Optimal. Leaf size=233 \[ \frac {a^2 \left (c^2-10 c d-12 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{20 d f}+\frac {3}{8} a^2 x (2 c+d) \left (2 c^2+3 c d+2 d^2\right )+\frac {a^2 \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right ) \sin (e+f x) \cos (e+f x)}{40 f}+\frac {a^2 \left (c^4-10 c^3 d-44 c^2 d^2-40 c d^3-12 d^4\right ) \cos (e+f x)}{10 d f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}+\frac {a^2 (c-10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f} \]
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Rubi [A] time = 0.31, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2763, 2753, 2734} \[ \frac {a^2 \left (-44 c^2 d^2-10 c^3 d+c^4-40 c d^3-12 d^4\right ) \cos (e+f x)}{10 d f}+\frac {a^2 \left (c^2-10 c d-12 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{20 d f}+\frac {a^2 \left (-20 c^2 d+2 c^3-57 c d^2-30 d^3\right ) \sin (e+f x) \cos (e+f x)}{40 f}+\frac {3}{8} a^2 x (2 c+d) \left (2 c^2+3 c d+2 d^2\right )-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}+\frac {a^2 (c-10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 2753
Rule 2763
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3 \, dx &=-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}+\frac {\int \left (9 a^2 d-a^2 (c-10 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^3 \, dx}{5 d}\\ &=\frac {a^2 (c-10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}+\frac {\int (c+d \sin (e+f x))^2 \left (3 a^2 d (11 c+10 d)-3 a^2 \left (c^2-10 c d-12 d^2\right ) \sin (e+f x)\right ) \, dx}{20 d}\\ &=\frac {a^2 \left (c^2-10 c d-12 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{20 d f}+\frac {a^2 (c-10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}+\frac {\int (c+d \sin (e+f x)) \left (3 a^2 d \left (31 c^2+50 c d+24 d^2\right )-3 a^2 \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right ) \sin (e+f x)\right ) \, dx}{60 d}\\ &=\frac {3}{8} a^2 (2 c+d) \left (2 c^2+3 c d+2 d^2\right ) x+\frac {a^2 \left (c^4-10 c^3 d-44 c^2 d^2-40 c d^3-12 d^4\right ) \cos (e+f x)}{10 d f}+\frac {a^2 \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right ) \cos (e+f x) \sin (e+f x)}{40 f}+\frac {a^2 \left (c^2-10 c d-12 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{20 d f}+\frac {a^2 (c-10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}\\ \end {align*}
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Mathematica [A] time = 0.94, size = 204, normalized size = 0.88 \[ -\frac {a^2 \cos (e+f x) \left (30 \left (4 c^3+8 c^2 d+7 c d^2+2 d^3\right ) \sin ^{-1}\left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )+\sqrt {\cos ^2(e+f x)} \left (8 d \left (5 c^2+10 c d+3 d^2\right ) \sin ^2(e+f x)+5 \left (4 c^3+24 c^2 d+21 c d^2+6 d^3\right ) \sin (e+f x)+8 \left (10 c^3+25 c^2 d+20 c d^2+6 d^3\right )+10 d^2 (3 c+2 d) \sin ^3(e+f x)+8 d^3 \sin ^4(e+f x)\right )\right )}{40 f \sqrt {\cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 217, normalized size = 0.93 \[ -\frac {8 \, a^{2} d^{3} \cos \left (f x + e\right )^{5} - 40 \, {\left (a^{2} c^{2} d + 2 \, a^{2} c d^{2} + a^{2} d^{3}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (4 \, a^{2} c^{3} + 8 \, a^{2} c^{2} d + 7 \, a^{2} c d^{2} + 2 \, a^{2} d^{3}\right )} f x + 80 \, {\left (a^{2} c^{3} + 3 \, a^{2} c^{2} d + 3 \, a^{2} c d^{2} + a^{2} d^{3}\right )} \cos \left (f x + e\right ) - 5 \, {\left (2 \, {\left (3 \, a^{2} c d^{2} + 2 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{3} - {\left (4 \, a^{2} c^{3} + 24 \, a^{2} c^{2} d + 27 \, a^{2} c d^{2} + 10 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{40 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 324, normalized size = 1.39 \[ -\frac {a^{2} d^{3} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac {a^{2} d^{3} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} - \frac {3 \, a^{2} c d^{2} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} + \frac {1}{8} \, {\left (4 \, a^{2} c^{3} + 24 \, a^{2} c^{2} d + 9 \, a^{2} c d^{2} + 6 \, a^{2} d^{3}\right )} x + \frac {1}{2} \, {\left (2 \, a^{2} c^{3} + 3 \, a^{2} c d^{2}\right )} x + \frac {{\left (12 \, a^{2} c^{2} d + 24 \, a^{2} c d^{2} + 5 \, a^{2} d^{3}\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {{\left (16 \, a^{2} c^{3} + 18 \, a^{2} c^{2} d + 36 \, a^{2} c d^{2} + 5 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )}{8 \, f} - \frac {3 \, {\left (4 \, a^{2} c^{2} d + a^{2} d^{3}\right )} \cos \left (f x + e\right )}{4 \, f} + \frac {{\left (3 \, a^{2} c d^{2} + 2 \, a^{2} d^{3}\right )} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} - \frac {{\left (a^{2} c^{3} + 6 \, a^{2} c^{2} d + 3 \, a^{2} c d^{2} + 2 \, a^{2} d^{3}\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 329, normalized size = 1.41 \[ \frac {a^{2} c^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-a^{2} c^{2} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 a^{2} c \,d^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {a^{2} d^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-2 a^{2} c^{3} \cos \left (f x +e \right )+6 a^{2} c^{2} d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-2 a^{2} c \,d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+2 a^{2} d^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+a^{2} c^{3} \left (f x +e \right )-3 a^{2} c^{2} d \cos \left (f x +e \right )+3 a^{2} c \,d^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {a^{2} d^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 318, normalized size = 1.36 \[ \frac {120 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{3} + 480 \, {\left (f x + e\right )} a^{2} c^{3} + 480 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c^{2} d + 720 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{2} d + 960 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c d^{2} + 45 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c d^{2} + 360 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c d^{2} - 32 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{2} d^{3} + 160 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} d^{3} + 30 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} d^{3} - 960 \, a^{2} c^{3} \cos \left (f x + e\right ) - 1440 \, a^{2} c^{2} d \cos \left (f x + e\right )}{480 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.42, size = 611, normalized size = 2.62 \[ \frac {3\,a^2\,\mathrm {atan}\left (\frac {3\,a^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,c+d\right )\,\left (2\,c^2+3\,c\,d+2\,d^2\right )}{4\,\left (3\,a^2\,c^3+6\,a^2\,c^2\,d+\frac {21\,a^2\,c\,d^2}{4}+\frac {3\,a^2\,d^3}{2}\right )}\right )\,\left (2\,c+d\right )\,\left (2\,c^2+3\,c\,d+2\,d^2\right )}{4\,f}-\frac {3\,a^2\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {f\,x}{2}\right )\,\left (4\,c^3+8\,c^2\,d+7\,c\,d^2+2\,d^3\right )}{4\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (4\,a^2\,c^3+6\,d\,a^2\,c^2\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (a^2\,c^3+6\,a^2\,c^2\,d+\frac {21\,a^2\,c\,d^2}{4}+\frac {3\,a^2\,d^3}{2}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (2\,a^2\,c^3+12\,a^2\,c^2\,d+\frac {33\,a^2\,c\,d^2}{2}+7\,a^2\,d^3\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (2\,a^2\,c^3+12\,a^2\,c^2\,d+\frac {33\,a^2\,c\,d^2}{2}+7\,a^2\,d^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (16\,a^2\,c^3+36\,a^2\,c^2\,d+24\,a^2\,c\,d^2+4\,a^2\,d^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (16\,a^2\,c^3+44\,a^2\,c^2\,d+40\,a^2\,c\,d^2+12\,a^2\,d^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (24\,a^2\,c^3+64\,a^2\,c^2\,d+56\,a^2\,c\,d^2+20\,a^2\,d^3\right )+4\,a^2\,c^3+\frac {12\,a^2\,d^3}{5}+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (a^2\,c^3+6\,a^2\,c^2\,d+\frac {21\,a^2\,c\,d^2}{4}+\frac {3\,a^2\,d^3}{2}\right )+8\,a^2\,c\,d^2+10\,a^2\,c^2\,d}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.89, size = 729, normalized size = 3.13 \[ \begin {cases} \frac {a^{2} c^{3} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {a^{2} c^{3} x \cos ^{2}{\left (e + f x \right )}}{2} + a^{2} c^{3} x - \frac {a^{2} c^{3} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {2 a^{2} c^{3} \cos {\left (e + f x \right )}}{f} + 3 a^{2} c^{2} d x \sin ^{2}{\left (e + f x \right )} + 3 a^{2} c^{2} d x \cos ^{2}{\left (e + f x \right )} - \frac {3 a^{2} c^{2} d \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {3 a^{2} c^{2} d \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {2 a^{2} c^{2} d \cos ^{3}{\left (e + f x \right )}}{f} - \frac {3 a^{2} c^{2} d \cos {\left (e + f x \right )}}{f} + \frac {9 a^{2} c d^{2} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {9 a^{2} c d^{2} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {3 a^{2} c d^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {9 a^{2} c d^{2} x \cos ^{4}{\left (e + f x \right )}}{8} + \frac {3 a^{2} c d^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {15 a^{2} c d^{2} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {6 a^{2} c d^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {9 a^{2} c d^{2} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {3 a^{2} c d^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {4 a^{2} c d^{2} \cos ^{3}{\left (e + f x \right )}}{f} + \frac {3 a^{2} d^{3} x \sin ^{4}{\left (e + f x \right )}}{4} + \frac {3 a^{2} d^{3} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{2} + \frac {3 a^{2} d^{3} x \cos ^{4}{\left (e + f x \right )}}{4} - \frac {a^{2} d^{3} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {5 a^{2} d^{3} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{4 f} - \frac {4 a^{2} d^{3} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {a^{2} d^{3} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {3 a^{2} d^{3} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{4 f} - \frac {8 a^{2} d^{3} \cos ^{5}{\left (e + f x \right )}}{15 f} - \frac {2 a^{2} d^{3} \cos ^{3}{\left (e + f x \right )}}{3 f} & \text {for}\: f \neq 0 \\x \left (c + d \sin {\relax (e )}\right )^{3} \left (a \sin {\relax (e )} + a\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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