Optimal. Leaf size=85 \[ \frac {a^2 c^3 \cos ^5(e+f x)}{5 f}+\frac {a^2 c^3 \sin (e+f x) \cos ^3(e+f x)}{4 f}+\frac {3 a^2 c^3 \sin (e+f x) \cos (e+f x)}{8 f}+\frac {3}{8} a^2 c^3 x \]
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Rubi [A] time = 0.10, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2736, 2669, 2635, 8} \[ \frac {a^2 c^3 \cos ^5(e+f x)}{5 f}+\frac {a^2 c^3 \sin (e+f x) \cos ^3(e+f x)}{4 f}+\frac {3 a^2 c^3 \sin (e+f x) \cos (e+f x)}{8 f}+\frac {3}{8} a^2 c^3 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2669
Rule 2736
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3 \, dx &=\left (a^2 c^2\right ) \int \cos ^4(e+f x) (c-c \sin (e+f x)) \, dx\\ &=\frac {a^2 c^3 \cos ^5(e+f x)}{5 f}+\left (a^2 c^3\right ) \int \cos ^4(e+f x) \, dx\\ &=\frac {a^2 c^3 \cos ^5(e+f x)}{5 f}+\frac {a^2 c^3 \cos ^3(e+f x) \sin (e+f x)}{4 f}+\frac {1}{4} \left (3 a^2 c^3\right ) \int \cos ^2(e+f x) \, dx\\ &=\frac {a^2 c^3 \cos ^5(e+f x)}{5 f}+\frac {3 a^2 c^3 \cos (e+f x) \sin (e+f x)}{8 f}+\frac {a^2 c^3 \cos ^3(e+f x) \sin (e+f x)}{4 f}+\frac {1}{8} \left (3 a^2 c^3\right ) \int 1 \, dx\\ &=\frac {3}{8} a^2 c^3 x+\frac {a^2 c^3 \cos ^5(e+f x)}{5 f}+\frac {3 a^2 c^3 \cos (e+f x) \sin (e+f x)}{8 f}+\frac {a^2 c^3 \cos ^3(e+f x) \sin (e+f x)}{4 f}\\ \end {align*}
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Mathematica [A] time = 1.64, size = 69, normalized size = 0.81 \[ \frac {a^2 c^3 (40 \sin (2 (e+f x))+5 \sin (4 (e+f x))+20 \cos (e+f x)+10 \cos (3 (e+f x))+2 \cos (5 (e+f x))+60 e+60 f x)}{160 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 71, normalized size = 0.84 \[ \frac {8 \, a^{2} c^{3} \cos \left (f x + e\right )^{5} + 15 \, a^{2} c^{3} f x + 5 \, {\left (2 \, a^{2} c^{3} \cos \left (f x + e\right )^{3} + 3 \, a^{2} c^{3} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{40 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 112, normalized size = 1.32 \[ \frac {3}{8} \, a^{2} c^{3} x + \frac {a^{2} c^{3} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac {a^{2} c^{3} \cos \left (3 \, f x + 3 \, e\right )}{16 \, f} + \frac {a^{2} c^{3} \cos \left (f x + e\right )}{8 \, f} + \frac {a^{2} c^{3} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} + \frac {a^{2} c^{3} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.29, size = 159, normalized size = 1.87 \[ \frac {\frac {c^{3} a^{2} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+c^{3} a^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {2 c^{3} a^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-2 c^{3} a^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+c^{3} a^{2} \cos \left (f x +e \right )+a^{2} c^{3} \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.00, size = 158, normalized size = 1.86 \[ \frac {32 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{2} c^{3} + 320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c^{3} + 15 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{3} - 240 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{3} + 480 \, {\left (f x + e\right )} a^{2} c^{3} + 480 \, a^{2} c^{3} \cos \left (f x + e\right )}{480 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.00, size = 220, normalized size = 2.59 \[ \frac {3\,a^2\,c^3\,x}{8}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (\frac {a^2\,c^3\,\left (75\,e+75\,f\,x+80\right )}{40}-\frac {15\,a^2\,c^3\,\left (e+f\,x\right )}{8}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {a^2\,c^3\,\left (150\,e+150\,f\,x+160\right )}{40}-\frac {15\,a^2\,c^3\,\left (e+f\,x\right )}{4}\right )+\frac {a^2\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{2}-\frac {a^2\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{2}-\frac {5\,a^2\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{4}+\frac {a^2\,c^3\,\left (15\,e+15\,f\,x+16\right )}{40}+\frac {5\,a^2\,c^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{4}-\frac {3\,a^2\,c^3\,\left (e+f\,x\right )}{8}}{f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.04, size = 340, normalized size = 4.00 \[ \begin {cases} \frac {3 a^{2} c^{3} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {3 a^{2} c^{3} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} - a^{2} c^{3} x \sin ^{2}{\left (e + f x \right )} + \frac {3 a^{2} c^{3} x \cos ^{4}{\left (e + f x \right )}}{8} - a^{2} c^{3} x \cos ^{2}{\left (e + f x \right )} + a^{2} c^{3} x + \frac {a^{2} c^{3} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {5 a^{2} c^{3} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} + \frac {4 a^{2} c^{3} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {2 a^{2} c^{3} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {3 a^{2} c^{3} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} + \frac {a^{2} c^{3} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {8 a^{2} c^{3} \cos ^{5}{\left (e + f x \right )}}{15 f} - \frac {4 a^{2} c^{3} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {a^{2} c^{3} \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (a \sin {\relax (e )} + a\right )^{2} \left (- c \sin {\relax (e )} + c\right )^{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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