Optimal. Leaf size=134 \[ -\frac{a^2 A \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{a A \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}{3 f}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{3/2}}{4 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.348195, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.075, Rules used = {2973, 2740, 2738} \[ -\frac{a^2 A \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{a A \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}{3 f}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{3/2}}{4 f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2973
Rule 2740
Rule 2738
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx &=-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{4 f}+A \int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2} \, dx\\ &=-\frac{a A \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}{3 f}-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{4 f}+\frac{1}{3} (2 a A) \int \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx\\ &=-\frac{a^2 A \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{3 f \sqrt{a+a \sin (e+f x)}}-\frac{a A \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}{3 f}-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{4 f}\\ \end{align*}
Mathematica [A] time = 0.780104, size = 96, normalized size = 0.72 \[ -\frac{c (\sin (e+f x)-1) \sec ^3(e+f x) (a (\sin (e+f x)+1))^{3/2} \sqrt{c-c \sin (e+f x)} (8 A (9 \sin (e+f x)+\sin (3 (e+f x)))-12 B \cos (2 (e+f x))-3 B \cos (4 (e+f x)))}{96 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.27, size = 86, normalized size = 0.6 \begin{align*}{\frac{ \left ( 3\,B \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) +4\,A \left ( \cos \left ( fx+e \right ) \right ) ^{2}+3\,B\sin \left ( fx+e \right ) +8\,A \right ) \sin \left ( fx+e \right ) }{12\,f \left ( \cos \left ( fx+e \right ) \right ) ^{3}} \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{3}{2}}} \left ( a \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{3}{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{\left (B \sin \left (f x + e\right ) + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.73974, size = 216, normalized size = 1.61 \begin{align*} -\frac{{\left (3 \, B a c \cos \left (f x + e\right )^{4} - 3 \, B a c - 4 \,{\left (A a c \cos \left (f x + e\right )^{2} + 2 \, A a c\right )} \sin \left (f x + e\right )\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c}}{12 \, f \cos \left (f x + e\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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]