3.3.0 ∫ (A+B sin(e+fx))(c+d sin(e+fx))3 ___________________________________________________

Integrand size = 35, antiderivative size = 228 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx=\frac {d \left (2 A (3 c-2 d) d+B \left (6 c^2-12 c d+7 d^2\right )\right ) x}{2 a^2}+\frac {2 d \left (A \left (c^2+6 c d-5 d^2\right )+B \left (2 c^2-15 c d+8 d^2\right )\right ) \cos (e+f x)}{3 a^2 f}+\frac {d^2 (B (4 c-21 d)+2 A (c+6 d)) \cos (e+f x) \sin (e+f x)}{6 a^2 f}-\frac {(2 B (c-4 d)+A (c+5 d)) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a^2 f (1+\sin (e+f x))}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{3 f (a+a \sin (e+f x))^2} \]

Rubi [A] (veriﬁed)

Time = 0.35 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {3056, 2813} \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx=\frac {2 d \left (A \left (c^2+6 c d-5 d^2\right )+B \left (2 c^2-15 c d+8 d^2\right )\right ) \cos (e+f x)}{3 a^2 f}+\frac {d x \left (2 A d (3 c-2 d)+B \left (6 c^2-12 c d+7 d^2\right )\right )}{2 a^2}+\frac {d^2 (2 A (c+6 d)+B (4 c-21 d)) \sin (e+f x) \cos (e+f x)}{6 a^2 f}-\frac {(A (c+5 d)+2 B (c-4 d)) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a^2 f (\sin (e+f x)+1)}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{3 f (a \sin (e+f x)+a)^2} \]

Rubi steps \begin{align*} \text {integral}& = -\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*}

Mathematica [B] (veriﬁed)

Leaf count is larger than twice the leaf count of optimal. \(547\) vs. \(2(228)=456\).

Time = 2.41 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.40 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (3 \left (8 A d \left (6 c^2+d^2 (5-6 e-6 f x)+3 c d (-4+3 e+3 f x)\right )+B \left (16 c^3+24 c^2 d (-4+3 e+3 f x)-24 c d^2 (-5+6 e+6 f x)+7 d^3 (-7+12 e+12 f x)\right )\right ) \cos \left (\frac {1}{2} (e+f x)\right )-\left (4 A \left (4 c^3+24 c^2 d+d^3 (41-12 e-12 f x)+6 c d^2 (-10+3 e+3 f x)\right )+B \left (32 c^3+24 c^2 d (-10+3 e+3 f x)-12 c d^2 (-41+12 e+12 f x)+d^3 (-239+84 e+84 f x)\right )\right ) \cos \left (\frac {3}{2} (e+f x)\right )+3 \left (d^2 (12 B c+4 A d-5 B d) \cos \left (\frac {5}{2} (e+f x)\right )+B d^3 \cos \left (\frac {7}{2} (e+f x)\right )+2 \left (8 A c^3+8 B c^3+24 A c^2 d-72 B c^2 d-72 A c d^2+108 B c d^2+36 A d^3-50 B d^3+48 B c^2 d e+48 A c d^2 e-96 B c d^2 e-32 A d^3 e+56 B d^3 e+48 B c^2 d f x+48 A c d^2 f x-96 B c d^2 f x-32 A d^3 f x+56 B d^3 f x+d \left (8 A d (3 c (e+f x)-2 d (1+e+f x))+B \left (24 c^2 (e+f x)-48 c d (1+e+f x)+d^2 (27+28 e+28 f x)\right )\right ) \cos (e+f x)+2 d^2 (-6 B c-2 A d+3 B d) \cos (2 (e+f x))+B d^3 \cos (3 (e+f x))\right ) \sin \left (\frac {1}{2} (e+f x)\right )\right )\right )}{48 a^2 f (1+\sin (e+f x))^2} \]

Maple [A] (veriﬁed)

Time = 1.05 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.49


method	result	size

derivativedivides	\(\frac {-\frac {2 \left (A \,c^{3}-3 d^{2} c A +2 A \,d^{3}-3 c^{2} d B +6 d^{2} c B -3 d^{3} B \right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-2 A \,c^{3}+6 c^{2} d A -6 d^{2} c A +2 A \,d^{3}+2 B \,c^{3}-6 c^{2} d B +6 d^{2} c B -2 d^{3} B}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (2 A \,c^{3}-6 c^{2} d A +6 d^{2} c A -2 A \,d^{3}-2 B \,c^{3}+6 c^{2} d B -6 d^{2} c B +2 d^{3} B \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+2 d \left (\frac {\frac {B \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) d^{2}}{2}+\left (-A \,d^{2}-3 c d B +2 d^{2} B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {B \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d^{2}}{2}-A \,d^{2}-3 c d B +2 d^{2} B}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {\left (6 A c d -4 A \,d^{2}+6 B \,c^{2}-12 c d B +7 d^{2} B \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )}{a^{2} f}\)	\(340\)
default	\(\frac {-\frac {2 \left (A \,c^{3}-3 d^{2} c A +2 A \,d^{3}-3 c^{2} d B +6 d^{2} c B -3 d^{3} B \right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-2 A \,c^{3}+6 c^{2} d A -6 d^{2} c A +2 A \,d^{3}+2 B \,c^{3}-6 c^{2} d B +6 d^{2} c B -2 d^{3} B}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (2 A \,c^{3}-6 c^{2} d A +6 d^{2} c A -2 A \,d^{3}-2 B \,c^{3}+6 c^{2} d B -6 d^{2} c B +2 d^{3} B \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+2 d \left (\frac {\frac {B \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) d^{2}}{2}+\left (-A \,d^{2}-3 c d B +2 d^{2} B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {B \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d^{2}}{2}-A \,d^{2}-3 c d B +2 d^{2} B}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {\left (6 A c d -4 A \,d^{2}+6 B \,c^{2}-12 c d B +7 d^{2} B \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )}{a^{2} f}\)	\(340\)
parallelrisch	\(\frac {\left (\left (720 f x A -1260 f x B +696 A -993 B \right ) d^{3}-1080 \left (f x A -2 f x B +\frac {7}{15} A -\frac {29}{15} B \right ) c \,d^{2}-72 c^{2} \left (15 f x B +A +7 B \right ) d +216 \left (A -\frac {B}{9}\right ) c^{3}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+\left (\left (-240 f x A +420 f x B +388 A -619 B \right ) d^{3}+360 c \left (f x A -2 f x B -\frac {23}{15} A +\frac {97}{30} B \right ) d^{2}+264 c^{2} \left (\frac {15}{11} f x B +A -\frac {23}{11} B \right ) d +8 c^{3} \left (A +11 B \right )\right ) \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+\left (\left (720 f x A -1260 f x B -24 A +177 B \right ) d^{3}-1080 \left (f x A -2 f x B -\frac {1}{5} A +\frac {1}{15} B \right ) c \,d^{2}-72 c^{2} \left (15 f x B +A -3 B \right ) d -24 c^{3} \left (A +B \right )\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+\left (\left (240 f x A -420 f x B +612 A -891 B \right ) d^{3}-360 c \left (f x A -2 f x B +\frac {9}{5} A -\frac {51}{10} B \right ) d^{2}+216 \left (-\frac {5}{3} f x B +A -3 B \right ) c^{2} d +72 c^{3} \left (A +B \right )\right ) \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )-60 \left (\left (\left (A -\frac {5 B}{4}\right ) d +3 B c \right ) \cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+\left (\left (-A +\frac {5 B}{4}\right ) d -3 B c \right ) \sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+\frac {d B \left (\cos \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )+\sin \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )\right )}{4}\right ) d^{2}}{120 f \,a^{2} \left (-3 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+\cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )-\sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )-3 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}\)	\(446\)
risch	\(\frac {3 x \,d^{2} A c}{a^{2}}-\frac {2 x \,d^{3} A}{a^{2}}+\frac {3 x d B \,c^{2}}{a^{2}}-\frac {6 x \,d^{2} c B}{a^{2}}+\frac {7 x \,d^{3} B}{2 a^{2}}+\frac {i d^{3} B \,{\mathrm e}^{2 i \left (f x +e \right )}}{8 a^{2} f}-\frac {d^{3} {\mathrm e}^{i \left (f x +e \right )} A}{2 a^{2} f}-\frac {3 d^{2} {\mathrm e}^{i \left (f x +e \right )} B c}{2 a^{2} f}+\frac {d^{3} {\mathrm e}^{i \left (f x +e \right )} B}{a^{2} f}-\frac {d^{3} {\mathrm e}^{-i \left (f x +e \right )} A}{2 a^{2} f}-\frac {3 d^{2} {\mathrm e}^{-i \left (f x +e \right )} B c}{2 a^{2} f}+\frac {d^{3} {\mathrm e}^{-i \left (f x +e \right )} B}{a^{2} f}-\frac {i d^{3} B \,{\mathrm e}^{-2 i \left (f x +e \right )}}{8 a^{2} f}-\frac {2 \left (-6 c^{2} d A +15 d^{2} c A +15 c^{2} d B -24 d^{2} c B +9 i A \,c^{2} d \,{\mathrm e}^{i \left (f x +e \right )}+3 i B \,c^{3} {\mathrm e}^{i \left (f x +e \right )}-27 i B \,c^{2} d \,{\mathrm e}^{i \left (f x +e \right )}-18 A c \,d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-18 B \,c^{2} d \,{\mathrm e}^{2 i \left (f x +e \right )}+11 d^{3} B +27 B c \,d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+9 A \,d^{3} {\mathrm e}^{2 i \left (f x +e \right )}-12 B \,d^{3} {\mathrm e}^{2 i \left (f x +e \right )}-A \,c^{3}-2 B \,c^{3}+3 B \,c^{3} {\mathrm e}^{2 i \left (f x +e \right )}+15 i A \,d^{3} {\mathrm e}^{i \left (f x +e \right )}+45 i B c \,d^{2} {\mathrm e}^{i \left (f x +e \right )}-21 i B \,d^{3} {\mathrm e}^{i \left (f x +e \right )}+3 i A \,c^{3} {\mathrm e}^{i \left (f x +e \right )}-27 i A c \,d^{2} {\mathrm e}^{i \left (f x +e \right )}+9 A \,c^{2} d \,{\mathrm e}^{2 i \left (f x +e \right )}-8 A \,d^{3}\right )}{3 f \,a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3}}\)	\(541\)
norman	\(\text {Expression too large to display}\)	\(1351\)

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 584 vs. \(2 (218) = 436\).

Time = 0.27 (sec) , antiderivative size = 584, normalized size of antiderivative = 2.56 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx=-\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 )}} \]

Sympy [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 14612 vs. \(2 (216) = 432\).

Time = 7.60 (sec) , antiderivative size = 14612, normalized size of antiderivative = 64.09 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx=\text {Too large to display} \]

Maxima [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1382 vs. \(2 (218) = 436\).

Time = 0.36 (sec) , antiderivative size = 1382, normalized size of antiderivative = 6.06 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx=\text {Too large to display} \]

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 472 vs. \(2 (218) = 436\).

Time = 0.32 (sec) , antiderivative size = 472, normalized size of antiderivative = 2.07 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx=\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} \]

Mupad [B] (veriﬁcation not implemented)

Time = 17.17 (sec) , antiderivative size = 663, normalized size of antiderivative = 2.91 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx=\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 )} \]

\(\int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx\) [272]

Optimal result