Optimal. Leaf size=192 \[ \frac{2 c^2 (7 A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)}}{105 f}+\frac{c^3 (7 A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{105 f \sqrt{c-c \sin (e+f x)}}+\frac{c (7 A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2}}{42 f}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{5/2}}{7 f} \]
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Rubi [A] time = 0.459797, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.075, Rules used = {2973, 2740, 2738} \[ \frac{2 c^2 (7 A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)}}{105 f}+\frac{c^3 (7 A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{105 f \sqrt{c-c \sin (e+f x)}}+\frac{c (7 A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2}}{42 f}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{5/2}}{7 f} \]
Antiderivative was successfully verified.
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Rule 2973
Rule 2740
Rule 2738
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx &=-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2}}{7 f}+\frac{1}{7} (7 A+B) \int (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx\\ &=\frac{(7 A+B) c \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{42 f}-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2}}{7 f}+\frac{1}{21} (2 (7 A+B) c) \int (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx\\ &=\frac{2 (7 A+B) c^2 \cos (e+f x) (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)}}{105 f}+\frac{(7 A+B) c \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{42 f}-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2}}{7 f}+\frac{1}{105} \left (4 (7 A+B) c^2\right ) \int (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)} \, dx\\ &=\frac{(7 A+B) c^3 \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{105 f \sqrt{c-c \sin (e+f x)}}+\frac{2 (7 A+B) c^2 \cos (e+f x) (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)}}{105 f}+\frac{(7 A+B) c \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{42 f}-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2}}{7 f}\\ \end{align*}
Mathematica [A] time = 2.19684, size = 232, normalized size = 1.21 \[ \frac{a^3 c^2 (\sin (e+f x)-1)^2 (\sin (e+f x)+1)^3 \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (-525 (A+B) \cos (2 (e+f x))-210 (A+B) \cos (4 (e+f x))+4200 A \sin (e+f x)+700 A \sin (3 (e+f x))+84 A \sin (5 (e+f x))-35 A \cos (6 (e+f x))+525 B \sin (e+f x)-35 B \sin (3 (e+f x))-63 B \sin (5 (e+f x))-15 B \sin (7 (e+f x))-35 B \cos (6 (e+f x)))}{6720 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^5 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.359, size = 203, normalized size = 1.1 \begin{align*}{\frac{ \left ( -30\,B \left ( \cos \left ( fx+e \right ) \right ) ^{6}+35\,A \left ( \cos \left ( fx+e \right ) \right ) ^{4}\sin \left ( fx+e \right ) +35\,B\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}+42\,A \left ( \cos \left ( fx+e \right ) \right ) ^{4}+6\,B \left ( \cos \left ( fx+e \right ) \right ) ^{4}+35\,A \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) +35\,B \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) +56\,A \left ( \cos \left ( fx+e \right ) \right ) ^{2}+8\,B \left ( \cos \left ( fx+e \right ) \right ) ^{2}+35\,A\sin \left ( fx+e \right ) +35\,B\sin \left ( fx+e \right ) +112\,A+16\,B \right ) \sin \left ( fx+e \right ) }{210\,f \left ( 1+\sin \left ( fx+e \right ) \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{5}} \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{5}{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.
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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{7}{2}}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73989, size = 373, normalized size = 1.94 \begin{align*} -\frac{{\left (35 \,{\left (A + B\right )} a^{3} c^{2} \cos \left (f x + e\right )^{6} - 35 \,{\left (A + B\right )} a^{3} c^{2} + 2 \,{\left (15 \, B a^{3} c^{2} \cos \left (f x + e\right )^{6} - 3 \,{\left (7 \, A + B\right )} a^{3} c^{2} \cos \left (f x + e\right )^{4} - 4 \,{\left (7 \, A + B\right )} a^{3} c^{2} \cos \left (f x + e\right )^{2} - 8 \,{\left (7 \, A + B\right )} a^{3} c^{2}\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}}{210 \, f \cos \left (f x + e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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