Optimal. Leaf size=179 \[ -\frac{2 a^4 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{35 f \sqrt{a \sin (e+f x)+a}}-\frac{4 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{35 f}-\frac{a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}{7 f}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.366398, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2740, 2738} \[ -\frac{2 a^4 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{35 f \sqrt{a \sin (e+f x)+a}}-\frac{4 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{35 f}-\frac{a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}{7 f}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2740
Rule 2738
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2} \, dx &=-\frac{a \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f}+\frac{1}{7} (6 a) \int (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2} \, dx\\ &=-\frac{a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{7 f}-\frac{a \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f}+\frac{1}{7} \left (4 a^2\right ) \int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2} \, dx\\ &=-\frac{4 a^3 \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}{35 f}-\frac{a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{7 f}-\frac{a \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f}+\frac{1}{35} \left (8 a^3\right ) \int \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx\\ &=-\frac{2 a^4 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{35 f \sqrt{a+a \sin (e+f x)}}-\frac{4 a^3 \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}{35 f}-\frac{a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{7 f}-\frac{a \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f}\\ \end{align*}
Mathematica [A] time = 1.01169, size = 87, normalized size = 0.49 \[ \frac{a^3 c^3 (1225 \sin (e+f x)+245 \sin (3 (e+f x))+49 \sin (5 (e+f x))+5 \sin (7 (e+f x))) \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)}}{2240 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.156, size = 77, normalized size = 0.4 \begin{align*}{\frac{ \left ( 5\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}+6\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}+8\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+16 \right ) \sin \left ( fx+e \right ) }{35\,f \left ( \cos \left ( fx+e \right ) \right ) ^{7}} \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{7}{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 (a \sin \left (f x + e\right ) + a\right )}^{\frac{7}{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 = 1.1831, size = 244, normalized size = 1.36 \begin{align*} \frac{{\left (5 \, a^{3} c^{3} \cos \left (f x + e\right )^{6} + 6 \, a^{3} c^{3} \cos \left (f x + e\right )^{4} + 8 \, a^{3} c^{3} \cos \left (f x + e\right )^{2} + 16 \, a^{3} c^{3}\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )}{35 \, 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]