Optimal. Leaf size=146 \[ -\frac{a^2 (3 A-B) \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{30 f \sqrt{a \sin (e+f x)+a}}-\frac{a (3 A-B) \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{15 f}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}{6 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.358177, antiderivative size = 146, 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} \[ -\frac{a^2 (3 A-B) \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{30 f \sqrt{a \sin (e+f x)+a}}-\frac{a (3 A-B) \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{15 f}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}{6 f} \]
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))^{7/2} \, dx &=-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{6 f}+\frac{1}{3} (3 A-B) \int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2} \, dx\\ &=-\frac{a (3 A-B) \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}{15 f}-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{6 f}+\frac{1}{15} (2 a (3 A-B)) \int \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx\\ &=-\frac{a^2 (3 A-B) \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{30 f \sqrt{a+a \sin (e+f x)}}-\frac{a (3 A-B) \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}{15 f}-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{6 f}\\ \end{align*}
Mathematica [A] time = 1.59013, size = 205, normalized size = 1.4 \[ -\frac{c^3 (\sin (e+f x)-1)^3 (a (\sin (e+f x)+1))^{3/2} \sqrt{c-c \sin (e+f x)} (15 (16 A-11 B) \cos (2 (e+f x))+30 (2 A-B) \cos (4 (e+f x))+840 A \sin (e+f x)+60 A \sin (3 (e+f x))-12 A \sin (5 (e+f x))-240 B \sin (e+f x)+40 B \sin (3 (e+f x))+24 B \sin (5 (e+f x))+5 B \cos (6 (e+f x)))}{960 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^7 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.324, size = 185, normalized size = 1.3 \begin{align*}{\frac{ \left ( 5\,B\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}+6\,A \left ( \cos \left ( fx+e \right ) \right ) ^{4}-12\,B \left ( \cos \left ( fx+e \right ) \right ) ^{4}+15\,A \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) -10\,B \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) -12\,A \left ( \cos \left ( fx+e \right ) \right ) ^{2}+4\,B \left ( \cos \left ( fx+e \right ) \right ) ^{2}+15\,A\sin \left ( fx+e \right ) -10\,B\sin \left ( fx+e \right ) -24\,A+8\,B \right ) \sin \left ( fx+e \right ) }{30\,f \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{2}+2\,\sin \left ( fx+e \right ) -2 \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{3}} \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{7}{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{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.87883, size = 356, normalized size = 2.44 \begin{align*} \frac{{\left (5 \, B a c^{3} \cos \left (f x + e\right )^{6} + 15 \,{\left (A - B\right )} a c^{3} \cos \left (f x + e\right )^{4} - 5 \,{\left (3 \, A - 2 \, B\right )} a c^{3} - 2 \,{\left (3 \,{\left (A - 2 \, B\right )} a c^{3} \cos \left (f x + e\right )^{4} - 2 \,{\left (3 \, A - B\right )} a c^{3} \cos \left (f x + e\right )^{2} - 4 \,{\left (3 \, A - B\right )} a c^{3}\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}}{30 \, 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(-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]