Optimal. Leaf size=132 \[ \frac{a^2 c^2 \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac{2 a^2 \cos ^5(e+f x)}{1155 c f (c-c \sin (e+f x))^5}+\frac{2 a^2 \cos ^5(e+f x)}{231 f (c-c \sin (e+f x))^6}+\frac{a^2 c \cos ^5(e+f x)}{33 f (c-c \sin (e+f x))^7} \]
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Rubi [A] time = 0.232794, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {2736, 2672, 2671} \[ \frac{a^2 c^2 \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac{2 a^2 \cos ^5(e+f x)}{1155 c f (c-c \sin (e+f x))^5}+\frac{2 a^2 \cos ^5(e+f x)}{231 f (c-c \sin (e+f x))^6}+\frac{a^2 c \cos ^5(e+f x)}{33 f (c-c \sin (e+f x))^7} \]
Antiderivative was successfully verified.
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Rule 2736
Rule 2672
Rule 2671
Rubi steps
\begin{align*} \int \frac{(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^6} \, dx &=\left (a^2 c^2\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^8} \, dx\\ &=\frac{a^2 c^2 \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac{1}{11} \left (3 a^2 c\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^7} \, dx\\ &=\frac{a^2 c^2 \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac{a^2 c \cos ^5(e+f x)}{33 f (c-c \sin (e+f x))^7}+\frac{1}{33} \left (2 a^2\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^6} \, dx\\ &=\frac{a^2 c^2 \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac{a^2 c \cos ^5(e+f x)}{33 f (c-c \sin (e+f x))^7}+\frac{2 a^2 \cos ^5(e+f x)}{231 f (c-c \sin (e+f x))^6}+\frac{\left (2 a^2\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^5} \, dx}{231 c}\\ &=\frac{a^2 c^2 \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac{a^2 c \cos ^5(e+f x)}{33 f (c-c \sin (e+f x))^7}+\frac{2 a^2 \cos ^5(e+f x)}{231 f (c-c \sin (e+f x))^6}+\frac{2 a^2 \cos ^5(e+f x)}{1155 c f (c-c \sin (e+f x))^5}\\ \end{align*}
Mathematica [A] time = 0.672689, size = 133, normalized size = 1.01 \[ \frac{a^2 \left (2541 \sin \left (\frac{1}{2} (e+f x)\right )+1155 \sin \left (\frac{3}{2} (e+f x)\right )-165 \sin \left (\frac{5}{2} (e+f x)\right )+11 \sin \left (\frac{9}{2} (e+f x)\right )+2079 \cos \left (\frac{1}{2} (e+f x)\right )-825 \cos \left (\frac{3}{2} (e+f x)\right )-55 \cos \left (\frac{7}{2} (e+f x)\right )+\cos \left (\frac{11}{2} (e+f x)\right )\right )}{9240 c^6 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{11}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.107, size = 178, normalized size = 1.4 \begin{align*} 2\,{\frac{{a}^{2}}{f{c}^{6}} \left ( -288\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-8}-{\frac{932}{5\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{5}}}-88\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-4}-{\frac{128}{11\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{11}}}-292\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-6}-30\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-3}-7\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-2}-64\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-10}-{\frac{2376}{7\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{7}}}- \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-1}-{\frac{512}{3\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{9}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.456, size = 1798, normalized size = 13.62 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.39429, size = 832, normalized size = 6.3 \begin{align*} -\frac{2 \, a^{2} \cos \left (f x + e\right )^{6} + 12 \, a^{2} \cos \left (f x + e\right )^{5} - 25 \, a^{2} \cos \left (f x + e\right )^{4} - 70 \, a^{2} \cos \left (f x + e\right )^{3} - 245 \, a^{2} \cos \left (f x + e\right )^{2} + 210 \, a^{2} \cos \left (f x + e\right ) + 420 \, a^{2} -{\left (2 \, a^{2} \cos \left (f x + e\right )^{5} - 10 \, a^{2} \cos \left (f x + e\right )^{4} - 35 \, a^{2} \cos \left (f x + e\right )^{3} + 35 \, a^{2} \cos \left (f x + e\right )^{2} - 210 \, a^{2} \cos \left (f x + e\right ) - 420 \, a^{2}\right )} \sin \left (f x + e\right )}{1155 \,{\left (c^{6} f \cos \left (f x + e\right )^{6} - 5 \, c^{6} f \cos \left (f x + e\right )^{5} - 18 \, c^{6} f \cos \left (f x + e\right )^{4} + 20 \, c^{6} f \cos \left (f x + e\right )^{3} + 48 \, c^{6} f \cos \left (f x + e\right )^{2} - 16 \, c^{6} f \cos \left (f x + e\right ) - 32 \, c^{6} f +{\left (c^{6} f \cos \left (f x + e\right )^{5} + 6 \, c^{6} f \cos \left (f x + e\right )^{4} - 12 \, c^{6} f \cos \left (f x + e\right )^{3} - 32 \, c^{6} f \cos \left (f x + e\right )^{2} + 16 \, c^{6} f \cos \left (f x + e\right ) + 32 \, c^{6} f\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 [A] time = 2.1786, size = 265, normalized size = 2.01 \begin{align*} -\frac{2 \,{\left (1155 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{10} - 3465 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{9} + 13860 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} - 23100 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 37422 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 32802 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 27060 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 11220 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 4895 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 517 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 152 \, a^{2}\right )}}{1155 \, c^{6} f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1\right )}^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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