Optimal. Leaf size=94 \[ \frac{a (A+B) \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac{a B \cos (e+f x)}{2 c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}} \]
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Rubi [A] time = 0.33662, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {2971, 2738} \[ \frac{a (A+B) \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac{a B \cos (e+f x)}{2 c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2971
Rule 2738
Rubi steps
\begin{align*} \int \frac{\sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx &=(A+B) \int \frac{\sqrt{a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx-\frac{B \int \frac{\sqrt{a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{5/2}} \, dx}{c}\\ &=\frac{a (A+B) \cos (e+f x)}{3 f \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac{a B \cos (e+f x)}{2 c f \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.627621, size = 103, normalized size = 1.1 \[ \frac{\sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (2 A+3 B \sin (e+f x)-B)}{6 c^4 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^7 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.364, size = 205, normalized size = 2.2 \begin{align*} -{\frac{ \left ( 2\,A \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) +2\,A \left ( \cos \left ( fx+e \right ) \right ) ^{3}-B \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) -B \left ( \cos \left ( fx+e \right ) \right ) ^{3}+6\,A\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) -8\,A \left ( \cos \left ( fx+e \right ) \right ) ^{2}-3\,B\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +4\,B \left ( \cos \left ( fx+e \right ) \right ) ^{2}-14\,A\sin \left ( fx+e \right ) -8\,A\cos \left ( fx+e \right ) +4\,B\sin \left ( fx+e \right ) +B\cos \left ( fx+e \right ) +14\,A-4\,B \right ) \sin \left ( fx+e \right ) }{6\,f \left ( 1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) \right ) }\sqrt{a \left ( 1+\sin \left ( fx+e \right ) \right ) } \left ( -c \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 \frac{{\left (B \sin \left (f x + e\right ) + A\right )} \sqrt{a \sin \left (f x + e\right ) + a}}{{\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 = 1.75888, size = 263, normalized size = 2.8 \begin{align*} -\frac{{\left (3 \, B \sin \left (f x + e\right ) + 2 \, A - B\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c}}{6 \,{\left (3 \, c^{4} f \cos \left (f x + e\right )^{3} - 4 \, c^{4} f \cos \left (f x + e\right ) -{\left (c^{4} f \cos \left (f x + e\right )^{3} - 4 \, c^{4} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\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 \frac{{\left (B \sin \left (f x + e\right ) + A\right )} \sqrt{a \sin \left (f x + e\right ) + a}}{{\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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