Optimal. Leaf size=180 \[ \frac {11 a^3 c^6 \cos ^7(e+f x)}{56 f}+\frac {11 a^3 \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{72 f}+\frac {11 a^3 c^6 \sin (e+f x) \cos ^5(e+f x)}{48 f}+\frac {55 a^3 c^6 \sin (e+f x) \cos ^3(e+f x)}{192 f}+\frac {55 a^3 c^6 \sin (e+f x) \cos (e+f x)}{128 f}+\frac {55}{128} a^3 c^6 x+\frac {a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f} \]
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Rubi [A] time = 0.21, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {11 a^3 c^6 \cos ^7(e+f x)}{56 f}+\frac {a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f}+\frac {11 a^3 \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{72 f}+\frac {11 a^3 c^6 \sin (e+f x) \cos ^5(e+f x)}{48 f}+\frac {55 a^3 c^6 \sin (e+f x) \cos ^3(e+f x)}{192 f}+\frac {55 a^3 c^6 \sin (e+f x) \cos (e+f x)}{128 f}+\frac {55}{128} a^3 c^6 x \]
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))^3 (c-c \sin (e+f x))^6 \, dx &=\left (a^3 c^3\right ) \int \cos ^6(e+f x) (c-c \sin (e+f x))^3 \, dx\\ &=\frac {a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f}+\frac {1}{9} \left (11 a^3 c^4\right ) \int \cos ^6(e+f x) (c-c \sin (e+f x))^2 \, dx\\ &=\frac {a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f}+\frac {11 a^3 \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{72 f}+\frac {1}{8} \left (11 a^3 c^5\right ) \int \cos ^6(e+f x) (c-c \sin (e+f x)) \, dx\\ &=\frac {11 a^3 c^6 \cos ^7(e+f x)}{56 f}+\frac {a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f}+\frac {11 a^3 \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{72 f}+\frac {1}{8} \left (11 a^3 c^6\right ) \int \cos ^6(e+f x) \, dx\\ &=\frac {11 a^3 c^6 \cos ^7(e+f x)}{56 f}+\frac {11 a^3 c^6 \cos ^5(e+f x) \sin (e+f x)}{48 f}+\frac {a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f}+\frac {11 a^3 \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{72 f}+\frac {1}{48} \left (55 a^3 c^6\right ) \int \cos ^4(e+f x) \, dx\\ &=\frac {11 a^3 c^6 \cos ^7(e+f x)}{56 f}+\frac {55 a^3 c^6 \cos ^3(e+f x) \sin (e+f x)}{192 f}+\frac {11 a^3 c^6 \cos ^5(e+f x) \sin (e+f x)}{48 f}+\frac {a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f}+\frac {11 a^3 \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{72 f}+\frac {1}{64} \left (55 a^3 c^6\right ) \int \cos ^2(e+f x) \, dx\\ &=\frac {11 a^3 c^6 \cos ^7(e+f x)}{56 f}+\frac {55 a^3 c^6 \cos (e+f x) \sin (e+f x)}{128 f}+\frac {55 a^3 c^6 \cos ^3(e+f x) \sin (e+f x)}{192 f}+\frac {11 a^3 c^6 \cos ^5(e+f x) \sin (e+f x)}{48 f}+\frac {a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f}+\frac {11 a^3 \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{72 f}+\frac {1}{128} \left (55 a^3 c^6\right ) \int 1 \, dx\\ &=\frac {55}{128} a^3 c^6 x+\frac {11 a^3 c^6 \cos ^7(e+f x)}{56 f}+\frac {55 a^3 c^6 \cos (e+f x) \sin (e+f x)}{128 f}+\frac {55 a^3 c^6 \cos ^3(e+f x) \sin (e+f x)}{192 f}+\frac {11 a^3 c^6 \cos ^5(e+f x) \sin (e+f x)}{48 f}+\frac {a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f}+\frac {11 a^3 \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{72 f}\\ \end {align*}
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Mathematica [A] time = 2.14, size = 109, normalized size = 0.61 \[ \frac {a^3 c^6 (18144 \sin (2 (e+f x))+1512 \sin (4 (e+f x))-672 \sin (6 (e+f x))-189 \sin (8 (e+f x))+16632 \cos (e+f x)+9744 \cos (3 (e+f x))+3024 \cos (5 (e+f x))+324 \cos (7 (e+f x))-28 \cos (9 (e+f x))+27720 e+27720 f x)}{64512 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 119, normalized size = 0.66 \[ -\frac {896 \, a^{3} c^{6} \cos \left (f x + e\right )^{9} - 4608 \, a^{3} c^{6} \cos \left (f x + e\right )^{7} - 3465 \, a^{3} c^{6} f x + 21 \, {\left (144 \, a^{3} c^{6} \cos \left (f x + e\right )^{7} - 88 \, a^{3} c^{6} \cos \left (f x + e\right )^{5} - 110 \, a^{3} c^{6} \cos \left (f x + e\right )^{3} - 165 \, a^{3} c^{6} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{8064 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 196, normalized size = 1.09 \[ \frac {55}{128} \, a^{3} c^{6} x - \frac {a^{3} c^{6} \cos \left (9 \, f x + 9 \, e\right )}{2304 \, f} + \frac {9 \, a^{3} c^{6} \cos \left (7 \, f x + 7 \, e\right )}{1792 \, f} + \frac {3 \, a^{3} c^{6} \cos \left (5 \, f x + 5 \, e\right )}{64 \, f} + \frac {29 \, a^{3} c^{6} \cos \left (3 \, f x + 3 \, e\right )}{192 \, f} + \frac {33 \, a^{3} c^{6} \cos \left (f x + e\right )}{128 \, f} - \frac {3 \, a^{3} c^{6} \sin \left (8 \, f x + 8 \, e\right )}{1024 \, f} - \frac {a^{3} c^{6} \sin \left (6 \, f x + 6 \, e\right )}{96 \, f} + \frac {3 \, a^{3} c^{6} \sin \left (4 \, f x + 4 \, e\right )}{128 \, f} + \frac {9 \, a^{3} c^{6} \sin \left (2 \, f x + 2 \, e\right )}{32 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 297, normalized size = 1.65 \[ \frac {-\frac {c^{6} a^{3} \left (\frac {128}{35}+\sin ^{8}\left (f x +e \right )+\frac {8 \left (\sin ^{6}\left (f x +e \right )\right )}{7}+\frac {48 \left (\sin ^{4}\left (f x +e \right )\right )}{35}+\frac {64 \left (\sin ^{2}\left (f x +e \right )\right )}{35}\right ) \cos \left (f x +e \right )}{9}-3 c^{6} a^{3} \left (-\frac {\left (\sin ^{7}\left (f x +e \right )+\frac {7 \left (\sin ^{5}\left (f x +e \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (f x +e \right )\right )}{24}+\frac {35 \sin \left (f x +e \right )}{16}\right ) \cos \left (f x +e \right )}{8}+\frac {35 f x}{128}+\frac {35 e}{128}\right )+8 c^{6} a^{3} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+\frac {6 c^{6} a^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-6 c^{6} a^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {8 c^{6} a^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+3 c^{6} a^{3} \cos \left (f x +e \right )+c^{6} a^{3} \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 301, normalized size = 1.67 \[ -\frac {1024 \, {\left (35 \, \cos \left (f x + e\right )^{9} - 180 \, \cos \left (f x + e\right )^{7} + 378 \, \cos \left (f x + e\right )^{5} - 420 \, \cos \left (f x + e\right )^{3} + 315 \, \cos \left (f x + e\right )\right )} a^{3} c^{6} - 129024 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{3} c^{6} - 860160 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} c^{6} + 315 \, {\left (128 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 840 \, f x + 840 \, e + 3 \, \sin \left (8 \, f x + 8 \, e\right ) + 168 \, \sin \left (4 \, f x + 4 \, e\right ) - 768 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{6} - 13440 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{6} + 60480 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{6} - 322560 \, {\left (f x + e\right )} a^{3} c^{6} - 967680 \, a^{3} c^{6} \cos \left (f x + e\right )}{322560 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.31, size = 403, normalized size = 2.24 \[ \frac {a^3\,c^6\,\left (3465\,e+9198\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+18432\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+79716\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+138240\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-4284\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+387072\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+176148\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+290304\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+645120\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}-176148\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}+236544\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}+4284\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{13}+129024\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}-79716\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{15}+48384\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{16}-9198\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{17}+3465\,f\,x+31185\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (e+f\,x\right )+124740\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (e+f\,x\right )+291060\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (e+f\,x\right )+436590\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (e+f\,x\right )+436590\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,\left (e+f\,x\right )+291060\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}\,\left (e+f\,x\right )+124740\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}\,\left (e+f\,x\right )+31185\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{16}\,\left (e+f\,x\right )+3465\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{18}\,\left (e+f\,x\right )+7424\right )}{8064\,f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 26.39, size = 838, normalized size = 4.66 \[ \begin {cases} - \frac {105 a^{3} c^{6} x \sin ^{8}{\left (e + f x \right )}}{128} - \frac {105 a^{3} c^{6} x \sin ^{6}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{32} + \frac {5 a^{3} c^{6} x \sin ^{6}{\left (e + f x \right )}}{2} - \frac {315 a^{3} c^{6} x \sin ^{4}{\left (e + f x \right )} \cos ^{4}{\left (e + f x \right )}}{64} + \frac {15 a^{3} c^{6} x \sin ^{4}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{2} - \frac {9 a^{3} c^{6} x \sin ^{4}{\left (e + f x \right )}}{4} - \frac {105 a^{3} c^{6} x \sin ^{2}{\left (e + f x \right )} \cos ^{6}{\left (e + f x \right )}}{32} + \frac {15 a^{3} c^{6} x \sin ^{2}{\left (e + f x \right )} \cos ^{4}{\left (e + f x \right )}}{2} - \frac {9 a^{3} c^{6} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{2} - \frac {105 a^{3} c^{6} x \cos ^{8}{\left (e + f x \right )}}{128} + \frac {5 a^{3} c^{6} x \cos ^{6}{\left (e + f x \right )}}{2} - \frac {9 a^{3} c^{6} x \cos ^{4}{\left (e + f x \right )}}{4} + a^{3} c^{6} x - \frac {a^{3} c^{6} \sin ^{8}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {279 a^{3} c^{6} \sin ^{7}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{128 f} - \frac {8 a^{3} c^{6} \sin ^{6}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {511 a^{3} c^{6} \sin ^{5}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{128 f} - \frac {11 a^{3} c^{6} \sin ^{5}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {16 a^{3} c^{6} \sin ^{4}{\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{5 f} + \frac {6 a^{3} c^{6} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {385 a^{3} c^{6} \sin ^{3}{\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{128 f} - \frac {20 a^{3} c^{6} \sin ^{3}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {15 a^{3} c^{6} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{4 f} - \frac {64 a^{3} c^{6} \sin ^{2}{\left (e + f x \right )} \cos ^{7}{\left (e + f x \right )}}{35 f} + \frac {8 a^{3} c^{6} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} - \frac {8 a^{3} c^{6} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {105 a^{3} c^{6} \sin {\left (e + f x \right )} \cos ^{7}{\left (e + f x \right )}}{128 f} - \frac {5 a^{3} c^{6} \sin {\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{2 f} + \frac {9 a^{3} c^{6} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{4 f} - \frac {128 a^{3} c^{6} \cos ^{9}{\left (e + f x \right )}}{315 f} + \frac {16 a^{3} c^{6} \cos ^{5}{\left (e + f x \right )}}{5 f} - \frac {16 a^{3} c^{6} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {3 a^{3} c^{6} \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (a \sin {\relax (e )} + a\right )^{3} \left (- c \sin {\relax (e )} + c\right )^{6} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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