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

Integrand size = 35, antiderivative size = 220 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{a+a \sin (e+f x)} \, dx=\frac {\left (3 A d \left (2 c^2-2 c d+d^2\right )+B \left (2 c^3-6 c^2 d+9 c d^2-3 d^3\right )\right ) x}{2 a}+\frac {2 d \left (3 A \left (c^2-3 c d+d^2\right )-B \left (7 c^2-9 c d+4 d^2\right )\right ) \cos (e+f x)}{3 a f}+\frac {d^2 (6 A c-11 B c-9 A d+9 B d) \cos (e+f x) \sin (e+f x)}{6 a f}+\frac {(3 A-4 B) d \cos (e+f x) (c+d \sin (e+f x))^2}{3 a f}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a+a \sin (e+f x))} \]

Rubi [A] (veriﬁed)

Time = 0.25 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {3056, 2832, 2813} \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{a+a \sin (e+f x)} \, dx=\frac {2 d \left (3 A \left (c^2-3 c d+d^2\right )-B \left (7 c^2-9 c d+4 d^2\right )\right ) \cos (e+f x)}{3 a f}+\frac {x \left (3 A d \left (2 c^2-2 c d+d^2\right )+B \left (2 c^3-6 c^2 d+9 c d^2-3 d^3\right )\right )}{2 a}+\frac {d^2 (6 A c-9 A d-11 B c+9 B d) \sin (e+f x) \cos (e+f x)}{6 a f}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a \sin (e+f x)+a)}+\frac {d (3 A-4 B) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a f} \]

Rubi steps \begin{align*} \text {integral}& = -\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a+a \sin (e+f x))}+\frac {\int (c+d \sin (e+f x))^2 (a (B (c-3 d)+3 A d)-a (3 A-4 B) d \sin (e+f x)) \, dx}{a^2} \\ & = \frac {(3 A-4 B) d \cos (e+f x) (c+d \sin (e+f x))^2}{3 a f}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a+a \sin (e+f x))}+\frac {\int (c+d \sin (e+f x)) \left (a \left (3 A (3 c-2 d) d+B \left (3 c^2-9 c d+8 d^2\right )\right )-a d (6 A c-11 B c-9 A d+9 B d) \sin (e+f x)\right ) \, dx}{3 a^2} \\ & = \frac {\left (3 A d \left (2 c^2-2 c d+d^2\right )+B \left (2 c^3-6 c^2 d+9 c d^2-3 d^3\right )\right ) x}{2 a}+\frac {2 d \left (3 A \left (c^2-3 c d+d^2\right )-B \left (7 c^2-9 c d+4 d^2\right )\right ) \cos (e+f x)}{3 a f}+\frac {d^2 (6 A c-11 B c-9 A d+9 B d) \cos (e+f x) \sin (e+f x)}{6 a f}+\frac {(3 A-4 B) d \cos (e+f x) (c+d \sin (e+f x))^2}{3 a f}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a+a \sin (e+f x))} \\ \end{align*}

Mathematica [B] (veriﬁed)

Leaf count is larger than twice the leaf count of optimal. \(788\) vs. \(2(220)=440\).

Time = 7.02 (sec) , antiderivative size = 788, normalized size of antiderivative = 3.58 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{a+a \sin (e+f x)} \, 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 (4 A d \left (6 c^2 (e+f x)-3 c d (1+2 e+2 f x)+d^2 (1+3 e+3 f x)\right )+B \left (8 c^3 (e+f x)-12 c^2 d (1+2 e+2 f x)+12 c d^2 (1+3 e+3 f x)-d^3 (7+12 e+12 f x)\right )\right ) \cos \left (\frac {1}{2} (e+f x)\right )+9 d \left (A d (-4 c+d)+B \left (-4 c^2+3 c d-2 d^2\right )\right ) \cos \left (\frac {3}{2} (e+f x)\right )+9 B c d^2 \cos \left (\frac {5}{2} (e+f x)\right )+3 A d^3 \cos \left (\frac {5}{2} (e+f x)\right )-2 B d^3 \cos \left (\frac {5}{2} (e+f x)\right )+B d^3 \cos \left (\frac {7}{2} (e+f x)\right )+48 A c^3 \sin \left (\frac {1}{2} (e+f x)\right )-48 B c^3 \sin \left (\frac {1}{2} (e+f x)\right )-144 A c^2 d \sin \left (\frac {1}{2} (e+f x)\right )+180 B c^2 d \sin \left (\frac {1}{2} (e+f x)\right )+180 A c d^2 \sin \left (\frac {1}{2} (e+f x)\right )-180 B c d^2 \sin \left (\frac {1}{2} (e+f x)\right )-60 A d^3 \sin \left (\frac {1}{2} (e+f x)\right )+69 B d^3 \sin \left (\frac {1}{2} (e+f x)\right )+24 B c^3 e \sin \left (\frac {1}{2} (e+f x)\right )+72 A c^2 d e \sin \left (\frac {1}{2} (e+f x)\right )-72 B c^2 d e \sin \left (\frac {1}{2} (e+f x)\right )-72 A c d^2 e \sin \left (\frac {1}{2} (e+f x)\right )+108 B c d^2 e \sin \left (\frac {1}{2} (e+f x)\right )+36 A d^3 e \sin \left (\frac {1}{2} (e+f x)\right )-36 B d^3 e \sin \left (\frac {1}{2} (e+f x)\right )+24 B c^3 f x \sin \left (\frac {1}{2} (e+f x)\right )+72 A c^2 d f x \sin \left (\frac {1}{2} (e+f x)\right )-72 B c^2 d f x \sin \left (\frac {1}{2} (e+f x)\right )-72 A c d^2 f x \sin \left (\frac {1}{2} (e+f x)\right )+108 B c d^2 f x \sin \left (\frac {1}{2} (e+f x)\right )+36 A d^3 f x \sin \left (\frac {1}{2} (e+f x)\right )-36 B d^3 f x \sin \left (\frac {1}{2} (e+f x)\right )-36 B c^2 d \sin \left (\frac {3}{2} (e+f x)\right )-36 A c d^2 \sin \left (\frac {3}{2} (e+f x)\right )+27 B c d^2 \sin \left (\frac {3}{2} (e+f x)\right )+9 A d^3 \sin \left (\frac {3}{2} (e+f x)\right )-18 B d^3 \sin \left (\frac {3}{2} (e+f x)\right )-9 B c d^2 \sin \left (\frac {5}{2} (e+f x)\right )-3 A d^3 \sin \left (\frac {5}{2} (e+f x)\right )+2 B d^3 \sin \left (\frac {5}{2} (e+f x)\right )+B d^3 \sin \left (\frac {7}{2} (e+f x)\right )\right )}{24 a f (1+\sin (e+f x))} \]

Maple [A] (veriﬁed)

Time = 0.86 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.53


method	result	size

derivativedivides	\(\frac {\frac {2 \left (\left (\frac {1}{2} A \,d^{3}+\frac {3}{2} d^{2} c B -\frac {1}{2} d^{3} B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-3 d^{2} c A +A \,d^{3}-3 c^{2} d B +3 d^{2} c B -d^{3} B \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-6 d^{2} c A +2 A \,d^{3}-6 c^{2} d B +6 d^{2} c B -4 d^{3} B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-\frac {1}{2} A \,d^{3}-\frac {3}{2} d^{2} c B +\frac {1}{2} d^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-3 d^{2} c A +A \,d^{3}-3 c^{2} d B +3 d^{2} c B -\frac {5 d^{3} B}{3}\right )}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3}}+\left (6 c^{2} d A -6 d^{2} c A +3 A \,d^{3}+2 B \,c^{3}-6 c^{2} d B +9 d^{2} c B -3 d^{3} B \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {2 \left (A \,c^{3}-3 c^{2} d A +3 d^{2} c A -A \,d^{3}-B \,c^{3}+3 c^{2} d B -3 d^{2} c B +d^{3} B \right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}{a f}\)	\(337\)
default	\(\frac {\frac {2 \left (\left (\frac {1}{2} A \,d^{3}+\frac {3}{2} d^{2} c B -\frac {1}{2} d^{3} B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-3 d^{2} c A +A \,d^{3}-3 c^{2} d B +3 d^{2} c B -d^{3} B \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-6 d^{2} c A +2 A \,d^{3}-6 c^{2} d B +6 d^{2} c B -4 d^{3} B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-\frac {1}{2} A \,d^{3}-\frac {3}{2} d^{2} c B +\frac {1}{2} d^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-3 d^{2} c A +A \,d^{3}-3 c^{2} d B +3 d^{2} c B -\frac {5 d^{3} B}{3}\right )}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3}}+\left (6 c^{2} d A -6 d^{2} c A +3 A \,d^{3}+2 B \,c^{3}-6 c^{2} d B +9 d^{2} c B -3 d^{3} B \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {2 \left (A \,c^{3}-3 c^{2} d A +3 d^{2} c A -A \,d^{3}-B \,c^{3}+3 c^{2} d B -3 d^{2} c B +d^{3} B \right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}{a f}\)	\(337\)
parallelrisch	\(\frac {\left (\left (36 f x A -36 f x B +84 A -109 B \right ) d^{3}-72 c \left (\frac {\left (-3 f x -7\right ) B}{2}+A \left (f x +\frac {7}{2}\right )\right ) d^{2}+72 \left (\left (-f x -\frac {7}{2}\right ) B +A \left (f x +2\right )\right ) c^{2} d -48 \left (\left (-\frac {f x}{2}-1\right ) B +A \right ) c^{3}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+\left (\left (36 f x A -36 f x B +12 A -19 B \right ) d^{3}-72 c \left (\frac {\left (-3 f x -1\right ) B}{2}+A \left (f x +\frac {1}{2}\right )\right ) d^{2}+72 \left (\left (-f x -\frac {1}{2}\right ) B +f x A \right ) c^{2} d +24 B \,c^{3} f x \right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-36 \left (\left (\frac {\left (-\frac {A}{2}+B \right ) d^{2}}{2}+c \left (A -\frac {3 B}{4}\right ) d +B \,c^{2}\right ) \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+\left (\frac {\left (-\frac {A}{2}+B \right ) d^{2}}{2}+c \left (A -\frac {3 B}{4}\right ) d +B \,c^{2}\right ) \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )-\frac {\left (\left (\left (A -\frac {2 B}{3}\right ) d +3 B c \right ) \cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+\left (\left (-A +\frac {2 B}{3}\right ) d -3 B c \right ) \sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+\frac {d B \left (\cos \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )+\sin \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )\right )}{3}\right ) d}{12}\right ) d}{24 \left (\sin \left (\frac {f x}{2}+\frac {e}{2}\right )+\cos \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a f}\)	\(350\)
risch	\(-\frac {d^{3} \sin \left (2 f x +2 e \right ) A}{4 a f}+\frac {3 x A \,d^{3}}{2 a}+\frac {x B \,c^{3}}{a}-\frac {3 x \,d^{3} B}{2 a}+\frac {3 x \,c^{2} d A}{a}-\frac {3 x \,d^{2} c A}{a}-\frac {3 x \,c^{2} d B}{a}+\frac {9 x \,d^{2} c B}{2 a}+\frac {B \,d^{3} \cos \left (3 f x +3 e \right )}{12 a f}+\frac {d^{3} \sin \left (2 f x +2 e \right ) B}{4 a f}-\frac {3 d^{2} \sin \left (2 f x +2 e \right ) B c}{4 a f}+\frac {d^{3} {\mathrm e}^{i \left (f x +e \right )} A}{2 a f}-\frac {7 d^{3} {\mathrm e}^{i \left (f x +e \right )} B}{8 a f}+\frac {d^{3} {\mathrm e}^{-i \left (f x +e \right )} A}{2 a f}-\frac {7 d^{3} {\mathrm e}^{-i \left (f x +e \right )} B}{8 a f}-\frac {2 A \,c^{3}}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}+\frac {2 A \,d^{3}}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}+\frac {2 B \,c^{3}}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}-\frac {2 d^{3} B}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}-\frac {6 c^{2} d B}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}+\frac {6 d^{2} c B}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}-\frac {3 d^{2} {\mathrm e}^{i \left (f x +e \right )} A c}{2 a f}-\frac {3 d \,{\mathrm e}^{i \left (f x +e \right )} B \,c^{2}}{2 a f}+\frac {3 d^{2} {\mathrm e}^{i \left (f x +e \right )} c B}{2 a f}-\frac {3 d^{2} {\mathrm e}^{-i \left (f x +e \right )} A c}{2 a f}-\frac {3 d \,{\mathrm e}^{-i \left (f x +e \right )} B \,c^{2}}{2 a f}+\frac {3 d^{2} {\mathrm e}^{-i \left (f x +e \right )} c B}{2 a f}+\frac {6 c^{2} d A}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}-\frac {6 d^{2} c A}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}\)	\(588\)
norman	\(\text {Expression too large to display}\)	\(1198\)

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 470 vs. \(2 (212) = 424\).

Time = 0.28 (sec) , antiderivative size = 470, normalized size of antiderivative = 2.14 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{a+a \sin (e+f x)} \, dx=\frac {2 \, B d^{3} \cos \left (f x + e\right )^{4} - 6 \, {\left (A - B\right )} c^{3} + 18 \, {\left (A - B\right )} c^{2} d - 18 \, {\left (A - B\right )} c d^{2} + 6 \, {\left (A - B\right )} d^{3} + {\left (9 \, B c d^{2} + {\left (3 \, A - B\right )} d^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (2 \, B c^{3} + 6 \, {\left (A - B\right )} c^{2} d - 3 \, {\left (2 \, A - 3 \, B\right )} c d^{2} + 3 \, {\left (A - B\right )} d^{3}\right )} f x - 6 \, {\left (3 \, B c^{2} d + 3 \, {\left (A - B\right )} c d^{2} - {\left (A - 2 \, B\right )} d^{3}\right )} \cos \left (f x + e\right )^{2} - 3 \, {\left (2 \, {\left (A - B\right )} c^{3} - 6 \, {\left (A - 2 \, B\right )} c^{2} d + 3 \, {\left (4 \, A - 3 \, B\right )} c d^{2} - {\left (3 \, A - 5 \, B\right )} d^{3} - {\left (2 \, B c^{3} + 6 \, {\left (A - B\right )} c^{2} d - 3 \, {\left (2 \, A - 3 \, B\right )} c d^{2} + 3 \, {\left (A - B\right )} d^{3}\right )} f x\right )} \cos \left (f x + e\right ) + {\left (2 \, B d^{3} \cos \left (f x + e\right )^{3} + 6 \, {\left (A - B\right )} c^{3} - 18 \, {\left (A - B\right )} c^{2} d + 18 \, {\left (A - B\right )} c d^{2} - 6 \, {\left (A - B\right )} d^{3} + 3 \, {\left (2 \, B c^{3} + 6 \, {\left (A - B\right )} c^{2} d - 3 \, {\left (2 \, A - 3 \, B\right )} c d^{2} + 3 \, {\left (A - B\right )} d^{3}\right )} f x - 3 \, {\left (3 \, B c d^{2} + {\left (A - B\right )} d^{3}\right )} \cos \left (f x + e\right )^{2} - 3 \, {\left (6 \, B c^{2} d + 3 \, {\left (2 \, A - B\right )} c d^{2} - {\left (A - 3 \, B\right )} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{6 \, {\left (a f \cos \left (f x + e\right ) + a f \sin \left (f x + e\right ) + a f\right )}} \]

Sympy [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 14644 vs. \(2 (204) = 408\).

Time = 3.87 (sec) , antiderivative size = 14644, normalized size of antiderivative = 66.56 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{a+a \sin (e+f x)} \, dx=\text {Too large to display} \]

Maxima [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1124 vs. \(2 (212) = 424\).

Time = 0.32 (sec) , antiderivative size = 1124, normalized size of antiderivative = 5.11 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{a+a \sin (e+f x)} \, dx=\text {Too large to display} \]

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 460 vs. \(2 (212) = 424\).

Time = 0.29 (sec) , antiderivative size = 460, normalized size of antiderivative = 2.09 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{a+a \sin (e+f x)} \, dx=\frac {\frac {3 \, {\left (2 \, B c^{3} + 6 \, A c^{2} d - 6 \, B c^{2} d - 6 \, A c d^{2} + 9 \, B c d^{2} + 3 \, A d^{3} - 3 \, B d^{3}\right )} {\left (f x + e\right )}}{a} - \frac {12 \, {\left (A c^{3} - B c^{3} - 3 \, A c^{2} d + 3 \, B c^{2} d + 3 \, A c d^{2} - 3 \, B c d^{2} - A d^{3} + B d^{3}\right )}}{a {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}} + \frac {2 \, {\left (9 \, B c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3 \, A d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 3 \, B d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 18 \, B c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 18 \, A c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 18 \, B c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 6 \, A d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 6 \, B d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 36 \, B c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 36 \, A c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 36 \, B c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 12 \, A d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 24 \, B d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 9 \, B c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, A d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, B d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 18 \, B c^{2} d - 18 \, A c d^{2} + 18 \, B c d^{2} + 6 \, A d^{3} - 10 \, B d^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{3} a}}{6 \, f} \]

Mupad [B] (veriﬁcation not implemented)

Time = 14.29 (sec) , antiderivative size = 839, normalized size of antiderivative = 3.81 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{a+a \sin (e+f x)} \, dx=\text {Too large to display} \]

\(\int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{a+a \sin (e+f x)} \, dx\) [265]

Optimal result