Optimal. Leaf size=118 \[ \frac {7 a^2 c^4 \cos ^5(e+f x)}{30 f}+\frac {a^2 \cos ^5(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{6 f}+\frac {7 a^2 c^4 \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac {7 a^2 c^4 \sin (e+f x) \cos (e+f x)}{16 f}+\frac {7}{16} a^2 c^4 x \]
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Rubi [A] time = 0.14, antiderivative size = 118, normalized size of antiderivative = 1.00, 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, 2678, 2669, 2635, 8} \[ \frac {7 a^2 c^4 \cos ^5(e+f x)}{30 f}+\frac {a^2 \cos ^5(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{6 f}+\frac {7 a^2 c^4 \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac {7 a^2 c^4 \sin (e+f x) \cos (e+f x)}{16 f}+\frac {7}{16} a^2 c^4 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2669
Rule 2678
Rule 2736
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^4 \, dx &=\left (a^2 c^2\right ) \int \cos ^4(e+f x) (c-c \sin (e+f x))^2 \, dx\\ &=\frac {a^2 \cos ^5(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{6 f}+\frac {1}{6} \left (7 a^2 c^3\right ) \int \cos ^4(e+f x) (c-c \sin (e+f x)) \, dx\\ &=\frac {7 a^2 c^4 \cos ^5(e+f x)}{30 f}+\frac {a^2 \cos ^5(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{6 f}+\frac {1}{6} \left (7 a^2 c^4\right ) \int \cos ^4(e+f x) \, dx\\ &=\frac {7 a^2 c^4 \cos ^5(e+f x)}{30 f}+\frac {7 a^2 c^4 \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac {a^2 \cos ^5(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{6 f}+\frac {1}{8} \left (7 a^2 c^4\right ) \int \cos ^2(e+f x) \, dx\\ &=\frac {7 a^2 c^4 \cos ^5(e+f x)}{30 f}+\frac {7 a^2 c^4 \cos (e+f x) \sin (e+f x)}{16 f}+\frac {7 a^2 c^4 \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac {a^2 \cos ^5(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{6 f}+\frac {1}{16} \left (7 a^2 c^4\right ) \int 1 \, dx\\ &=\frac {7}{16} a^2 c^4 x+\frac {7 a^2 c^4 \cos ^5(e+f x)}{30 f}+\frac {7 a^2 c^4 \cos (e+f x) \sin (e+f x)}{16 f}+\frac {7 a^2 c^4 \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac {a^2 \cos ^5(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{6 f}\\ \end {align*}
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Mathematica [A] time = 0.73, size = 79, normalized size = 0.67 \[ \frac {a^2 c^4 (255 \sin (2 (e+f x))+15 \sin (4 (e+f x))-5 \sin (6 (e+f x))+240 \cos (e+f x)+120 \cos (3 (e+f x))+24 \cos (5 (e+f x))+420 e+420 f x)}{960 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 87, normalized size = 0.74 \[ \frac {96 \, a^{2} c^{4} \cos \left (f x + e\right )^{5} + 105 \, a^{2} c^{4} f x - 5 \, {\left (8 \, a^{2} c^{4} \cos \left (f x + e\right )^{5} - 14 \, a^{2} c^{4} \cos \left (f x + e\right )^{3} - 21 \, a^{2} c^{4} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{240 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 133, normalized size = 1.13 \[ \frac {7}{16} \, a^{2} c^{4} x + \frac {a^{2} c^{4} \cos \left (5 \, f x + 5 \, e\right )}{40 \, f} + \frac {a^{2} c^{4} \cos \left (3 \, f x + 3 \, e\right )}{8 \, f} + \frac {a^{2} c^{4} \cos \left (f x + e\right )}{4 \, f} - \frac {a^{2} c^{4} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac {a^{2} c^{4} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} + \frac {17 \, a^{2} c^{4} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 211, normalized size = 1.79 \[ \frac {c^{4} a^{2} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+\frac {2 c^{4} a^{2} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-c^{4} a^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {4 c^{4} a^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-c^{4} a^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+2 c^{4} a^{2} \cos \left (f x +e \right )+c^{4} a^{2} \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 209, normalized size = 1.77 \[ \frac {128 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{2} c^{4} + 1280 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c^{4} + 5 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{4} - 30 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{4} - 240 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{4} + 960 \, {\left (f x + e\right )} a^{2} c^{4} + 1920 \, a^{2} c^{4} \cos \left (f x + e\right )}{960 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.90, size = 284, normalized size = 2.41 \[ \frac {a^2\,c^4\,\left (105\,e+270\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+192\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+890\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+1920\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-660\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+1920\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+660\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+960\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-890\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9+960\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}-270\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}+105\,f\,x+630\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (e+f\,x\right )+1575\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (e+f\,x\right )+2100\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (e+f\,x\right )+1575\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (e+f\,x\right )+630\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,\left (e+f\,x\right )+105\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}\,\left (e+f\,x\right )+192\right )}{240\,f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.93, size = 530, normalized size = 4.49 \[ \begin {cases} \frac {5 a^{2} c^{4} x \sin ^{6}{\left (e + f x \right )}}{16} + \frac {15 a^{2} c^{4} x \sin ^{4}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{16} - \frac {3 a^{2} c^{4} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {15 a^{2} c^{4} x \sin ^{2}{\left (e + f x \right )} \cos ^{4}{\left (e + f x \right )}}{16} - \frac {3 a^{2} c^{4} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} - \frac {a^{2} c^{4} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {5 a^{2} c^{4} x \cos ^{6}{\left (e + f x \right )}}{16} - \frac {3 a^{2} c^{4} x \cos ^{4}{\left (e + f x \right )}}{8} - \frac {a^{2} c^{4} x \cos ^{2}{\left (e + f x \right )}}{2} + a^{2} c^{4} x - \frac {11 a^{2} c^{4} \sin ^{5}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{16 f} + \frac {2 a^{2} c^{4} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {5 a^{2} c^{4} \sin ^{3}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{6 f} + \frac {5 a^{2} c^{4} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} + \frac {8 a^{2} c^{4} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {4 a^{2} c^{4} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {5 a^{2} c^{4} \sin {\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{16 f} + \frac {3 a^{2} c^{4} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} + \frac {a^{2} c^{4} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {16 a^{2} c^{4} \cos ^{5}{\left (e + f x \right )}}{15 f} - \frac {8 a^{2} c^{4} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {2 a^{2} c^{4} \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (a \sin {\relax (e )} + a\right )^{2} \left (- c \sin {\relax (e )} + c\right )^{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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