Optimal. Leaf size=138 \[ -\frac{64 a^3 (7 c+5 d) \cos (e+f x)}{105 f \sqrt{a \sin (e+f x)+a}}-\frac{16 a^2 (7 c+5 d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{105 f}-\frac{2 a (7 c+5 d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 f}-\frac{2 d \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{7 f} \]
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Rubi [A] time = 0.107962, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2751, 2647, 2646} \[ -\frac{64 a^3 (7 c+5 d) \cos (e+f x)}{105 f \sqrt{a \sin (e+f x)+a}}-\frac{16 a^2 (7 c+5 d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{105 f}-\frac{2 a (7 c+5 d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 f}-\frac{2 d \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{7 f} \]
Antiderivative was successfully verified.
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Rule 2751
Rule 2647
Rule 2646
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x)) \, dx &=-\frac{2 d \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 f}+\frac{1}{7} (7 c+5 d) \int (a+a \sin (e+f x))^{5/2} \, dx\\ &=-\frac{2 a (7 c+5 d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 f}-\frac{2 d \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 f}+\frac{1}{35} (8 a (7 c+5 d)) \int (a+a \sin (e+f x))^{3/2} \, dx\\ &=-\frac{16 a^2 (7 c+5 d) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{105 f}-\frac{2 a (7 c+5 d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 f}-\frac{2 d \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 f}+\frac{1}{105} \left (32 a^2 (7 c+5 d)\right ) \int \sqrt{a+a \sin (e+f x)} \, dx\\ &=-\frac{64 a^3 (7 c+5 d) \cos (e+f x)}{105 f \sqrt{a+a \sin (e+f x)}}-\frac{16 a^2 (7 c+5 d) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{105 f}-\frac{2 a (7 c+5 d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 f}-\frac{2 d \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 f}\\ \end{align*}
Mathematica [A] time = 1.50671, size = 119, normalized size = 0.86 \[ -\frac{a^2 \sqrt{a (\sin (e+f x)+1)} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) ((392 c+505 d) \sin (e+f x)-6 (7 c+20 d) \cos (2 (e+f x))+1246 c-15 d \sin (3 (e+f x))+1040 d)}{210 f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.743, size = 99, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2+2\,\sin \left ( fx+e \right ) \right ){a}^{3} \left ( -1+\sin \left ( fx+e \right ) \right ) \left ( -15\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) d+ \left ( 98\,c+130\,d \right ) \sin \left ( fx+e \right ) + \left ( -21\,c-60\,d \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+322\,c+290\,d \right ) }{105\,f\cos \left ( fx+e \right ) }{\frac{1}{\sqrt{a+a\sin \left ( fx+e \right ) }}}} \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 (a \sin \left (f x + e\right ) + a\right )}^{\frac{5}{2}}{\left (d \sin \left (f x + e\right ) + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64812, size = 518, normalized size = 3.75 \begin{align*} \frac{2 \,{\left (15 \, a^{2} d \cos \left (f x + e\right )^{4} + 3 \,{\left (7 \, a^{2} c + 20 \, a^{2} d\right )} \cos \left (f x + e\right )^{3} - 224 \, a^{2} c - 160 \, a^{2} d -{\left (77 \, a^{2} c + 85 \, a^{2} d\right )} \cos \left (f x + e\right )^{2} - 2 \,{\left (161 \, a^{2} c + 145 \, a^{2} d\right )} \cos \left (f x + e\right ) +{\left (15 \, a^{2} d \cos \left (f x + e\right )^{3} + 224 \, a^{2} c + 160 \, a^{2} d - 3 \,{\left (7 \, a^{2} c + 15 \, a^{2} d\right )} \cos \left (f x + e\right )^{2} - 2 \,{\left (49 \, a^{2} c + 65 \, a^{2} d\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt{a \sin \left (f x + e\right ) + a}}{105 \,{\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\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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