Optimal. Leaf size=117 \[ \frac{a^3 A c^3 \sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac{5 a^3 A c^3 \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{5 a^3 A c^3 \sin (e+f x) \cos (e+f x)}{16 f}+\frac{5}{16} a^3 A c^3 x-\frac{a^3 B c^3 \cos ^7(e+f x)}{7 f} \]
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Rubi [A] time = 0.147686, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2967, 2669, 2635, 8} \[ \frac{a^3 A c^3 \sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac{5 a^3 A c^3 \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{5 a^3 A c^3 \sin (e+f x) \cos (e+f x)}{16 f}+\frac{5}{16} a^3 A c^3 x-\frac{a^3 B c^3 \cos ^7(e+f x)}{7 f} \]
Antiderivative was successfully verified.
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Rule 2967
Rule 2669
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx &=\left (a^3 c^3\right ) \int \cos ^6(e+f x) (A+B \sin (e+f x)) \, dx\\ &=-\frac{a^3 B c^3 \cos ^7(e+f x)}{7 f}+\left (a^3 A c^3\right ) \int \cos ^6(e+f x) \, dx\\ &=-\frac{a^3 B c^3 \cos ^7(e+f x)}{7 f}+\frac{a^3 A c^3 \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac{1}{6} \left (5 a^3 A c^3\right ) \int \cos ^4(e+f x) \, dx\\ &=-\frac{a^3 B c^3 \cos ^7(e+f x)}{7 f}+\frac{5 a^3 A c^3 \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac{a^3 A c^3 \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac{1}{8} \left (5 a^3 A c^3\right ) \int \cos ^2(e+f x) \, dx\\ &=-\frac{a^3 B c^3 \cos ^7(e+f x)}{7 f}+\frac{5 a^3 A c^3 \cos (e+f x) \sin (e+f x)}{16 f}+\frac{5 a^3 A c^3 \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac{a^3 A c^3 \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac{1}{16} \left (5 a^3 A c^3\right ) \int 1 \, dx\\ &=\frac{5}{16} a^3 A c^3 x-\frac{a^3 B c^3 \cos ^7(e+f x)}{7 f}+\frac{5 a^3 A c^3 \cos (e+f x) \sin (e+f x)}{16 f}+\frac{5 a^3 A c^3 \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac{a^3 A c^3 \cos ^5(e+f x) \sin (e+f x)}{6 f}\\ \end{align*}
Mathematica [A] time = 0.225704, size = 64, normalized size = 0.55 \[ \frac{a^3 c^3 \left (7 A (45 \sin (2 (e+f x))+9 \sin (4 (e+f x))+\sin (6 (e+f x))+60 e+60 f x)-192 B \cos ^7(e+f x)\right )}{1344 f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.024, size = 263, normalized size = 2.3 \begin{align*}{\frac{1}{f} \left ({\frac{B{a}^{3}{c}^{3}\cos \left ( fx+e \right ) }{7} \left ({\frac{16}{5}}+ \left ( \sin \left ( fx+e \right ) \right ) ^{6}+{\frac{6\, \left ( \sin \left ( fx+e \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{5}} \right ) }-A{a}^{3}{c}^{3} \left ( -{\frac{\cos \left ( fx+e \right ) }{6} \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{5}+{\frac{5\, \left ( \sin \left ( fx+e \right ) \right ) ^{3}}{4}}+{\frac{15\,\sin \left ( fx+e \right ) }{8}} \right ) }+{\frac{5\,fx}{16}}+{\frac{5\,e}{16}} \right ) -{\frac{3\,B{a}^{3}{c}^{3}\cos \left ( fx+e \right ) }{5} \left ({\frac{8}{3}}+ \left ( \sin \left ( fx+e \right ) \right ) ^{4}+{\frac{4\, \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{3}} \right ) }+3\,A{a}^{3}{c}^{3} \left ( -1/4\, \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{3}+3/2\,\sin \left ( fx+e \right ) \right ) \cos \left ( fx+e \right ) +3/8\,fx+3/8\,e \right ) +B{a}^{3}{c}^{3} \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) -3\,A{a}^{3}{c}^{3} \left ( -1/2\,\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +1/2\,fx+e/2 \right ) -B{a}^{3}{c}^{3}\cos \left ( fx+e \right ) +A{a}^{3}{c}^{3} \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.992634, size = 356, normalized size = 3.04 \begin{align*} -\frac{35 \,{\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 a^{3} c^{3} - 630 \,{\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 a^{3} c^{3} + 5040 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} c^{3} - 6720 \,{\left (f x + e\right )} A a^{3} c^{3} + 192 \,{\left (5 \, \cos \left (f x + e\right )^{7} - 21 \, \cos \left (f x + e\right )^{5} + 35 \, \cos \left (f x + e\right )^{3} - 35 \, \cos \left (f x + e\right )\right )} B a^{3} c^{3} + 1344 \,{\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 )} B a^{3} c^{3} + 6720 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{3} c^{3} + 6720 \, B a^{3} c^{3} \cos \left (f x + e\right )}{6720 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45975, size = 221, normalized size = 1.89 \begin{align*} -\frac{48 \, B a^{3} c^{3} \cos \left (f x + e\right )^{7} - 105 \, A a^{3} c^{3} f x - 7 \,{\left (8 \, A a^{3} c^{3} \cos \left (f x + e\right )^{5} + 10 \, A a^{3} c^{3} \cos \left (f x + e\right )^{3} + 15 \, A a^{3} c^{3} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{336 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 15.9223, size = 682, normalized size = 5.83 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30246, size = 219, normalized size = 1.87 \begin{align*} \frac{5}{16} \, A a^{3} c^{3} x - \frac{B a^{3} c^{3} \cos \left (7 \, f x + 7 \, e\right )}{448 \, f} - \frac{B a^{3} c^{3} \cos \left (5 \, f x + 5 \, e\right )}{64 \, f} - \frac{3 \, B a^{3} c^{3} \cos \left (3 \, f x + 3 \, e\right )}{64 \, f} - \frac{5 \, B a^{3} c^{3} \cos \left (f x + e\right )}{64 \, f} + \frac{A a^{3} c^{3} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac{3 \, A a^{3} c^{3} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} + \frac{15 \, A a^{3} c^{3} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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