Optimal. Leaf size=147 \[ \frac{a^2 c^3 (6 A-B) \cos ^5(e+f x)}{30 f}+\frac{a^2 c^3 (6 A-B) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{a^2 c^3 (6 A-B) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{1}{16} a^2 c^3 x (6 A-B)-\frac{a^2 B \cos ^5(e+f x) \left (c^3-c^3 \sin (e+f x)\right )}{6 f} \]
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Rubi [A] time = 0.216463, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.139, Rules used = {2967, 2860, 2669, 2635, 8} \[ \frac{a^2 c^3 (6 A-B) \cos ^5(e+f x)}{30 f}+\frac{a^2 c^3 (6 A-B) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{a^2 c^3 (6 A-B) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{1}{16} a^2 c^3 x (6 A-B)-\frac{a^2 B \cos ^5(e+f x) \left (c^3-c^3 \sin (e+f x)\right )}{6 f} \]
Antiderivative was successfully verified.
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Rule 2967
Rule 2860
Rule 2669
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx &=\left (a^2 c^2\right ) \int \cos ^4(e+f x) (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx\\ &=-\frac{a^2 B \cos ^5(e+f x) \left (c^3-c^3 \sin (e+f x)\right )}{6 f}+\frac{1}{6} \left (a^2 (6 A-B) c^2\right ) \int \cos ^4(e+f x) (c-c \sin (e+f x)) \, dx\\ &=\frac{a^2 (6 A-B) c^3 \cos ^5(e+f x)}{30 f}-\frac{a^2 B \cos ^5(e+f x) \left (c^3-c^3 \sin (e+f x)\right )}{6 f}+\frac{1}{6} \left (a^2 (6 A-B) c^3\right ) \int \cos ^4(e+f x) \, dx\\ &=\frac{a^2 (6 A-B) c^3 \cos ^5(e+f x)}{30 f}+\frac{a^2 (6 A-B) c^3 \cos ^3(e+f x) \sin (e+f x)}{24 f}-\frac{a^2 B \cos ^5(e+f x) \left (c^3-c^3 \sin (e+f x)\right )}{6 f}+\frac{1}{8} \left (a^2 (6 A-B) c^3\right ) \int \cos ^2(e+f x) \, dx\\ &=\frac{a^2 (6 A-B) c^3 \cos ^5(e+f x)}{30 f}+\frac{a^2 (6 A-B) c^3 \cos (e+f x) \sin (e+f x)}{16 f}+\frac{a^2 (6 A-B) c^3 \cos ^3(e+f x) \sin (e+f x)}{24 f}-\frac{a^2 B \cos ^5(e+f x) \left (c^3-c^3 \sin (e+f x)\right )}{6 f}+\frac{1}{16} \left (a^2 (6 A-B) c^3\right ) \int 1 \, dx\\ &=\frac{1}{16} a^2 (6 A-B) c^3 x+\frac{a^2 (6 A-B) c^3 \cos ^5(e+f x)}{30 f}+\frac{a^2 (6 A-B) c^3 \cos (e+f x) \sin (e+f x)}{16 f}+\frac{a^2 (6 A-B) c^3 \cos ^3(e+f x) \sin (e+f x)}{24 f}-\frac{a^2 B \cos ^5(e+f x) \left (c^3-c^3 \sin (e+f x)\right )}{6 f}\\ \end{align*}
Mathematica [A] time = 1.05085, size = 137, normalized size = 0.93 \[ \frac{a^2 c^3 (120 (A-B) \cos (e+f x)+60 (A-B) \cos (3 (e+f x))+240 A \sin (2 (e+f x))+30 A \sin (4 (e+f x))+12 A \cos (5 (e+f x))+360 A e+360 A f x-15 B \sin (2 (e+f x))+15 B \sin (4 (e+f x))+5 B \sin (6 (e+f x))-12 B \cos (5 (e+f x))-60 B e-60 B f x)}{960 f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.03, size = 365, normalized size = 2.5 \begin{align*}{\frac{1}{f} \left ({\frac{A{a}^{2}{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 ) }+A{a}^{2}{c}^{3} \left ( -{\frac{\cos \left ( fx+e \right ) }{4} \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{3}+{\frac{3\,\sin \left ( fx+e \right ) }{2}} \right ) }+{\frac{3\,fx}{8}}+{\frac{3\,e}{8}} \right ) -{\frac{2\,A{a}^{2}{c}^{3} \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) }{3}}-2\,A{a}^{2}{c}^{3} \left ( -1/2\,\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +1/2\,fx+e/2 \right ) -B{a}^{2}{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{B{a}^{2}{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 ) }+2\,B{a}^{2}{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 ) +{\frac{2\,B{a}^{2}{c}^{3} \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) }{3}}+A{a}^{2}{c}^{3}\cos \left ( fx+e \right ) -B{a}^{2}{c}^{3} \left ( -{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) +A{a}^{2}{c}^{3} \left ( fx+e \right ) -B{a}^{2}{c}^{3}\cos \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.988055, size = 486, normalized size = 3.31 \begin{align*} \frac{64 \,{\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 a^{2} c^{3} + 640 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{2} c^{3} + 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 a^{2} c^{3} - 480 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} c^{3} + 960 \,{\left (f x + e\right )} A a^{2} c^{3} - 64 \,{\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^{2} c^{3} - 640 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{2} c^{3} - 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 )} B a^{2} c^{3} + 60 \,{\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c^{3} - 240 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c^{3} + 960 \, A a^{2} c^{3} \cos \left (f x + e\right ) - 960 \, B a^{2} c^{3} \cos \left (f x + e\right )}{960 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50883, size = 257, normalized size = 1.75 \begin{align*} \frac{48 \,{\left (A - B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{5} + 15 \,{\left (6 \, A - B\right )} a^{2} c^{3} f x + 5 \,{\left (8 \, B a^{2} c^{3} \cos \left (f x + e\right )^{5} + 2 \,{\left (6 \, A - B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{3} + 3 \,{\left (6 \, A - B\right )} a^{2} c^{3} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{240 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 16.9081, size = 910, normalized size = 6.19 \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 = 1.16322, size = 281, normalized size = 1.91 \begin{align*} \frac{B a^{2} c^{3} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac{1}{16} \,{\left (6 \, A a^{2} c^{3} - B a^{2} c^{3}\right )} x + \frac{{\left (A a^{2} c^{3} - B a^{2} c^{3}\right )} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac{{\left (A a^{2} c^{3} - B a^{2} c^{3}\right )} \cos \left (3 \, f x + 3 \, e\right )}{16 \, f} + \frac{{\left (A a^{2} c^{3} - B a^{2} c^{3}\right )} \cos \left (f x + e\right )}{8 \, f} + \frac{{\left (2 \, A a^{2} c^{3} + B a^{2} c^{3}\right )} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} + \frac{{\left (16 \, A a^{2} c^{3} - B a^{2} c^{3}\right )} \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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