Optimal. Leaf size=236 \[ -\frac{a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{11/2}}{15 c f}-\frac{4 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{11/2}}{105 c f}-\frac{4 a^4 \cos (e+f x) (c-c \sin (e+f x))^{11/2}}{315 c f \sqrt{a \sin (e+f x)+a}}-\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{11/2}}{10 c f}-\frac{4 a \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{11/2}}{45 c f} \]
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Rubi [A] time = 0.724664, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.079, Rules used = {2841, 2740, 2738} \[ -\frac{a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{11/2}}{15 c f}-\frac{4 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{11/2}}{105 c f}-\frac{4 a^4 \cos (e+f x) (c-c \sin (e+f x))^{11/2}}{315 c f \sqrt{a \sin (e+f x)+a}}-\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{11/2}}{10 c f}-\frac{4 a \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{11/2}}{45 c f} \]
Antiderivative was successfully verified.
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Rule 2841
Rule 2740
Rule 2738
Rubi steps
\begin{align*} \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2} \, dx &=\frac{\int (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{11/2} \, dx}{a c}\\ &=-\frac{\cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{11/2}}{10 c f}+\frac{4 \int (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{11/2} \, dx}{5 c}\\ &=-\frac{4 a \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{11/2}}{45 c f}-\frac{\cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{11/2}}{10 c f}+\frac{(8 a) \int (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{11/2} \, dx}{15 c}\\ &=-\frac{a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{11/2}}{15 c f}-\frac{4 a \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{11/2}}{45 c f}-\frac{\cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{11/2}}{10 c f}+\frac{\left (4 a^2\right ) \int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{11/2} \, dx}{15 c}\\ &=-\frac{4 a^3 \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{11/2}}{105 c f}-\frac{a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{11/2}}{15 c f}-\frac{4 a \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{11/2}}{45 c f}-\frac{\cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{11/2}}{10 c f}+\frac{\left (8 a^3\right ) \int \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{11/2} \, dx}{105 c}\\ &=-\frac{4 a^4 \cos (e+f x) (c-c \sin (e+f x))^{11/2}}{315 c f \sqrt{a+a \sin (e+f x)}}-\frac{4 a^3 \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{11/2}}{105 c f}-\frac{a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{11/2}}{15 c f}-\frac{4 a \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{11/2}}{45 c f}-\frac{\cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{11/2}}{10 c f}\\ \end{align*}
Mathematica [A] time = 5.70095, size = 209, normalized size = 0.89 \[ \frac{a^3 c^4 (\sin (e+f x)-1)^4 (\sin (e+f x)+1)^3 \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (158760 \sin (e+f x)+35280 \sin (3 (e+f x))+9072 \sin (5 (e+f x))+1620 \sin (7 (e+f x))+140 \sin (9 (e+f x))+13230 \cos (2 (e+f x))+7560 \cos (4 (e+f x))+2835 \cos (6 (e+f x))+630 \cos (8 (e+f x))+63 \cos (10 (e+f x)))}{322560 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.31, size = 169, normalized size = 0.7 \begin{align*}{\frac{\sin \left ( fx+e \right ) \left ( 63\, \left ( \cos \left ( fx+e \right ) \right ) ^{10}+7\,\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{8}+70\, \left ( \cos \left ( fx+e \right ) \right ) ^{8}+17\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}\sin \left ( fx+e \right ) +80\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}+33\,\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}+96\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}+65\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) +128\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+193\,\sin \left ( fx+e \right ) +193 \right ) }{630\,f \left ( \cos \left ( fx+e \right ) \right ) ^{9}} \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{9}{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(-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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Fricas [A] time = 2.13492, size = 351, normalized size = 1.49 \begin{align*} \frac{{\left (63 \, a^{3} c^{4} \cos \left (f x + e\right )^{10} - 63 \, a^{3} c^{4} + 2 \,{\left (35 \, a^{3} c^{4} \cos \left (f x + e\right )^{8} + 40 \, a^{3} c^{4} \cos \left (f x + e\right )^{6} + 48 \, a^{3} c^{4} \cos \left (f x + e\right )^{4} + 64 \, a^{3} c^{4} \cos \left (f x + e\right )^{2} + 128 \, a^{3} c^{4}\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}}{630 \, 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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