Optimal. Leaf size=198 \[ -\frac{2 a^2 (7 A-B) \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{105 f}-\frac{a^3 (7 A-B) \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{105 f \sqrt{a \sin (e+f x)+a}}-\frac{a (7 A-B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}{42 f}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f} \]
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Rubi [A] time = 0.476191, antiderivative size = 198, 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} \[ -\frac{2 a^2 (7 A-B) \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{105 f}-\frac{a^3 (7 A-B) \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{105 f \sqrt{a \sin (e+f x)+a}}-\frac{a (7 A-B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}{42 f}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f} \]
Antiderivative was successfully verified.
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Rule 2973
Rule 2740
Rule 2738
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^{5/2} (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))^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f}+\frac{1}{7} (7 A-B) \int (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2} \, dx\\ &=-\frac{a (7 A-B) \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{42 f}-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f}+\frac{1}{21} (2 a (7 A-B)) \int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2} \, dx\\ &=-\frac{2 a^2 (7 A-B) \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}{105 f}-\frac{a (7 A-B) \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{42 f}-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f}+\frac{1}{105} \left (4 a^2 (7 A-B)\right ) \int \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx\\ &=-\frac{a^3 (7 A-B) \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{105 f \sqrt{a+a \sin (e+f x)}}-\frac{2 a^2 (7 A-B) \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}{105 f}-\frac{a (7 A-B) \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{42 f}-\frac{B \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 = 2.85785, size = 223, normalized size = 1.13 \[ -\frac{c^3 (\sin (e+f x)-1)^3 (a (\sin (e+f x)+1))^{5/2} \sqrt{c-c \sin (e+f x)} (525 (A-B) \cos (2 (e+f x))+210 (A-B) \cos (4 (e+f x))+4200 A \sin (e+f x)+700 A \sin (3 (e+f x))+84 A \sin (5 (e+f x))+35 A \cos (6 (e+f x))-525 B \sin (e+f x)+35 B \sin (3 (e+f x))+63 B \sin (5 (e+f x))+15 B \sin (7 (e+f x))-35 B \cos (6 (e+f x)))}{6720 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^7 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.368, size = 203, normalized size = 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{7}{2}}} \left ( a \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{5}{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{5}{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 = 2.34748, size = 371, normalized size = 1.87 \begin{align*} \frac{{\left (35 \,{\left (A - B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{6} - 35 \,{\left (A - B\right )} a^{2} c^{3} + 2 \,{\left (15 \, B a^{2} c^{3} \cos \left (f x + e\right )^{6} + 3 \,{\left (7 \, A - B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{4} + 4 \,{\left (7 \, A - B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{2} + 8 \,{\left (7 \, A - B\right )} a^{2} 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}}{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(-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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