Optimal. Leaf size=226 \[ -\frac{2 a^4 A \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{35 f \sqrt{a \sin (e+f x)+a}}-\frac{4 a^3 A \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{35 f}-\frac{a^2 A \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}{7 f}-\frac{a A \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{7/2}}{8 f} \]
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Rubi [A] time = 0.559084, antiderivative size = 226, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.075, Rules used = {2973, 2740, 2738} \[ -\frac{2 a^4 A \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{35 f \sqrt{a \sin (e+f x)+a}}-\frac{4 a^3 A \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{35 f}-\frac{a^2 A \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}{7 f}-\frac{a A \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{7/2}}{8 f} \]
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))^{7/2} \, dx &=-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2}}{8 f}+A \int (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2} \, dx\\ &=-\frac{a A \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f}-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2}}{8 f}+\frac{1}{7} (6 a A) \int (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2} \, dx\\ &=-\frac{a^2 A \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{7 f}-\frac{a A \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f}-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2}}{8 f}+\frac{1}{7} \left (4 a^2 A\right ) \int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2} \, dx\\ &=-\frac{4 a^3 A \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}{35 f}-\frac{a^2 A \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{7 f}-\frac{a A \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f}-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2}}{8 f}+\frac{1}{35} \left (8 a^3 A\right ) \int \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx\\ &=-\frac{2 a^4 A \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 A \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}{35 f}-\frac{a^2 A \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{7 f}-\frac{a A \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f}-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2}}{8 f}\\ \end{align*}
Mathematica [A] time = 1.673, size = 135, normalized size = 0.6 \[ \frac{a^3 c^3 \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (19600 A \sin (e+f x)+3920 A \sin (3 (e+f x))+784 A \sin (5 (e+f x))+80 A \sin (7 (e+f x))-1960 B \cos (2 (e+f x))-980 B \cos (4 (e+f x))-280 B \cos (6 (e+f x))-35 B \cos (8 (e+f x)))}{35840 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.33, size = 142, normalized size = 0.6 \begin{align*}{\frac{ \left ( 35\,B \left ( \cos \left ( fx+e \right ) \right ) ^{6}\sin \left ( fx+e \right ) +40\,A \left ( \cos \left ( fx+e \right ) \right ) ^{6}+35\,B\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}+48\,A \left ( \cos \left ( fx+e \right ) \right ) ^{4}+35\,B \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) +64\,A \left ( \cos \left ( fx+e \right ) \right ) ^{2}+35\,B\sin \left ( fx+e \right ) +128\,A \right ) \sin \left ( fx+e \right ) }{280\,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.
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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{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81399, size = 324, normalized size = 1.43 \begin{align*} -\frac{{\left (35 \, B a^{3} c^{3} \cos \left (f x + e\right )^{8} - 35 \, B a^{3} c^{3} - 8 \,{\left (5 \, A a^{3} c^{3} \cos \left (f x + e\right )^{6} + 6 \, A a^{3} c^{3} \cos \left (f x + e\right )^{4} + 8 \, A a^{3} c^{3} \cos \left (f x + e\right )^{2} + 16 \, A a^{3} 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}}{280 \, 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*} \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{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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