Optimal. Leaf size=201 \[ \frac {a^3 (20 A c+15 A d+15 B c+13 B d) \cos ^3(e+f x)}{60 f}-\frac {a^3 (20 A c+15 A d+15 B c+13 B d) \cos (e+f x)}{5 f}-\frac {3 a^3 (20 A c+15 A d+15 B c+13 B d) \sin (e+f x) \cos (e+f x)}{40 f}+\frac {1}{8} a^3 x (20 A c+15 A d+15 B c+13 B d)-\frac {(5 A d+5 B c-B d) \cos (e+f x) (a \sin (e+f x)+a)^3}{20 f}-\frac {B d \cos (e+f x) (a \sin (e+f x)+a)^4}{5 a f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.33, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {2968, 3023, 2751, 2645, 2638, 2635, 8, 2633} \[ \frac {a^3 (20 A c+15 A d+15 B c+13 B d) \cos ^3(e+f x)}{60 f}-\frac {a^3 (20 A c+15 A d+15 B c+13 B d) \cos (e+f x)}{5 f}-\frac {3 a^3 (20 A c+15 A d+15 B c+13 B d) \sin (e+f x) \cos (e+f x)}{40 f}+\frac {1}{8} a^3 x (20 A c+15 A d+15 B c+13 B d)-\frac {(5 A d+5 B c-B d) \cos (e+f x) (a \sin (e+f x)+a)^3}{20 f}-\frac {B d \cos (e+f x) (a \sin (e+f x)+a)^4}{5 a f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2633
Rule 2635
Rule 2638
Rule 2645
Rule 2751
Rule 2968
Rule 3023
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx &=\int (a+a \sin (e+f x))^3 \left (A c+(B c+A d) \sin (e+f x)+B d \sin ^2(e+f x)\right ) \, dx\\ &=-\frac {B d \cos (e+f x) (a+a \sin (e+f x))^4}{5 a f}+\frac {\int (a+a \sin (e+f x))^3 (a (5 A c+4 B d)+a (5 B c+5 A d-B d) \sin (e+f x)) \, dx}{5 a}\\ &=-\frac {(5 B c+5 A d-B d) \cos (e+f x) (a+a \sin (e+f x))^3}{20 f}-\frac {B d \cos (e+f x) (a+a \sin (e+f x))^4}{5 a f}+\frac {1}{20} (20 A c+15 B c+15 A d+13 B d) \int (a+a \sin (e+f x))^3 \, dx\\ &=-\frac {(5 B c+5 A d-B d) \cos (e+f x) (a+a \sin (e+f x))^3}{20 f}-\frac {B d \cos (e+f x) (a+a \sin (e+f x))^4}{5 a f}+\frac {1}{20} (20 A c+15 B c+15 A d+13 B d) \int \left (a^3+3 a^3 \sin (e+f x)+3 a^3 \sin ^2(e+f x)+a^3 \sin ^3(e+f x)\right ) \, dx\\ &=\frac {1}{20} a^3 (20 A c+15 B c+15 A d+13 B d) x-\frac {(5 B c+5 A d-B d) \cos (e+f x) (a+a \sin (e+f x))^3}{20 f}-\frac {B d \cos (e+f x) (a+a \sin (e+f x))^4}{5 a f}+\frac {1}{20} \left (a^3 (20 A c+15 B c+15 A d+13 B d)\right ) \int \sin ^3(e+f x) \, dx+\frac {1}{20} \left (3 a^3 (20 A c+15 B c+15 A d+13 B d)\right ) \int \sin (e+f x) \, dx+\frac {1}{20} \left (3 a^3 (20 A c+15 B c+15 A d+13 B d)\right ) \int \sin ^2(e+f x) \, dx\\ &=\frac {1}{20} a^3 (20 A c+15 B c+15 A d+13 B d) x-\frac {3 a^3 (20 A c+15 B c+15 A d+13 B d) \cos (e+f x)}{20 f}-\frac {3 a^3 (20 A c+15 B c+15 A d+13 B d) \cos (e+f x) \sin (e+f x)}{40 f}-\frac {(5 B c+5 A d-B d) \cos (e+f x) (a+a \sin (e+f x))^3}{20 f}-\frac {B d \cos (e+f x) (a+a \sin (e+f x))^4}{5 a f}+\frac {1}{40} \left (3 a^3 (20 A c+15 B c+15 A d+13 B d)\right ) \int 1 \, dx-\frac {\left (a^3 (20 A c+15 B c+15 A d+13 B d)\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (e+f x)\right )}{20 f}\\ &=\frac {1}{8} a^3 (20 A c+15 B c+15 A d+13 B d) x-\frac {a^3 (20 A c+15 B c+15 A d+13 B d) \cos (e+f x)}{5 f}+\frac {a^3 (20 A c+15 B c+15 A d+13 B d) \cos ^3(e+f x)}{60 f}-\frac {3 a^3 (20 A c+15 B c+15 A d+13 B d) \cos (e+f x) \sin (e+f x)}{40 f}-\frac {(5 B c+5 A d-B d) \cos (e+f x) (a+a \sin (e+f x))^3}{20 f}-\frac {B d \cos (e+f x) (a+a \sin (e+f x))^4}{5 a f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.91, size = 156, normalized size = 0.78 \[ \frac {\cos (e+f x) \left (-\frac {1}{4} a^4 (5 A d+5 B c-B d) (\sin (e+f x)+1)^3-\frac {a^4 (20 A c+15 A d+15 B c+13 B d) \left (30 \sin ^{-1}\left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )+\left (2 \sin ^2(e+f x)+9 \sin (e+f x)+22\right ) \sqrt {\cos ^2(e+f x)}\right )}{24 \sqrt {\cos ^2(e+f x)}}-B d (a \sin (e+f x)+a)^4\right )}{5 a f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 178, normalized size = 0.89 \[ -\frac {24 \, B a^{3} d \cos \left (f x + e\right )^{5} - 40 \, {\left ({\left (A + 3 \, B\right )} a^{3} c + {\left (3 \, A + 5 \, B\right )} a^{3} d\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (5 \, {\left (4 \, A + 3 \, B\right )} a^{3} c + {\left (15 \, A + 13 \, B\right )} a^{3} d\right )} f x + 480 \, {\left ({\left (A + B\right )} a^{3} c + {\left (A + B\right )} a^{3} d\right )} \cos \left (f x + e\right ) - 15 \, {\left (2 \, {\left (B a^{3} c + {\left (A + 3 \, B\right )} a^{3} d\right )} \cos \left (f x + e\right )^{3} - {\left ({\left (12 \, A + 17 \, B\right )} a^{3} c + {\left (17 \, A + 19 \, B\right )} a^{3} d\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{120 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 217, normalized size = 1.08 \[ -\frac {B a^{3} d \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac {1}{8} \, {\left (20 \, A a^{3} c + 15 \, B a^{3} c + 15 \, A a^{3} d + 13 \, B a^{3} d\right )} x + \frac {{\left (4 \, A a^{3} c + 12 \, B a^{3} c + 12 \, A a^{3} d + 17 \, B a^{3} d\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {{\left (30 \, A a^{3} c + 26 \, B a^{3} c + 26 \, A a^{3} d + 23 \, B a^{3} d\right )} \cos \left (f x + e\right )}{8 \, f} + \frac {{\left (B a^{3} c + A a^{3} d + 3 \, B a^{3} d\right )} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} - \frac {{\left (3 \, A a^{3} c + 4 \, B a^{3} c + 4 \, A a^{3} d + 4 \, B a^{3} d\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.51, size = 414, normalized size = 2.06 \[ \frac {-\frac {a^{3} A c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+a^{3} A d \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+B \,a^{3} c \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {B \,a^{3} d \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+3 a^{3} A c \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-a^{3} A d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )-B \,a^{3} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 B \,a^{3} d \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-3 a^{3} A c \cos \left (f x +e \right )+3 a^{3} A d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+3 B \,a^{3} c \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B \,a^{3} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+a^{3} A c \left (f x +e \right )-a^{3} A d \cos \left (f x +e \right )-B \,a^{3} c \cos \left (f x +e \right )+B \,a^{3} d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.43, size = 398, normalized size = 1.98 \[ \frac {160 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{3} c + 360 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} c + 480 \, {\left (f x + e\right )} A a^{3} c + 480 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{3} c + 15 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} c + 360 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} c + 480 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{3} d + 15 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} d + 360 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} d - 32 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} B a^{3} d + 480 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{3} d + 45 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} d + 120 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} d - 1440 \, A a^{3} c \cos \left (f x + e\right ) - 480 \, B a^{3} c \cos \left (f x + e\right ) - 480 \, A a^{3} d \cos \left (f x + e\right )}{480 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 14.62, size = 550, normalized size = 2.74 \[ \frac {a^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (20\,A\,c+15\,A\,d+15\,B\,c+13\,B\,d\right )}{4\,\left (5\,A\,a^3\,c+\frac {15\,A\,a^3\,d}{4}+\frac {15\,B\,a^3\,c}{4}+\frac {13\,B\,a^3\,d}{4}\right )}\right )\,\left (20\,A\,c+15\,A\,d+15\,B\,c+13\,B\,d\right )}{4\,f}-\frac {a^3\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {f\,x}{2}\right )\,\left (20\,A\,c+15\,A\,d+15\,B\,c+13\,B\,d\right )}{4\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (6\,A\,a^3\,c+\frac {19\,A\,a^3\,d}{2}+\frac {19\,B\,a^3\,c}{2}+\frac {25\,B\,a^3\,d}{2}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (3\,A\,a^3\,c+\frac {15\,A\,a^3\,d}{4}+\frac {15\,B\,a^3\,c}{4}+\frac {13\,B\,a^3\,d}{4}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (6\,A\,a^3\,c+\frac {19\,A\,a^3\,d}{2}+\frac {19\,B\,a^3\,c}{2}+\frac {25\,B\,a^3\,d}{2}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (28\,A\,a^3\,c+20\,A\,a^3\,d+20\,B\,a^3\,c+12\,B\,a^3\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {92\,A\,a^3\,c}{3}+28\,A\,a^3\,d+28\,B\,a^3\,c+\frac {76\,B\,a^3\,d}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {136\,A\,a^3\,c}{3}+40\,A\,a^3\,d+40\,B\,a^3\,c+\frac {116\,B\,a^3\,d}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (6\,A\,a^3\,c+2\,A\,a^3\,d+2\,B\,a^3\,c\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (3\,A\,a^3\,c+\frac {15\,A\,a^3\,d}{4}+\frac {15\,B\,a^3\,c}{4}+\frac {13\,B\,a^3\,d}{4}\right )+\frac {22\,A\,a^3\,c}{3}+6\,A\,a^3\,d+6\,B\,a^3\,c+\frac {76\,B\,a^3\,d}{15}}{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.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 5.28, size = 960, normalized size = 4.78 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________