Optimal. Leaf size=79 \[ -\frac {a (c+d) \cos ^3(e+f x)}{3 f}+\frac {a (4 c+d) \sin (e+f x) \cos (e+f x)}{8 f}+\frac {1}{8} a x (4 c+d)-\frac {a d \sin (e+f x) \cos ^3(e+f x)}{4 f} \]
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Rubi [A] time = 0.09, antiderivative size = 84, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2860, 2669, 2635, 8} \[ -\frac {a (4 c+d) \cos ^3(e+f x)}{12 f}+\frac {a (4 c+d) \sin (e+f x) \cos (e+f x)}{8 f}+\frac {1}{8} a x (4 c+d)-\frac {d \cos ^3(e+f x) (a \sin (e+f x)+a)}{4 f} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2669
Rule 2860
Rubi steps
\begin {align*} \int \cos ^2(e+f x) (a+a \sin (e+f x)) (c+d \sin (e+f x)) \, dx &=-\frac {d \cos ^3(e+f x) (a+a \sin (e+f x))}{4 f}+\frac {1}{4} (4 c+d) \int \cos ^2(e+f x) (a+a \sin (e+f x)) \, dx\\ &=-\frac {a (4 c+d) \cos ^3(e+f x)}{12 f}-\frac {d \cos ^3(e+f x) (a+a \sin (e+f x))}{4 f}+\frac {1}{4} (a (4 c+d)) \int \cos ^2(e+f x) \, dx\\ &=-\frac {a (4 c+d) \cos ^3(e+f x)}{12 f}+\frac {a (4 c+d) \cos (e+f x) \sin (e+f x)}{8 f}-\frac {d \cos ^3(e+f x) (a+a \sin (e+f x))}{4 f}+\frac {1}{8} (a (4 c+d)) \int 1 \, dx\\ &=\frac {1}{8} a (4 c+d) x-\frac {a (4 c+d) \cos ^3(e+f x)}{12 f}+\frac {a (4 c+d) \cos (e+f x) \sin (e+f x)}{8 f}-\frac {d \cos ^3(e+f x) (a+a \sin (e+f x))}{4 f}\\ \end {align*}
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Mathematica [A] time = 0.60, size = 64, normalized size = 0.81 \[ -\frac {a (24 (c+d) \cos (e+f x)+8 (c+d) \cos (3 (e+f x))-12 f x (4 c+d)-24 c \sin (2 (e+f x))+3 d \sin (4 (e+f x)))}{96 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 72, normalized size = 0.91 \[ -\frac {8 \, {\left (a c + a d\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (4 \, a c + a d\right )} f x + 3 \, {\left (2 \, a d \cos \left (f x + e\right )^{3} - {\left (4 \, a c + a d\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{24 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 87, normalized size = 1.10 \[ \frac {1}{8} \, {\left (4 \, a c + a d\right )} x - \frac {a d \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} + \frac {a c \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} - \frac {{\left (a c + a d\right )} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} - \frac {{\left (a c + a d\right )} \cos \left (f x + e\right )}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 96, normalized size = 1.22 \[ \frac {d a \left (-\frac {\sin \left (f x +e \right ) \left (\cos ^{3}\left (f x +e \right )\right )}{4}+\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{8}+\frac {f x}{8}+\frac {e}{8}\right )-\frac {\left (\cos ^{3}\left (f x +e \right )\right ) a c}{3}-\frac {d a \left (\cos ^{3}\left (f x +e \right )\right )}{3}+c a \left (\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 74, normalized size = 0.94 \[ -\frac {32 \, a c \cos \left (f x + e\right )^{3} + 32 \, a d \cos \left (f x + e\right )^{3} - 24 \, {\left (2 \, f x + 2 \, e + \sin \left (2 \, f x + 2 \, e\right )\right )} a c - 3 \, {\left (4 \, f x + 4 \, e - \sin \left (4 \, f x + 4 \, e\right )\right )} a d}{96 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.24, size = 276, normalized size = 3.49 \[ \frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (4\,c+d\right )}{4\,\left (a\,c+\frac {a\,d}{4}\right )}\right )\,\left (4\,c+d\right )}{4\,f}-\frac {\left (a\,c-\frac {a\,d}{4}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+\left (2\,a\,c+2\,a\,d\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+\left (a\,c+\frac {7\,a\,d}{4}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+\left (2\,a\,c+2\,a\,d\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+\left (-a\,c-\frac {7\,a\,d}{4}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+\left (\frac {2\,a\,c}{3}+\frac {2\,a\,d}{3}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+\left (\frac {a\,d}{4}-a\,c\right )\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+\frac {2\,a\,c}{3}+\frac {2\,a\,d}{3}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}-\frac {a\,\left (4\,c+d\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {f\,x}{2}\right )}{4\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.99, size = 199, normalized size = 2.52 \[ \begin {cases} \frac {a c x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {a c x \cos ^{2}{\left (e + f x \right )}}{2} + \frac {a c \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {a c \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {a d x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {a d x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {a d x \cos ^{4}{\left (e + f x \right )}}{8} + \frac {a d \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {a d \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {a d \cos ^{3}{\left (e + f x \right )}}{3 f} & \text {for}\: f \neq 0 \\x \left (c + d \sin {\relax (e )}\right ) \left (a \sin {\relax (e )} + a\right ) \cos ^{2}{\relax (e )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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