 d^4}\\ &=\frac{2 c \sqrt{c \sin (a+b x)}}{9 b d (d \cos (a+b x))^{9/2}}-\frac{2 c \sqrt{c \sin (a+b x)}}{45 b d^3 (d \cos (a+b x))^{5/2}}-\frac{8 c \sqrt{c \sin (a+b x)}}{45 b d^5 \sqrt{d \cos (a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.298347, size = 57, normalized size = 0.54 \[ \frac{2 (2 \cos (2 (a+b x))+7) \sec ^5(a+b x) (c \sin (a+b x))^{5/2} \sqrt{d \cos (a+b x)}}{45 b c d^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.072, size = 50, normalized size = 0.5 \begin{align*}{\frac{ \left ( 8\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}+10 \right ) \cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }{45\,b} \left ( c\sin \left ( bx+a \right ) \right ) ^{{\frac{3}{2}}} \left ( d\cos \left ( bx+a \right ) \right ) ^{-{\frac{11}{2}}}} \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 \frac{\left (c \sin \left (b x + a\right )\right )^{\frac{3}{2}}}{\left (d \cos \left (b x + a\right )\right )^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 4.18239, size = 159, normalized size = 1.5 \begin{align*} -\frac{2 \,{\left (4 \, c \cos \left (b x + a\right )^{4} + c \cos \left (b x + a\right )^{2} - 5 \, c\right )} \sqrt{d \cos \left (b x + a\right )} \sqrt{c \sin \left (b x + a\right )}}{45 \, b d^{6} \cos \left (b x + a\right )^{5}} \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(-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]