Optimal. Leaf size=89 \[ \frac {a^2 A c^2 \sin (e+f x) \cos ^3(e+f x)}{4 f}+\frac {3 a^2 A c^2 \sin (e+f x) \cos (e+f x)}{8 f}+\frac {3}{8} a^2 A c^2 x-\frac {a^2 B c^2 \cos ^5(e+f x)}{5 f} \]
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Rubi [A] time = 0.14, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2967, 2669, 2635, 8} \[ \frac {a^2 A c^2 \sin (e+f x) \cos ^3(e+f x)}{4 f}+\frac {3 a^2 A c^2 \sin (e+f x) \cos (e+f x)}{8 f}+\frac {3}{8} a^2 A c^2 x-\frac {a^2 B c^2 \cos ^5(e+f x)}{5 f} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2669
Rule 2967
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx &=\left (a^2 c^2\right ) \int \cos ^4(e+f x) (A+B \sin (e+f x)) \, dx\\ &=-\frac {a^2 B c^2 \cos ^5(e+f x)}{5 f}+\left (a^2 A c^2\right ) \int \cos ^4(e+f x) \, dx\\ &=-\frac {a^2 B c^2 \cos ^5(e+f x)}{5 f}+\frac {a^2 A c^2 \cos ^3(e+f x) \sin (e+f x)}{4 f}+\frac {1}{4} \left (3 a^2 A c^2\right ) \int \cos ^2(e+f x) \, dx\\ &=-\frac {a^2 B c^2 \cos ^5(e+f x)}{5 f}+\frac {3 a^2 A c^2 \cos (e+f x) \sin (e+f x)}{8 f}+\frac {a^2 A c^2 \cos ^3(e+f x) \sin (e+f x)}{4 f}+\frac {1}{8} \left (3 a^2 A c^2\right ) \int 1 \, dx\\ &=\frac {3}{8} a^2 A c^2 x-\frac {a^2 B c^2 \cos ^5(e+f x)}{5 f}+\frac {3 a^2 A c^2 \cos (e+f x) \sin (e+f x)}{8 f}+\frac {a^2 A c^2 \cos ^3(e+f x) \sin (e+f x)}{4 f}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 54, normalized size = 0.61 \[ \frac {a^2 c^2 \left (5 A (12 (e+f x)+8 \sin (2 (e+f x))+\sin (4 (e+f x)))-32 B \cos ^5(e+f x)\right )}{160 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 75, normalized size = 0.84 \[ -\frac {8 \, B a^{2} c^{2} \cos \left (f x + e\right )^{5} - 15 \, A a^{2} c^{2} f x - 5 \, {\left (2 \, A a^{2} c^{2} \cos \left (f x + e\right )^{3} + 3 \, A a^{2} c^{2} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{40 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 118, normalized size = 1.33 \[ \frac {3}{8} \, A a^{2} c^{2} x - \frac {B a^{2} c^{2} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} - \frac {B a^{2} c^{2} \cos \left (3 \, f x + 3 \, e\right )}{16 \, f} - \frac {B a^{2} c^{2} \cos \left (f x + e\right )}{8 \, f} + \frac {A a^{2} c^{2} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} + \frac {A a^{2} c^{2} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.48, size = 166, normalized size = 1.87 \[ \frac {-\frac {B \,a^{2} c^{2} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+a^{2} A \,c^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+\frac {2 B \,a^{2} c^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-2 a^{2} A \,c^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B \,a^{2} c^{2} \cos \left (f x +e \right )+a^{2} A \,c^{2} \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 164, normalized size = 1.84 \[ \frac {15 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} c^{2} - 240 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} c^{2} + 480 \, {\left (f x + e\right )} A a^{2} c^{2} - 32 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} B a^{2} c^{2} - 320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{2} c^{2} - 480 \, B a^{2} c^{2} \cos \left (f x + e\right )}{480 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.21, size = 238, normalized size = 2.67 \[ \frac {3\,A\,a^2\,c^2\,x}{8}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (\frac {a^2\,c^2\,\left (80\,B-75\,A\,\left (e+f\,x\right )\right )}{40}+\frac {15\,A\,a^2\,c^2\,\left (e+f\,x\right )}{8}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {a^2\,c^2\,\left (160\,B-150\,A\,\left (e+f\,x\right )\right )}{40}+\frac {15\,A\,a^2\,c^2\,\left (e+f\,x\right )}{4}\right )+\frac {a^2\,c^2\,\left (16\,B-15\,A\,\left (e+f\,x\right )\right )}{40}+\frac {3\,A\,a^2\,c^2\,\left (e+f\,x\right )}{8}-\frac {A\,a^2\,c^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{2}+\frac {A\,a^2\,c^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{2}+\frac {5\,A\,a^2\,c^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{4}-\frac {5\,A\,a^2\,c^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{4}}{f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.70, size = 372, normalized size = 4.18 \[ \begin {cases} \frac {3 A a^{2} c^{2} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {3 A a^{2} c^{2} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} - A a^{2} c^{2} x \sin ^{2}{\left (e + f x \right )} + \frac {3 A a^{2} c^{2} x \cos ^{4}{\left (e + f x \right )}}{8} - A a^{2} c^{2} x \cos ^{2}{\left (e + f x \right )} + A a^{2} c^{2} x - \frac {5 A a^{2} c^{2} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {3 A a^{2} c^{2} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} + \frac {A a^{2} c^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {B a^{2} c^{2} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {4 B a^{2} c^{2} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {2 B a^{2} c^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {8 B a^{2} c^{2} \cos ^{5}{\left (e + f x \right )}}{15 f} + \frac {4 B a^{2} c^{2} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {B a^{2} c^{2} \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (A + B \sin {\relax (e )}\right ) \left (a \sin {\relax (e )} + a\right )^{2} \left (- c \sin {\relax (e )} + c\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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