Optimal. Leaf size=135 \[ \frac{35 c^4 \cos (e+f x)}{2 a^2 f}-\frac{2 a^3 c^4 \cos ^7(e+f x)}{3 f (a \sin (e+f x)+a)^5}+\frac{14 a^4 c^4 \cos ^5(e+f x)}{3 f \left (a^2 \sin (e+f x)+a^2\right )^3}+\frac{35 c^4 \cos ^3(e+f x)}{6 f \left (a^2 \sin (e+f x)+a^2\right )}+\frac{35 c^4 x}{2 a^2} \]
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Rubi [A] time = 0.224106, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {2736, 2680, 2679, 2682, 8} \[ \frac{35 c^4 \cos (e+f x)}{2 a^2 f}-\frac{2 a^3 c^4 \cos ^7(e+f x)}{3 f (a \sin (e+f x)+a)^5}+\frac{14 a^4 c^4 \cos ^5(e+f x)}{3 f \left (a^2 \sin (e+f x)+a^2\right )^3}+\frac{35 c^4 \cos ^3(e+f x)}{6 f \left (a^2 \sin (e+f x)+a^2\right )}+\frac{35 c^4 x}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 2736
Rule 2680
Rule 2679
Rule 2682
Rule 8
Rubi steps
\begin{align*} \int \frac{(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx &=\left (a^4 c^4\right ) \int \frac{\cos ^8(e+f x)}{(a+a \sin (e+f x))^6} \, dx\\ &=-\frac{2 a^3 c^4 \cos ^7(e+f x)}{3 f (a+a \sin (e+f x))^5}-\frac{1}{3} \left (7 a^2 c^4\right ) \int \frac{\cos ^6(e+f x)}{(a+a \sin (e+f x))^4} \, dx\\ &=-\frac{2 a^3 c^4 \cos ^7(e+f x)}{3 f (a+a \sin (e+f x))^5}+\frac{14 a c^4 \cos ^5(e+f x)}{3 f (a+a \sin (e+f x))^3}+\frac{1}{3} \left (35 c^4\right ) \int \frac{\cos ^4(e+f x)}{(a+a \sin (e+f x))^2} \, dx\\ &=-\frac{2 a^3 c^4 \cos ^7(e+f x)}{3 f (a+a \sin (e+f x))^5}+\frac{14 a c^4 \cos ^5(e+f x)}{3 f (a+a \sin (e+f x))^3}+\frac{35 c^4 \cos ^3(e+f x)}{6 f \left (a^2+a^2 \sin (e+f x)\right )}+\frac{\left (35 c^4\right ) \int \frac{\cos ^2(e+f x)}{a+a \sin (e+f x)} \, dx}{2 a}\\ &=\frac{35 c^4 \cos (e+f x)}{2 a^2 f}-\frac{2 a^3 c^4 \cos ^7(e+f x)}{3 f (a+a \sin (e+f x))^5}+\frac{14 a c^4 \cos ^5(e+f x)}{3 f (a+a \sin (e+f x))^3}+\frac{35 c^4 \cos ^3(e+f x)}{6 f \left (a^2+a^2 \sin (e+f x)\right )}+\frac{\left (35 c^4\right ) \int 1 \, dx}{2 a^2}\\ &=\frac{35 c^4 x}{2 a^2}+\frac{35 c^4 \cos (e+f x)}{2 a^2 f}-\frac{2 a^3 c^4 \cos ^7(e+f x)}{3 f (a+a \sin (e+f x))^5}+\frac{14 a c^4 \cos ^5(e+f x)}{3 f (a+a \sin (e+f x))^3}+\frac{35 c^4 \cos ^3(e+f x)}{6 f \left (a^2+a^2 \sin (e+f x)\right )}\\ \end{align*}
Mathematica [A] time = 0.503876, size = 243, normalized size = 1.8 \[ \frac{(c-c \sin (e+f x))^4 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \left (128 \sin \left (\frac{1}{2} (e+f x)\right )+210 (e+f x) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3+72 \cos (e+f x) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3-3 \sin (2 (e+f x)) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3-640 \sin \left (\frac{1}{2} (e+f x)\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2-64 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )\right )}{12 f (a \sin (e+f x)+a)^2 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.093, size = 229, normalized size = 1.7 \begin{align*}{\frac{{c}^{4}}{{a}^{2}f} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{2} \right ) ^{-2}}+12\,{\frac{{c}^{4} \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2}}{{a}^{2}f \left ( 1+ \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{{c}^{4}}{{a}^{2}f}\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{2} \right ) ^{-2}}+12\,{\frac{{c}^{4}}{{a}^{2}f \left ( 1+ \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2} \right ) ^{2}}}+35\,{\frac{{c}^{4}\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) }{{a}^{2}f}}-{\frac{64\,{c}^{4}}{3\,{a}^{2}f} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) +1 \right ) ^{-3}}+32\,{\frac{{c}^{4}}{{a}^{2}f \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{2}}}+32\,{\frac{{c}^{4}}{{a}^{2}f \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.35734, size = 1219, normalized size = 9.03 \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 [A] time = 1.43884, size = 513, normalized size = 3.8 \begin{align*} -\frac{3 \, c^{4} \cos \left (f x + e\right )^{4} - 30 \, c^{4} \cos \left (f x + e\right )^{3} + 210 \, c^{4} f x - 32 \, c^{4} -{\left (105 \, c^{4} f x - 193 \, c^{4}\right )} \cos \left (f x + e\right )^{2} +{\left (105 \, c^{4} f x + 194 \, c^{4}\right )} \cos \left (f x + e\right ) +{\left (3 \, c^{4} \cos \left (f x + e\right )^{3} + 210 \, c^{4} f x + 33 \, c^{4} \cos \left (f x + e\right )^{2} + 32 \, c^{4} +{\left (105 \, c^{4} f x + 226 \, c^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{6 \,{\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f -{\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} 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 = 2.22172, size = 205, normalized size = 1.52 \begin{align*} \frac{\frac{105 \,{\left (f x + e\right )} c^{4}}{a^{2}} + \frac{6 \,{\left (c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 12 \, c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 12 \, c^{4}\right )}}{{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 1\right )}^{2} a^{2}} + \frac{64 \,{\left (3 \, c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 9 \, c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 4 \, c^{4}\right )}}{a^{2}{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1\right )}^{3}}}{6 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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