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

Optimal result

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

1/2*(3*A*d*(2*c^2-2*c*d+d^2)+B*(2*c^3-6*c^2*d+9*c*d^2-3*d^3))*x/a+2/3*d*(3 
*A*(c^2-3*c*d+d^2)-B*(7*c^2-9*c*d+4*d^2))*cos(f*x+e)/a/f+1/6*d^2*(6*A*c-9* 
A*d-11*B*c+9*B*d)*cos(f*x+e)*sin(f*x+e)/a/f+1/3*(3*A-4*B)*d*cos(f*x+e)*(c+ 
d*sin(f*x+e))^2/a/f-(A-B)*cos(f*x+e)*(c+d*sin(f*x+e))^3/f/(a+a*sin(f*x+e))
 

Mathematica [B] (verified)

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

Time = 7.89 (sec) , antiderivative size = 788, normalized size of antiderivative = 3.58 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{a+a \sin (e+f x)} \, dx =\text {Too large to display} \] Input:

Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3)/(a + a*Sin[e + f*x 
]),x]
 

Output:

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(3*(4*A*d*(6*c^2*(e + f*x) - 3*c*d* 
(1 + 2*e + 2*f*x) + d^2*(1 + 3*e + 3*f*x)) + B*(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)) 
)*Cos[(e + f*x)/2] + 9*d*(A*d*(-4*c + d) + B*(-4*c^2 + 3*c*d - 2*d^2))*Cos 
[(3*(e + f*x))/2] + 9*B*c*d^2*Cos[(5*(e + f*x))/2] + 3*A*d^3*Cos[(5*(e + f 
*x))/2] - 2*B*d^3*Cos[(5*(e + f*x))/2] + B*d^3*Cos[(7*(e + f*x))/2] + 48*A 
*c^3*Sin[(e + f*x)/2] - 48*B*c^3*Sin[(e + f*x)/2] - 144*A*c^2*d*Sin[(e + f 
*x)/2] + 180*B*c^2*d*Sin[(e + f*x)/2] + 180*A*c*d^2*Sin[(e + f*x)/2] - 180 
*B*c*d^2*Sin[(e + f*x)/2] - 60*A*d^3*Sin[(e + f*x)/2] + 69*B*d^3*Sin[(e + 
f*x)/2] + 24*B*c^3*e*Sin[(e + f*x)/2] + 72*A*c^2*d*e*Sin[(e + f*x)/2] - 72 
*B*c^2*d*e*Sin[(e + f*x)/2] - 72*A*c*d^2*e*Sin[(e + f*x)/2] + 108*B*c*d^2* 
e*Sin[(e + f*x)/2] + 36*A*d^3*e*Sin[(e + f*x)/2] - 36*B*d^3*e*Sin[(e + f*x 
)/2] + 24*B*c^3*f*x*Sin[(e + f*x)/2] + 72*A*c^2*d*f*x*Sin[(e + f*x)/2] - 7 
2*B*c^2*d*f*x*Sin[(e + f*x)/2] - 72*A*c*d^2*f*x*Sin[(e + f*x)/2] + 108*B*c 
*d^2*f*x*Sin[(e + f*x)/2] + 36*A*d^3*f*x*Sin[(e + f*x)/2] - 36*B*d^3*f*x*S 
in[(e + f*x)/2] - 36*B*c^2*d*Sin[(3*(e + f*x))/2] - 36*A*c*d^2*Sin[(3*(e + 
 f*x))/2] + 27*B*c*d^2*Sin[(3*(e + f*x))/2] + 9*A*d^3*Sin[(3*(e + f*x))/2] 
 - 18*B*d^3*Sin[(3*(e + f*x))/2] - 9*B*c*d^2*Sin[(5*(e + f*x))/2] - 3*A*d^ 
3*Sin[(5*(e + f*x))/2] + 2*B*d^3*Sin[(5*(e + f*x))/2] + B*d^3*Sin[(7*(e + 
f*x))/2]))/(24*a*f*(1 + Sin[e + f*x]))
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3042, 3456, 3042, 3232, 3042, 3213}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3456

\(\displaystyle \frac {\int (c+d \sin (e+f x))^2 (a (B (c-3 d)+3 A d)-a (3 A-4 B) d \sin (e+f x))dx}{a^2}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a \sin (e+f x)+a)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int (c+d \sin (e+f x))^2 (a (B (c-3 d)+3 A d)-a (3 A-4 B) d \sin (e+f x))dx}{a^2}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a \sin (e+f x)+a)}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {\frac {1}{3} \int (c+d \sin (e+f x)) \left (a \left (3 A (3 c-2 d) d+B \left (3 c^2-9 d c+8 d^2\right )\right )-a d (6 A c-11 B c-9 A d+9 B d) \sin (e+f x)\right )dx+\frac {a d (3 A-4 B) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}}{a^2}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a \sin (e+f x)+a)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{3} \int (c+d \sin (e+f x)) \left (a \left (3 A (3 c-2 d) d+B \left (3 c^2-9 d c+8 d^2\right )\right )-a d (6 A c-11 B c-9 A d+9 B d) \sin (e+f x)\right )dx+\frac {a d (3 A-4 B) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}}{a^2}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a \sin (e+f x)+a)}\)

\(\Big \downarrow \) 3213

\(\displaystyle \frac {\frac {1}{3} \left (\frac {2 a d \left (3 A \left (c^2-3 c d+d^2\right )-B \left (7 c^2-9 c d+4 d^2\right )\right ) \cos (e+f x)}{f}+\frac {3}{2} a x \left (3 A d \left (2 c^2-2 c d+d^2\right )+B \left (2 c^3-6 c^2 d+9 c d^2-3 d^3\right )\right )+\frac {a d^2 (6 A c-9 A d-11 B c+9 B d) \sin (e+f x) \cos (e+f x)}{2 f}\right )+\frac {a d (3 A-4 B) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}}{a^2}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a \sin (e+f x)+a)}\)

Input:

Int[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3)/(a + a*Sin[e + f*x]),x]
 

Output:

-(((A - B)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(f*(a + a*Sin[e + f*x]))) 
+ ((a*(3*A - 4*B)*d*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*f) + ((3*a*(3* 
A*d*(2*c^2 - 2*c*d + d^2) + B*(2*c^3 - 6*c^2*d + 9*c*d^2 - 3*d^3))*x)/2 + 
(2*a*d*(3*A*(c^2 - 3*c*d + d^2) - B*(7*c^2 - 9*c*d + 4*d^2))*Cos[e + f*x]) 
/f + (a*d^2*(6*A*c - 11*B*c - 9*A*d + 9*B*d)*Cos[e + f*x]*Sin[e + f*x])/(2 
*f))/3)/a^2
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3213

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.) 
*(x_)]), x_Symbol] :> Simp[(2*a*c + b*d)*(x/2), x] + (-Simp[(b*c + a*d)*(Co 
s[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /; Free 
Q[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
 

rule 3232

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( 
f*(m + 1))), x] + Simp[1/(m + 1)   Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[b* 
d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ 
[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 
 0] && IntegerQ[2*m]
 

rule 3456

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + 
(f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim 
p[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/( 
a*f*(2*m + 1))), x] - Simp[1/(a*b*(2*m + 1))   Int[(a + b*Sin[e + f*x])^(m 
+ 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + 
 b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] & 
& NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (In 
tegerQ[2*n] || EqQ[c, 0])
 

Maple [A] (verified)

Time = 2.16 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.53

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

Input:

int((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3/(a+a*sin(f*x+e)),x,method=_RETURNV 
ERBOSE)
 

Output:

2/f/a*(((1/2*A*d^3+3/2*B*c*d^2-1/2*B*d^3)*tan(1/2*f*x+1/2*e)^5+(-3*A*c*d^2 
+A*d^3-3*B*c^2*d+3*B*c*d^2-B*d^3)*tan(1/2*f*x+1/2*e)^4+(-6*A*c*d^2+2*A*d^3 
-6*B*c^2*d+6*B*c*d^2-4*B*d^3)*tan(1/2*f*x+1/2*e)^2+(-1/2*A*d^3-3/2*B*c*d^2 
+1/2*B*d^3)*tan(1/2*f*x+1/2*e)-3*A*c*d^2+A*d^3-3*B*c^2*d+3*B*c*d^2-5/3*B*d 
^3)/(1+tan(1/2*f*x+1/2*e)^2)^3+1/2*(6*A*c^2*d-6*A*c*d^2+3*A*d^3+2*B*c^3-6* 
B*c^2*d+9*B*c*d^2-3*B*d^3)*arctan(tan(1/2*f*x+1/2*e))-(A*c^3-3*A*c^2*d+3*A 
*c*d^2-A*d^3-B*c^3+3*B*c^2*d-3*B*c*d^2+B*d^3)/(tan(1/2*f*x+1/2*e)+1))
 

Fricas [B] (verification not implemented)

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

Time = 0.15 (sec) , antiderivative size = 470, normalized size of antiderivative = 2.14 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{a+a \sin (e+f x)} \, dx=\frac {2 \, B d^{3} \cos \left (f x + e\right )^{4} - 6 \, {\left (A - B\right )} c^{3} + 18 \, {\left (A - B\right )} c^{2} d - 18 \, {\left (A - B\right )} c d^{2} + 6 \, {\left (A - B\right )} d^{3} + {\left (9 \, B c d^{2} + {\left (3 \, A - B\right )} d^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (2 \, B c^{3} + 6 \, {\left (A - B\right )} c^{2} d - 3 \, {\left (2 \, A - 3 \, B\right )} c d^{2} + 3 \, {\left (A - B\right )} d^{3}\right )} f x - 6 \, {\left (3 \, B c^{2} d + 3 \, {\left (A - B\right )} c d^{2} - {\left (A - 2 \, B\right )} d^{3}\right )} \cos \left (f x + e\right )^{2} - 3 \, {\left (2 \, {\left (A - B\right )} c^{3} - 6 \, {\left (A - 2 \, B\right )} c^{2} d + 3 \, {\left (4 \, A - 3 \, B\right )} c d^{2} - {\left (3 \, A - 5 \, B\right )} d^{3} - {\left (2 \, B c^{3} + 6 \, {\left (A - B\right )} c^{2} d - 3 \, {\left (2 \, A - 3 \, B\right )} c d^{2} + 3 \, {\left (A - B\right )} d^{3}\right )} f x\right )} \cos \left (f x + e\right ) + {\left (2 \, B d^{3} \cos \left (f x + e\right )^{3} + 6 \, {\left (A - B\right )} c^{3} - 18 \, {\left (A - B\right )} c^{2} d + 18 \, {\left (A - B\right )} c d^{2} - 6 \, {\left (A - B\right )} d^{3} + 3 \, {\left (2 \, B c^{3} + 6 \, {\left (A - B\right )} c^{2} d - 3 \, {\left (2 \, A - 3 \, B\right )} c d^{2} + 3 \, {\left (A - B\right )} d^{3}\right )} f x - 3 \, {\left (3 \, B c d^{2} + {\left (A - B\right )} d^{3}\right )} \cos \left (f x + e\right )^{2} - 3 \, {\left (6 \, B c^{2} d + 3 \, {\left (2 \, A - B\right )} c d^{2} - {\left (A - 3 \, B\right )} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{6 \, {\left (a f \cos \left (f x + e\right ) + a f \sin \left (f x + e\right ) + a f\right )}} \] Input:

integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3/(a+a*sin(f*x+e)),x, algorith 
m="fricas")
 

Output:

1/6*(2*B*d^3*cos(f*x + e)^4 - 6*(A - B)*c^3 + 18*(A - B)*c^2*d - 18*(A - B 
)*c*d^2 + 6*(A - B)*d^3 + (9*B*c*d^2 + (3*A - B)*d^3)*cos(f*x + e)^3 + 3*( 
2*B*c^3 + 6*(A - B)*c^2*d - 3*(2*A - 3*B)*c*d^2 + 3*(A - B)*d^3)*f*x - 6*( 
3*B*c^2*d + 3*(A - B)*c*d^2 - (A - 2*B)*d^3)*cos(f*x + e)^2 - 3*(2*(A - B) 
*c^3 - 6*(A - 2*B)*c^2*d + 3*(4*A - 3*B)*c*d^2 - (3*A - 5*B)*d^3 - (2*B*c^ 
3 + 6*(A - B)*c^2*d - 3*(2*A - 3*B)*c*d^2 + 3*(A - B)*d^3)*f*x)*cos(f*x + 
e) + (2*B*d^3*cos(f*x + e)^3 + 6*(A - B)*c^3 - 18*(A - B)*c^2*d + 18*(A - 
B)*c*d^2 - 6*(A - B)*d^3 + 3*(2*B*c^3 + 6*(A - B)*c^2*d - 3*(2*A - 3*B)*c* 
d^2 + 3*(A - B)*d^3)*f*x - 3*(3*B*c*d^2 + (A - B)*d^3)*cos(f*x + e)^2 - 3* 
(6*B*c^2*d + 3*(2*A - B)*c*d^2 - (A - 3*B)*d^3)*cos(f*x + e))*sin(f*x + e) 
)/(a*f*cos(f*x + e) + a*f*sin(f*x + e) + a*f)
 

Sympy [B] (verification not implemented)

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

Time = 4.31 (sec) , antiderivative size = 14644, normalized size of antiderivative = 66.56 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{a+a \sin (e+f x)} \, dx=\text {Too large to display} \] Input:

integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))**3/(a+a*sin(f*x+e)),x)
 

Output:

Piecewise((-12*A*c**3*tan(e/2 + f*x/2)**6/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a 
*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x 
/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*t 
an(e/2 + f*x/2) + 6*a*f) - 36*A*c**3*tan(e/2 + f*x/2)**4/(6*a*f*tan(e/2 + 
f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a* 
f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/ 
2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 36*A*c**3*tan(e/2 + f*x/2)**2/(6 
*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f* 
x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f 
*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 12*A*c**3/(6*a*f* 
tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)* 
*5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan( 
e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 18*A*c**2*d*f*x*tan(e/ 
2 + f*x/2)**7/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18* 
a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f* 
x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 1 
8*A*c**2*d*f*x*tan(e/2 + f*x/2)**6/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan( 
e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 
+ 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 
+ f*x/2) + 6*a*f) + 54*A*c**2*d*f*x*tan(e/2 + f*x/2)**5/(6*a*f*tan(e/2 ...
 

Maxima [B] (verification not implemented)

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

Time = 0.14 (sec) , antiderivative size = 1124, normalized size of antiderivative = 5.11 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{a+a \sin (e+f x)} \, dx=\text {Too large to display} \] Input:

integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3/(a+a*sin(f*x+e)),x, algorith 
m="maxima")
 

Output:

-1/3*(B*d^3*((7*sin(f*x + e)/(cos(f*x + e) + 1) + 39*sin(f*x + e)^2/(cos(f 
*x + e) + 1)^2 + 24*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 24*sin(f*x + e)^ 
4/(cos(f*x + e) + 1)^4 + 9*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 9*sin(f*x 
 + e)^6/(cos(f*x + e) + 1)^6 + 16)/(a + a*sin(f*x + e)/(cos(f*x + e) + 1) 
+ 3*a*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 3*a*sin(f*x + e)^3/(cos(f*x + 
e) + 1)^3 + 3*a*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 3*a*sin(f*x + e)^5/( 
cos(f*x + e) + 1)^5 + a*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + a*sin(f*x + 
e)^7/(cos(f*x + e) + 1)^7) + 9*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a) 
- 9*B*c*d^2*((sin(f*x + e)/(cos(f*x + e) + 1) + 5*sin(f*x + e)^2/(cos(f*x 
+ e) + 1)^2 + 3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*sin(f*x + e)^4/(co 
s(f*x + e) + 1)^4 + 4)/(a + a*sin(f*x + e)/(cos(f*x + e) + 1) + 2*a*sin(f* 
x + e)^2/(cos(f*x + e) + 1)^2 + 2*a*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 
a*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + a*sin(f*x + e)^5/(cos(f*x + e) + 1 
)^5) + 3*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a) - 3*A*d^3*((sin(f*x + 
e)/(cos(f*x + e) + 1) + 5*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 3*sin(f*x 
+ e)^3/(cos(f*x + e) + 1)^3 + 3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 4)/( 
a + a*sin(f*x + e)/(cos(f*x + e) + 1) + 2*a*sin(f*x + e)^2/(cos(f*x + e) + 
 1)^2 + 2*a*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + a*sin(f*x + e)^4/(cos(f* 
x + e) + 1)^4 + a*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 3*arctan(sin(f*x 
+ e)/(cos(f*x + e) + 1))/a) + 18*B*c^2*d*((sin(f*x + e)/(cos(f*x + e) +...
 

Giac [B] (verification not implemented)

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

Time = 0.22 (sec) , antiderivative size = 460, normalized size of antiderivative = 2.09 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{a+a \sin (e+f x)} \, dx=\frac {\frac {3 \, {\left (2 \, B c^{3} + 6 \, A c^{2} d - 6 \, B c^{2} d - 6 \, A c d^{2} + 9 \, B c d^{2} + 3 \, A d^{3} - 3 \, B d^{3}\right )} {\left (f x + e\right )}}{a} - \frac {12 \, {\left (A c^{3} - B c^{3} - 3 \, A c^{2} d + 3 \, B c^{2} d + 3 \, A c d^{2} - 3 \, B c d^{2} - A d^{3} + B d^{3}\right )}}{a {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}} + \frac {2 \, {\left (9 \, B c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3 \, A d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 3 \, B d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 18 \, B c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 18 \, A c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 18 \, B c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 6 \, A d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 6 \, B d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 36 \, B c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 36 \, A c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 36 \, B c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 12 \, A d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 24 \, B d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 9 \, B c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, A d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, B d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 18 \, B c^{2} d - 18 \, A c d^{2} + 18 \, B c d^{2} + 6 \, A d^{3} - 10 \, B d^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{3} a}}{6 \, f} \] Input:

integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3/(a+a*sin(f*x+e)),x, algorith 
m="giac")
 

Output:

1/6*(3*(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)*(f*x + e)/a - 12*(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)/(a*(tan(1/2*f*x + 1/2*e) + 1)) + 2*(9*B* 
c*d^2*tan(1/2*f*x + 1/2*e)^5 + 3*A*d^3*tan(1/2*f*x + 1/2*e)^5 - 3*B*d^3*ta 
n(1/2*f*x + 1/2*e)^5 - 18*B*c^2*d*tan(1/2*f*x + 1/2*e)^4 - 18*A*c*d^2*tan( 
1/2*f*x + 1/2*e)^4 + 18*B*c*d^2*tan(1/2*f*x + 1/2*e)^4 + 6*A*d^3*tan(1/2*f 
*x + 1/2*e)^4 - 6*B*d^3*tan(1/2*f*x + 1/2*e)^4 - 36*B*c^2*d*tan(1/2*f*x + 
1/2*e)^2 - 36*A*c*d^2*tan(1/2*f*x + 1/2*e)^2 + 36*B*c*d^2*tan(1/2*f*x + 1/ 
2*e)^2 + 12*A*d^3*tan(1/2*f*x + 1/2*e)^2 - 24*B*d^3*tan(1/2*f*x + 1/2*e)^2 
 - 9*B*c*d^2*tan(1/2*f*x + 1/2*e) - 3*A*d^3*tan(1/2*f*x + 1/2*e) + 3*B*d^3 
*tan(1/2*f*x + 1/2*e) - 18*B*c^2*d - 18*A*c*d^2 + 18*B*c*d^2 + 6*A*d^3 - 1 
0*B*d^3)/((tan(1/2*f*x + 1/2*e)^2 + 1)^3*a))/f
 

Mupad [B] (verification not implemented)

Time = 38.14 (sec) , antiderivative size = 839, normalized size of antiderivative = 3.81 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{a+a \sin (e+f x)} \, dx=\text {Too large to display} \] Input:

int(((A + B*sin(e + f*x))*(c + d*sin(e + f*x))^3)/(a + a*sin(e + f*x)),x)
 

Output:

-(12*A*c^3*cos(e/2 + (f*x)/2) - 18*A*d^3*cos(e/2 + (f*x)/2) - 12*B*c^3*cos 
(e/2 + (f*x)/2) + 18*B*d^3*cos(e/2 + (f*x)/2) + 6*A*d^3*cos(e/2 + (f*x)/2) 
^3 - 12*A*d^3*cos(e/2 + (f*x)/2)^5 - 6*B*d^3*cos(e/2 + (f*x)/2)^3 + 36*B*d 
^3*cos(e/2 + (f*x)/2)^5 - 16*B*d^3*cos(e/2 + (f*x)/2)^7 - 9*A*d^3*cos(e/2 
+ (f*x)/2)*(e + f*x) - 6*B*c^3*cos(e/2 + (f*x)/2)*(e + f*x) + 9*B*d^3*cos( 
e/2 + (f*x)/2)*(e + f*x) - 9*A*d^3*sin(e/2 + (f*x)/2)*(e + f*x) - 6*B*c^3* 
sin(e/2 + (f*x)/2)*(e + f*x) + 9*B*d^3*sin(e/2 + (f*x)/2)*(e + f*x) - 18*A 
*d^3*cos(e/2 + (f*x)/2)^2*sin(e/2 + (f*x)/2) + 12*A*d^3*cos(e/2 + (f*x)/2) 
^4*sin(e/2 + (f*x)/2) + 18*B*d^3*cos(e/2 + (f*x)/2)^2*sin(e/2 + (f*x)/2) + 
 12*B*d^3*cos(e/2 + (f*x)/2)^4*sin(e/2 + (f*x)/2) - 16*B*d^3*cos(e/2 + (f* 
x)/2)^6*sin(e/2 + (f*x)/2) + 36*A*c*d^2*cos(e/2 + (f*x)/2) - 36*A*c^2*d*co 
s(e/2 + (f*x)/2) - 54*B*c*d^2*cos(e/2 + (f*x)/2) + 36*B*c^2*d*cos(e/2 + (f 
*x)/2) + 36*A*c*d^2*cos(e/2 + (f*x)/2)^3 + 18*B*c*d^2*cos(e/2 + (f*x)/2)^3 
 + 36*B*c^2*d*cos(e/2 + (f*x)/2)^3 - 36*B*c*d^2*cos(e/2 + (f*x)/2)^5 + 18* 
A*c*d^2*cos(e/2 + (f*x)/2)*(e + f*x) - 18*A*c^2*d*cos(e/2 + (f*x)/2)*(e + 
f*x) - 27*B*c*d^2*cos(e/2 + (f*x)/2)*(e + f*x) + 18*B*c^2*d*cos(e/2 + (f*x 
)/2)*(e + f*x) + 18*A*c*d^2*sin(e/2 + (f*x)/2)*(e + f*x) - 18*A*c^2*d*sin( 
e/2 + (f*x)/2)*(e + f*x) - 27*B*c*d^2*sin(e/2 + (f*x)/2)*(e + f*x) + 18*B* 
c^2*d*sin(e/2 + (f*x)/2)*(e + f*x) + 36*A*c*d^2*cos(e/2 + (f*x)/2)^2*sin(e 
/2 + (f*x)/2) - 54*B*c*d^2*cos(e/2 + (f*x)/2)^2*sin(e/2 + (f*x)/2) + 36...
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 810, normalized size of antiderivative = 3.68 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{a+a \sin (e+f x)} \, dx =\text {Too large to display} \] Input:

int((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3/(a+a*sin(f*x+e)),x)
 

Output:

(2*cos(e + f*x)*sin(e + f*x)**3*b*d**3 + 3*cos(e + f*x)*sin(e + f*x)**2*a* 
d**3 + 9*cos(e + f*x)*sin(e + f*x)**2*b*c*d**2 - cos(e + f*x)*sin(e + f*x) 
**2*b*d**3 + 18*cos(e + f*x)*sin(e + f*x)*a*c*d**2 - 3*cos(e + f*x)*sin(e 
+ f*x)*a*d**3 + 18*cos(e + f*x)*sin(e + f*x)*b*c**2*d - 9*cos(e + f*x)*sin 
(e + f*x)*b*c*d**2 + 7*cos(e + f*x)*sin(e + f*x)*b*d**3 + 12*cos(e + f*x)* 
a*c**3 + 18*cos(e + f*x)*a*c**2*d*f*x - 36*cos(e + f*x)*a*c**2*d - 18*cos( 
e + f*x)*a*c*d**2*f*x + 36*cos(e + f*x)*a*c*d**2 + 9*cos(e + f*x)*a*d**3*f 
*x - 18*cos(e + f*x)*a*d**3 + 6*cos(e + f*x)*b*c**3*f*x - 12*cos(e + f*x)* 
b*c**3 - 18*cos(e + f*x)*b*c**2*d*f*x + 36*cos(e + f*x)*b*c**2*d + 27*cos( 
e + f*x)*b*c*d**2*f*x - 54*cos(e + f*x)*b*c*d**2 - 9*cos(e + f*x)*b*d**3*f 
*x + 18*cos(e + f*x)*b*d**3 + 2*sin(e + f*x)**4*b*d**3 + 3*sin(e + f*x)**3 
*a*d**3 + 9*sin(e + f*x)**3*b*c*d**2 - 3*sin(e + f*x)**3*b*d**3 + 18*sin(e 
 + f*x)**2*a*c*d**2 - 6*sin(e + f*x)**2*a*d**3 + 18*sin(e + f*x)**2*b*c**2 
*d - 18*sin(e + f*x)**2*b*c*d**2 + 8*sin(e + f*x)**2*b*d**3 - 18*sin(e + f 
*x)*a*c**2*d*f*x + 18*sin(e + f*x)*a*c*d**2*f*x + 18*sin(e + f*x)*a*c*d**2 
 - 9*sin(e + f*x)*a*d**3*f*x - 3*sin(e + f*x)*a*d**3 - 6*sin(e + f*x)*b*c* 
*3*f*x + 18*sin(e + f*x)*b*c**2*d*f*x + 18*sin(e + f*x)*b*c**2*d - 27*sin( 
e + f*x)*b*c*d**2*f*x - 9*sin(e + f*x)*b*c*d**2 + 9*sin(e + f*x)*b*d**3*f* 
x + 7*sin(e + f*x)*b*d**3 - 12*a*c**3 - 18*a*c**2*d*f*x + 36*a*c**2*d + 18 
*a*c*d**2*f*x - 36*a*c*d**2 - 9*a*d**3*f*x + 18*a*d**3 - 6*b*c**3*f*x +...