3.3.0 ∫ (A+B sin(e+fx))(c+d sin(e+fx))2 ___________________________________________________

Integrand size = 35, antiderivative size = 143 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{a+a \sin (e+f x)} \, dx=\frac {\left (2 A (2 c-d) d+B \left (2 c^2-4 c d+3 d^2\right )\right ) x}{2 a}+\frac {2 (A (c-d)-B (2 c-d)) d \cos (e+f x)}{a f}+\frac {(2 A-3 B) d^2 \cos (e+f x) \sin (e+f x)}{2 a f}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{f (a+a \sin (e+f x))} \]

Rubi [A] (veriﬁed)

Time = 0.14 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {3056, 2813} \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{a+a \sin (e+f x)} \, dx=\frac {x \left (-\left (d^2 (2 A-3 B)\right )+4 A c d+2 B c (c-2 d)\right )}{2 a}+\frac {2 d (A (c-d)-B (2 c-d)) \cos (e+f x)}{a f}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{f (a \sin (e+f x)+a)}+\frac {d^2 (2 A-3 B) \sin (e+f x) \cos (e+f x)}{2 a f} \]

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

Mathematica [A] (veriﬁed)

Time = 6.46 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.40 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{a+a \sin (e+f x)} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (8 (A-B) (c-d)^2 \sin \left (\frac {1}{2} (e+f x)\right )+2 \left (2 A (2 c-d) d+B \left (2 c^2-4 c d+3 d^2\right )\right ) (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+4 d (-A d+B (-2 c+d)) \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )-B d^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (2 (e+f x))\right )}{4 a f (1+\sin (e+f x))} \]

Maple [A] (veriﬁed)

Time = 0.63 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.35


method	result	size

derivativedivides	\(\frac {-\frac {2 \left (A \,c^{2}-2 A c d +A \,d^{2}-B \,c^{2}+2 c d B -d^{2} B \right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}+\frac {2 \left (\frac {B \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) d^{2}}{2}+\left (-A \,d^{2}-2 c d B +d^{2} B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {B \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d^{2}}{2}-A \,d^{2}-2 c d B +d^{2} B \right )}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\left (4 A c d -2 A \,d^{2}+2 B \,c^{2}-4 c d B +3 d^{2} B \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}\)	\(193\)
default	\(\frac {-\frac {2 \left (A \,c^{2}-2 A c d +A \,d^{2}-B \,c^{2}+2 c d B -d^{2} B \right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}+\frac {2 \left (\frac {B \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) d^{2}}{2}+\left (-A \,d^{2}-2 c d B +d^{2} B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {B \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d^{2}}{2}-A \,d^{2}-2 c d B +d^{2} B \right )}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\left (4 A c d -2 A \,d^{2}+2 B \,c^{2}-4 c d B +3 d^{2} B \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}\)	\(193\)
parallelrisch	\(\frac {\left (\left (-2 f x A +3 f x B -7 A +7 B \right ) d^{2}+4 c \left (\left (-f x -\frac {7}{2}\right ) B +A \left (f x +2\right )\right ) d -4 \left (\left (-\frac {f x}{2}-1\right ) B +A \right ) c^{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+\left (\left (-2 f x A +3 f x B -A +B \right ) d^{2}+4 c \left (\left (-f x -\frac {1}{2}\right ) B +f x A \right ) d +2 B \,c^{2} f x \right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-\left (\left (\left (A -\frac {3 B}{4}\right ) d +2 B c \right ) \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+\left (\left (A -\frac {3 B}{4}\right ) d +2 B c \right ) \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )-\frac {d B \left (\cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )-\sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )\right )}{4}\right ) d}{2 \left (\sin \left (\frac {f x}{2}+\frac {e}{2}\right )+\cos \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a f}\)	\(226\)
risch	\(\frac {2 x A c d}{a}-\frac {x A \,d^{2}}{a}+\frac {x B \,c^{2}}{a}-\frac {2 x c d B}{a}+\frac {3 x \,d^{2} B}{2 a}-\frac {d^{2} {\mathrm e}^{i \left (f x +e \right )} A}{2 a f}-\frac {d \,{\mathrm e}^{i \left (f x +e \right )} B c}{a f}+\frac {d^{2} {\mathrm e}^{i \left (f x +e \right )} B}{2 a f}-\frac {d^{2} {\mathrm e}^{-i \left (f x +e \right )} A}{2 a f}-\frac {d \,{\mathrm e}^{-i \left (f x +e \right )} B c}{a f}+\frac {d^{2} {\mathrm e}^{-i \left (f x +e \right )} B}{2 a f}-\frac {2 A \,c^{2}}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}+\frac {4 A c d}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}-\frac {2 A \,d^{2}}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}+\frac {2 B \,c^{2}}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}-\frac {4 c d B}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}+\frac {2 d^{2} B}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}-\frac {d^{2} B \sin \left (2 f x +2 e \right )}{4 a f}\)	\(348\)
norman	\(\frac {\frac {\left (2 A \,c^{2}-4 A c d -2 B \,c^{2}-2 d^{2} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}+\frac {\left (2 A \,c^{2}-4 A c d +2 A \,d^{2}-2 B \,c^{2}+4 c d B -3 d^{2} B \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {\left (6 A \,c^{2}-12 A c d +2 A \,d^{2}-6 B \,c^{2}+4 c d B -5 d^{2} B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {\left (6 A \,c^{2}-12 A c d +4 A \,d^{2}-6 B \,c^{2}+8 c d B -6 d^{2} B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {2 A \,d^{2}+4 c d B -d^{2} B}{a f}+\frac {\left (4 A c d -2 A \,d^{2}+2 B \,c^{2}-4 c d B +3 d^{2} B \right ) x}{2 a}-\frac {\left (4 A \,d^{2}+8 c d B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {\left (2 A \,d^{2}+4 c d B +d^{2} B \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {\left (4 A c d -2 A \,d^{2}+2 B \,c^{2}-4 c d B +3 d^{2} B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a}+\frac {3 \left (4 A c d -2 A \,d^{2}+2 B \,c^{2}-4 c d B +3 d^{2} B \right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {3 \left (4 A c d -2 A \,d^{2}+2 B \,c^{2}-4 c d B +3 d^{2} B \right ) x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {3 \left (4 A c d -2 A \,d^{2}+2 B \,c^{2}-4 c d B +3 d^{2} B \right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {3 \left (4 A c d -2 A \,d^{2}+2 B \,c^{2}-4 c d B +3 d^{2} B \right ) x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {\left (4 A c d -2 A \,d^{2}+2 B \,c^{2}-4 c d B +3 d^{2} B \right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {\left (4 A c d -2 A \,d^{2}+2 B \,c^{2}-4 c d B +3 d^{2} B \right ) x \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3}}\)	\(678\)

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (139) = 278\).

Time = 0.26 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.12 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{a+a \sin (e+f x)} \, dx=\frac {B d^{2} \cos \left (f x + e\right )^{3} - 2 \, {\left (A - B\right )} c^{2} + 4 \, {\left (A - B\right )} c d - 2 \, {\left (A - B\right )} d^{2} + {\left (2 \, B c^{2} + 4 \, {\left (A - B\right )} c d - {\left (2 \, A - 3 \, B\right )} d^{2}\right )} f x - 2 \, {\left (2 \, B c d + {\left (A - B\right )} d^{2}\right )} \cos \left (f x + e\right )^{2} - {\left (2 \, {\left (A - B\right )} c^{2} - 4 \, {\left (A - 2 \, B\right )} c d + {\left (4 \, A - 3 \, B\right )} d^{2} - {\left (2 \, B c^{2} + 4 \, {\left (A - B\right )} c d - {\left (2 \, A - 3 \, B\right )} d^{2}\right )} f x\right )} \cos \left (f x + e\right ) - {\left (B d^{2} \cos \left (f x + e\right )^{2} - 2 \, {\left (A - B\right )} c^{2} + 4 \, {\left (A - B\right )} c d - 2 \, {\left (A - B\right )} d^{2} - {\left (2 \, B c^{2} + 4 \, {\left (A - B\right )} c d - {\left (2 \, A - 3 \, B\right )} d^{2}\right )} f x + {\left (4 \, B c d + {\left (2 \, A - B\right )} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{2 \, {\left (a f \cos \left (f x + e\right ) + a f \sin \left (f x + e\right ) + a f\right )}} \]

Sympy [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5763 vs. \(2 (117) = 234\).

Time = 2.04 (sec) , antiderivative size = 5763, normalized size of antiderivative = 40.30 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{a+a \sin (e+f x)} \, dx=\text {Too large to display} \]

Maxima [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 606 vs. \(2 (139) = 278\).

Time = 0.29 (sec) , antiderivative size = 606, normalized size of antiderivative = 4.24 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{a+a \sin (e+f x)} \, dx=\frac {B d^{2} {\left (\frac {\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {5 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {3 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 4}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {2 \, a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {2 \, a \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {a \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} + \frac {3 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a}\right )} - 4 \, B c d {\left (\frac {\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 2}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac {\arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a}\right )} - 2 \, A d^{2} {\left (\frac {\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 2}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac {\arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a}\right )} + 2 \, B c^{2} {\left (\frac {\arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a} + \frac {1}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )} + 4 \, A c d {\left (\frac {\arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a} + \frac {1}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )} - \frac {2 \, A c^{2}}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}}{f} \]

Giac [A] (veriﬁcation not implemented)

Time = 0.29 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.50 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{a+a \sin (e+f x)} \, dx=\frac {\frac {{\left (2 \, B c^{2} + 4 \, A c d - 4 \, B c d - 2 \, A d^{2} + 3 \, B d^{2}\right )} {\left (f x + e\right )}}{a} - \frac {4 \, {\left (A c^{2} - B c^{2} - 2 \, A c d + 2 \, B c d + A d^{2} - B d^{2}\right )}}{a {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}} + \frac {2 \, {\left (B d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, B c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, A d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, B d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - B d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 4 \, B c d - 2 \, A d^{2} + 2 \, B d^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{2} a}}{2 \, f} \]

Mupad [B] (veriﬁcation not implemented)

Time = 16.96 (sec) , antiderivative size = 297, normalized size of antiderivative = 2.08 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{a+a \sin (e+f x)} \, dx=\frac {x\,\left (2\,B\,c^2-2\,A\,d^2+3\,B\,d^2+4\,A\,c\,d-4\,B\,c\,d\right )}{2\,a}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (2\,A\,d^2-3\,B\,d^2+4\,B\,c\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (2\,A\,c^2+2\,A\,d^2-2\,B\,c^2-3\,B\,d^2-4\,A\,c\,d+4\,B\,c\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (4\,A\,c^2+6\,A\,d^2-4\,B\,c^2-5\,B\,d^2-8\,A\,c\,d+12\,B\,c\,d\right )+2\,A\,c^2+4\,A\,d^2-2\,B\,c^2-4\,B\,d^2+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,A\,d^2-B\,d^2+4\,B\,c\,d\right )-4\,A\,c\,d+8\,B\,c\,d}{f\,\left (a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+2\,a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+2\,a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+a\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+a\right )} \]

\(\int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{a+a \sin (e+f x)} \, dx\) [266]

Optimal result