Optimal. Leaf size=228 \[ \frac {2 d \left (A \left (c^2+6 c d-5 d^2\right )+B \left (2 c^2-15 c d+8 d^2\right )\right ) \cos (e+f x)}{3 a^2 f}+\frac {d x \left (2 A d (3 c-2 d)+B \left (6 c^2-12 c d+7 d^2\right )\right )}{2 a^2}+\frac {d^2 (2 A (c+6 d)+B (4 c-21 d)) \sin (e+f x) \cos (e+f x)}{6 a^2 f}-\frac {(A (c+5 d)+2 B (c-4 d)) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a^2 f (\sin (e+f x)+1)}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{3 f (a \sin (e+f x)+a)^2} \]
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Rubi [A] time = 0.52, antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {2977, 2734} \[ \frac {2 d \left (A \left (c^2+6 c d-5 d^2\right )+B \left (2 c^2-15 c d+8 d^2\right )\right ) \cos (e+f x)}{3 a^2 f}+\frac {d x \left (2 A d (3 c-2 d)+B \left (6 c^2-12 c d+7 d^2\right )\right )}{2 a^2}+\frac {d^2 (2 A (c+6 d)+B (4 c-21 d)) \sin (e+f x) \cos (e+f x)}{6 a^2 f}-\frac {(A (c+5 d)+2 B (c-4 d)) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a^2 f (\sin (e+f x)+1)}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{3 f (a \sin (e+f x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 2977
Rubi steps
\begin {align*} \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx &=-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{3 f (a+a \sin (e+f x))^2}+\frac {\int \frac {(c+d \sin (e+f x))^2 (a (A c+2 B c+3 A d-3 B d)-a (2 A-5 B) d \sin (e+f x))}{a+a \sin (e+f x)} \, dx}{3 a^2}\\ &=-\frac {(2 B (c-4 d)+A (c+5 d)) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a^2 f (1+\sin (e+f x))}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{3 f (a+a \sin (e+f x))^2}+\frac {\int (c+d \sin (e+f x)) \left (a^2 d (9 B c+10 A d-16 B d)-a^2 d (B (4 c-21 d)+2 A (c+6 d)) \sin (e+f x)\right ) \, dx}{3 a^4}\\ &=\frac {d \left (2 A (3 c-2 d) d+B \left (6 c^2-12 c d+7 d^2\right )\right ) x}{2 a^2}+\frac {2 d \left (A \left (c^2+6 c d-5 d^2\right )+B \left (2 c^2-15 c d+8 d^2\right )\right ) \cos (e+f x)}{3 a^2 f}+\frac {d^2 (B (4 c-21 d)+2 A (c+6 d)) \cos (e+f x) \sin (e+f x)}{6 a^2 f}-\frac {(2 B (c-4 d)+A (c+5 d)) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a^2 f (1+\sin (e+f x))}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{3 f (a+a \sin (e+f x))^2}\\ \end {align*}
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Mathematica [B] time = 3.57, size = 547, normalized size = 2.40 \[ \frac {\left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) \left (3 \cos \left (\frac {1}{2} (e+f x)\right ) \left (8 A d \left (6 c^2+3 c d (3 e+3 f x-4)+d^2 (-6 e-6 f x+5)\right )+B \left (16 c^3+24 c^2 d (3 e+3 f x-4)-24 c d^2 (6 e+6 f x-5)+7 d^3 (12 e+12 f x-7)\right )\right )-\cos \left (\frac {3}{2} (e+f x)\right ) \left (4 A \left (4 c^3+24 c^2 d+6 c d^2 (3 e+3 f x-10)+d^3 (-12 e-12 f x+41)\right )+B \left (32 c^3+24 c^2 d (3 e+3 f x-10)-12 c d^2 (12 e+12 f x-41)+d^3 (84 e+84 f x-239)\right )\right )+3 \left (2 \sin \left (\frac {1}{2} (e+f x)\right ) \left (d \cos (e+f x) \left (8 A d (3 c (e+f x)-2 d (e+f x+1))+B \left (24 c^2 (e+f x)-48 c d (e+f x+1)+d^2 (28 e+28 f x+27)\right )\right )+2 d^2 (-2 A d-6 B c+3 B d) \cos (2 (e+f x))+8 A c^3+24 A c^2 d+48 A c d^2 e+48 A c d^2 f x-72 A c d^2-32 A d^3 e-32 A d^3 f x+36 A d^3+8 B c^3+48 B c^2 d e+48 B c^2 d f x-72 B c^2 d-96 B c d^2 e-96 B c d^2 f x+108 B c d^2+B d^3 \cos (3 (e+f x))+56 B d^3 e+56 B d^3 f x-50 B d^3\right )+d^2 (4 A d+12 B c-5 B d) \cos \left (\frac {5}{2} (e+f x)\right )+B d^3 \cos \left (\frac {7}{2} (e+f x)\right )\right )\right )}{48 a^2 f (\sin (e+f x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 584, normalized size = 2.56 \[ -\frac {3 \, B d^{3} \cos \left (f x + e\right )^{4} - 2 \, {\left (A - B\right )} c^{3} + 6 \, {\left (A - B\right )} c^{2} d - 6 \, {\left (A - B\right )} c d^{2} + 2 \, {\left (A - B\right )} d^{3} + 6 \, {\left (3 \, B c d^{2} + {\left (A - B\right )} d^{3}\right )} \cos \left (f x + e\right )^{3} + 6 \, {\left (6 \, B c^{2} d + 6 \, {\left (A - 2 \, B\right )} c d^{2} - {\left (4 \, A - 7 \, B\right )} d^{3}\right )} f x - {\left (2 \, {\left (A + 2 \, B\right )} c^{3} + 6 \, {\left (2 \, A - 5 \, B\right )} c^{2} d - 6 \, {\left (5 \, A - 11 \, B\right )} c d^{2} + {\left (22 \, A - 31 \, B\right )} d^{3} + 3 \, {\left (6 \, B c^{2} d + 6 \, {\left (A - 2 \, B\right )} c d^{2} - {\left (4 \, A - 7 \, B\right )} d^{3}\right )} f x\right )} \cos \left (f x + e\right )^{2} - {\left (2 \, {\left (2 \, A + B\right )} c^{3} + 6 \, {\left (A - 4 \, B\right )} c^{2} d - 6 \, {\left (4 \, A - 13 \, B\right )} c d^{2} + 2 \, {\left (13 \, A - 19 \, B\right )} d^{3} - 3 \, {\left (6 \, B c^{2} d + 6 \, {\left (A - 2 \, B\right )} c d^{2} - {\left (4 \, A - 7 \, B\right )} d^{3}\right )} f x\right )} \cos \left (f x + e\right ) + {\left (3 \, B d^{3} \cos \left (f x + e\right )^{3} + 2 \, {\left (A - B\right )} c^{3} - 6 \, {\left (A - B\right )} c^{2} d + 6 \, {\left (A - B\right )} c d^{2} - 2 \, {\left (A - B\right )} d^{3} + 6 \, {\left (6 \, B c^{2} d + 6 \, {\left (A - 2 \, B\right )} c d^{2} - {\left (4 \, A - 7 \, B\right )} d^{3}\right )} f x - 3 \, {\left (6 \, B c d^{2} + {\left (2 \, A - 3 \, B\right )} d^{3}\right )} \cos \left (f x + e\right )^{2} - {\left (2 \, {\left (A + 2 \, B\right )} c^{3} + 6 \, {\left (2 \, A - 5 \, B\right )} c^{2} d - 6 \, {\left (5 \, A - 14 \, B\right )} c d^{2} + 4 \, {\left (7 \, A - 10 \, B\right )} d^{3} - 3 \, {\left (6 \, B c^{2} d + 6 \, {\left (A - 2 \, B\right )} c d^{2} - {\left (4 \, A - 7 \, B\right )} d^{3}\right )} f x\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{6 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f - {\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} f\right )} \sin \left (f x + e\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 494, normalized size = 2.17 \[ \frac {\frac {3 \, {\left (6 \, B c^{2} d + 6 \, A c d^{2} - 12 \, B c d^{2} - 4 \, A d^{3} + 7 \, B d^{3}\right )} {\left (f x + e\right )}}{a^{2}} + \frac {6 \, {\left (B d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 6 \, B c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, A d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 4 \, B d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - B d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 6 \, B c d^{2} - 2 \, A d^{3} + 4 \, B d^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{2} a^{2}} - \frac {4 \, {\left (3 \, A c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 9 \, B c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 9 \, A c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 18 \, B c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 6 \, A d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 9 \, B d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, A c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, B c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 9 \, A c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 27 \, B c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 27 \, A c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 45 \, B c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 15 \, A d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 21 \, B d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, A c^{3} + B c^{3} + 3 \, A c^{2} d - 12 \, B c^{2} d - 12 \, A c d^{2} + 21 \, B c d^{2} + 7 \, A d^{3} - 10 \, B d^{3}\right )}}{a^{2} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}}}{6 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.43, size = 946, normalized size = 4.15 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.57, size = 1382, normalized size = 6.06 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 16.35, size = 663, normalized size = 2.91 \[ \frac {d\,\mathrm {atan}\left (\frac {d\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (6\,B\,c^2-4\,A\,d^2+7\,B\,d^2+6\,A\,c\,d-12\,B\,c\,d\right )}{7\,B\,d^3-4\,A\,d^3+6\,A\,c\,d^2-12\,B\,c\,d^2+6\,B\,c^2\,d}\right )\,\left (6\,B\,c^2-4\,A\,d^2+7\,B\,d^2+6\,A\,c\,d-12\,B\,c\,d\right )}{a^2\,f}-\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,A\,c^3+16\,A\,d^3+2\,B\,c^3-25\,B\,d^3-18\,A\,c\,d^2+6\,A\,c^2\,d+48\,B\,c\,d^2-18\,B\,c^2\,d\right )+\frac {4\,A\,c^3}{3}+\frac {20\,A\,d^3}{3}+\frac {2\,B\,c^3}{3}-\frac {32\,B\,d^3}{3}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (2\,A\,c^3+4\,A\,d^3-7\,B\,d^3-6\,A\,c\,d^2+12\,B\,c\,d^2-6\,B\,c^2\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (2\,A\,c^3+12\,A\,d^3+2\,B\,c^3-21\,B\,d^3-18\,A\,c\,d^2+6\,A\,c^2\,d+36\,B\,c\,d^2-18\,B\,c^2\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (4\,A\,c^3+28\,A\,d^3+4\,B\,c^3-42\,B\,d^3-36\,A\,c\,d^2+12\,A\,c^2\,d+84\,B\,c\,d^2-36\,B\,c^2\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {16\,A\,c^3}{3}+\frac {56\,A\,d^3}{3}+\frac {2\,B\,c^3}{3}-\frac {98\,B\,d^3}{3}-20\,A\,c\,d^2+2\,A\,c^2\,d+56\,B\,c\,d^2-20\,B\,c^2\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {14\,A\,c^3}{3}+\frac {64\,A\,d^3}{3}+\frac {4\,B\,c^3}{3}-\frac {97\,B\,d^3}{3}-22\,A\,c\,d^2+4\,A\,c^2\,d+64\,B\,c\,d^2-22\,B\,c^2\,d\right )-8\,A\,c\,d^2+2\,A\,c^2\,d+20\,B\,c\,d^2-8\,B\,c^2\,d}{f\,\left (a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+3\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+5\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+7\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+7\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+5\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+3\,a^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+a^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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