\(\int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^4} \, dx\) [700]

Optimal result

Integrand size = 25, antiderivative size = 325 \[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^4} \, dx=-\frac {(a c-b d) \left (10 a b c d-b^2 \left (3 c^2+2 d^2\right )-a^2 \left (2 c^2+3 d^2\right )\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\left (c^2-d^2\right )^{7/2} f}+\frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}+\frac {(b c-a d)^2 \left (2 b c^2+5 a c d-7 b d^2\right ) \cos (e+f x)}{6 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}-\frac {(b c-a d) \left (5 a b c d \left (c^2-7 d^2\right )+a^2 d^2 \left (11 c^2+4 d^2\right )+b^2 \left (2 c^4-5 c^2 d^2+18 d^4\right )\right ) \cos (e+f x)}{6 d^2 \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))} \] Output:

-(a*c-b*d)*(10*a*b*c*d-b^2*(3*c^2+2*d^2)-a^2*(2*c^2+3*d^2))*arctan((d+c*ta 
n(1/2*f*x+1/2*e))/(c^2-d^2)^(1/2))/(c^2-d^2)^(7/2)/f+1/3*(-a*d+b*c)^2*cos( 
f*x+e)*(a+b*sin(f*x+e))/d/(c^2-d^2)/f/(c+d*sin(f*x+e))^3+1/6*(-a*d+b*c)^2* 
(5*a*c*d+2*b*c^2-7*b*d^2)*cos(f*x+e)/d^2/(c^2-d^2)^2/f/(c+d*sin(f*x+e))^2- 
1/6*(-a*d+b*c)*(5*a*b*c*d*(c^2-7*d^2)+a^2*d^2*(11*c^2+4*d^2)+b^2*(2*c^4-5* 
c^2*d^2+18*d^4))*cos(f*x+e)/d^2/(c^2-d^2)^3/f/(c+d*sin(f*x+e))
 

Mathematica [A] (verified)

Time = 5.98 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^4} \, dx=\frac {\frac {6 \left (-3 a^2 b d \left (4 c^2+d^2\right )-b^3 d \left (3 c^2+2 d^2\right )+3 a b^2 c \left (c^2+4 d^2\right )+a^3 \left (2 c^3+3 c d^2\right )\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\left (c^2-d^2\right )^{7/2}}+\frac {2 (b c-a d)^3 \cos (e+f x)}{d^2 \left (-c^2+d^2\right ) (c+d \sin (e+f x))^3}+\frac {(b c-a d)^2 \left (4 b c^2+5 a c d-9 b d^2\right ) \cos (e+f x)}{d^2 \left (c^2-d^2\right )^2 (c+d \sin (e+f x))^2}+\frac {\left (-a^3 d^3 \left (11 c^2+4 d^2\right )+3 a^2 b c d^2 \left (2 c^2+13 d^2\right )+3 a b^2 d \left (c^4-10 c^2 d^2-6 d^4\right )+b^3 \left (2 c^5-5 c^3 d^2+18 c d^4\right )\right ) \cos (e+f x)}{d^2 \left (-c^2+d^2\right )^3 (c+d \sin (e+f x))}}{6 f} \] Input:

Integrate[(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^4,x]
 

Output:

((6*(-3*a^2*b*d*(4*c^2 + d^2) - b^3*d*(3*c^2 + 2*d^2) + 3*a*b^2*c*(c^2 + 4 
*d^2) + a^3*(2*c^3 + 3*c*d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - 
d^2]])/(c^2 - d^2)^(7/2) + (2*(b*c - a*d)^3*Cos[e + f*x])/(d^2*(-c^2 + d^2 
)*(c + d*Sin[e + f*x])^3) + ((b*c - a*d)^2*(4*b*c^2 + 5*a*c*d - 9*b*d^2)*C 
os[e + f*x])/(d^2*(c^2 - d^2)^2*(c + d*Sin[e + f*x])^2) + ((-(a^3*d^3*(11* 
c^2 + 4*d^2)) + 3*a^2*b*c*d^2*(2*c^2 + 13*d^2) + 3*a*b^2*d*(c^4 - 10*c^2*d 
^2 - 6*d^4) + b^3*(2*c^5 - 5*c^3*d^2 + 18*c*d^4))*Cos[e + f*x])/(d^2*(-c^2 
 + d^2)^3*(c + d*Sin[e + f*x])))/(6*f)
 

Rubi [A] (verified)

Time = 1.50 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.14, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 3271, 3042, 3500, 25, 3042, 3233, 27, 3042, 3139, 1083, 217}

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.

\(\displaystyle \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^4} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^4}dx\)

\(\Big \downarrow \) 3271

\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^3}-\frac {\int \frac {-3 c d a^3+7 b d^2 a^2-5 b^2 c d a+b^3 c^2-b \left (\left (2 c^2-3 d^2\right ) b^2+2 a c d b-a^2 d^2\right ) \sin ^2(e+f x)-\left (-2 d^2 a^3+7 b c d a^2+b^2 \left (c^2-9 d^2\right ) a+3 b^3 c d\right ) \sin (e+f x)}{(c+d \sin (e+f x))^3}dx}{3 d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^3}-\frac {\int \frac {-3 c d a^3+7 b d^2 a^2-5 b^2 c d a+b^3 c^2-b \left (\left (2 c^2-3 d^2\right ) b^2+2 a c d b-a^2 d^2\right ) \sin (e+f x)^2-\left (-2 d^2 a^3+7 b c d a^2+b^2 \left (c^2-9 d^2\right ) a+3 b^3 c d\right ) \sin (e+f x)}{(c+d \sin (e+f x))^3}dx}{3 d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 3500

\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^3}-\frac {-\frac {\int -\frac {2 d \left (-d \left (3 c^2+2 d^2\right ) a^3+15 b c d^2 a^2-b^2 \left (9 d^3+6 c^2 d\right ) a-b^3 \left (c^3-6 c d^2\right )\right )+\left (-\left (\left (2 c^4-3 d^2 c^2+6 d^4\right ) b^3\right )-3 a c d \left (c^2-6 d^2\right ) b^2-3 a^2 d^2 \left (2 c^2+3 d^2\right ) b+5 a^3 c d^3\right ) \sin (e+f x)}{(c+d \sin (e+f x))^2}dx}{2 d \left (c^2-d^2\right )}-\frac {\left (5 a c d+2 b c^2-7 b d^2\right ) (b c-a d)^2 \cos (e+f x)}{2 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{3 d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^3}-\frac {\frac {\int \frac {2 d \left (-d \left (3 c^2+2 d^2\right ) a^3+15 b c d^2 a^2-3 b^2 d \left (2 c^2+3 d^2\right ) a-b^3 \left (c^3-6 c d^2\right )\right )+\left (-\left (\left (2 c^4-3 d^2 c^2+6 d^4\right ) b^3\right )-3 a c d \left (c^2-6 d^2\right ) b^2-3 a^2 d^2 \left (2 c^2+3 d^2\right ) b+5 a^3 c d^3\right ) \sin (e+f x)}{(c+d \sin (e+f x))^2}dx}{2 d \left (c^2-d^2\right )}-\frac {(b c-a d)^2 \left (5 a c d+2 b c^2-7 b d^2\right ) \cos (e+f x)}{2 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{3 d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^3}-\frac {\frac {\int \frac {2 d \left (-d \left (3 c^2+2 d^2\right ) a^3+15 b c d^2 a^2-3 b^2 d \left (2 c^2+3 d^2\right ) a-b^3 \left (c^3-6 c d^2\right )\right )+\left (-\left (\left (2 c^4-3 d^2 c^2+6 d^4\right ) b^3\right )-3 a c d \left (c^2-6 d^2\right ) b^2-3 a^2 d^2 \left (2 c^2+3 d^2\right ) b+5 a^3 c d^3\right ) \sin (e+f x)}{(c+d \sin (e+f x))^2}dx}{2 d \left (c^2-d^2\right )}-\frac {(b c-a d)^2 \left (5 a c d+2 b c^2-7 b d^2\right ) \cos (e+f x)}{2 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{3 d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 3233

\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^3}-\frac {\frac {\frac {(b c-a d) \left (11 a^2 c^2 d^2+4 a^2 d^4+5 a b c^3 d-35 a b c d^3+2 b^2 c^4-5 b^2 c^2 d^2+18 b^2 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}-\frac {\int -\frac {3 d^2 (a c-b d) \left (-\left (\left (2 c^2+3 d^2\right ) a^2\right )+10 b c d a-b^2 \left (3 c^2+2 d^2\right )\right )}{c+d \sin (e+f x)}dx}{c^2-d^2}}{2 d \left (c^2-d^2\right )}-\frac {(b c-a d)^2 \left (5 a c d+2 b c^2-7 b d^2\right ) \cos (e+f x)}{2 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{3 d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^3}-\frac {\frac {\frac {3 d^2 (a c-b d) \left (-\left (a^2 \left (2 c^2+3 d^2\right )\right )+10 a b c d-b^2 \left (3 c^2+2 d^2\right )\right ) \int \frac {1}{c+d \sin (e+f x)}dx}{c^2-d^2}+\frac {(b c-a d) \left (11 a^2 c^2 d^2+4 a^2 d^4+5 a b c^3 d-35 a b c d^3+2 b^2 c^4-5 b^2 c^2 d^2+18 b^2 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 d \left (c^2-d^2\right )}-\frac {(b c-a d)^2 \left (5 a c d+2 b c^2-7 b d^2\right ) \cos (e+f x)}{2 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{3 d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^3}-\frac {\frac {\frac {3 d^2 (a c-b d) \left (-\left (a^2 \left (2 c^2+3 d^2\right )\right )+10 a b c d-b^2 \left (3 c^2+2 d^2\right )\right ) \int \frac {1}{c+d \sin (e+f x)}dx}{c^2-d^2}+\frac {(b c-a d) \left (11 a^2 c^2 d^2+4 a^2 d^4+5 a b c^3 d-35 a b c d^3+2 b^2 c^4-5 b^2 c^2 d^2+18 b^2 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 d \left (c^2-d^2\right )}-\frac {(b c-a d)^2 \left (5 a c d+2 b c^2-7 b d^2\right ) \cos (e+f x)}{2 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{3 d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 3139

\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^3}-\frac {\frac {\frac {6 d^2 (a c-b d) \left (-\left (a^2 \left (2 c^2+3 d^2\right )\right )+10 a b c d-b^2 \left (3 c^2+2 d^2\right )\right ) \int \frac {1}{c \tan ^2\left (\frac {1}{2} (e+f x)\right )+2 d \tan \left (\frac {1}{2} (e+f x)\right )+c}d\tan \left (\frac {1}{2} (e+f x)\right )}{f \left (c^2-d^2\right )}+\frac {(b c-a d) \left (11 a^2 c^2 d^2+4 a^2 d^4+5 a b c^3 d-35 a b c d^3+2 b^2 c^4-5 b^2 c^2 d^2+18 b^2 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 d \left (c^2-d^2\right )}-\frac {(b c-a d)^2 \left (5 a c d+2 b c^2-7 b d^2\right ) \cos (e+f x)}{2 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{3 d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^3}-\frac {\frac {\frac {(b c-a d) \left (11 a^2 c^2 d^2+4 a^2 d^4+5 a b c^3 d-35 a b c d^3+2 b^2 c^4-5 b^2 c^2 d^2+18 b^2 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}-\frac {12 d^2 (a c-b d) \left (-\left (a^2 \left (2 c^2+3 d^2\right )\right )+10 a b c d-b^2 \left (3 c^2+2 d^2\right )\right ) \int \frac {1}{-\left (2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )^2-4 \left (c^2-d^2\right )}d\left (2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )}{f \left (c^2-d^2\right )}}{2 d \left (c^2-d^2\right )}-\frac {(b c-a d)^2 \left (5 a c d+2 b c^2-7 b d^2\right ) \cos (e+f x)}{2 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{3 d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^3}-\frac {\frac {\frac {6 d^2 (a c-b d) \left (-\left (a^2 \left (2 c^2+3 d^2\right )\right )+10 a b c d-b^2 \left (3 c^2+2 d^2\right )\right ) \arctan \left (\frac {2 c \tan \left (\frac {1}{2} (e+f x)\right )+2 d}{2 \sqrt {c^2-d^2}}\right )}{f \left (c^2-d^2\right )^{3/2}}+\frac {(b c-a d) \left (11 a^2 c^2 d^2+4 a^2 d^4+5 a b c^3 d-35 a b c d^3+2 b^2 c^4-5 b^2 c^2 d^2+18 b^2 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 d \left (c^2-d^2\right )}-\frac {(b c-a d)^2 \left (5 a c d+2 b c^2-7 b d^2\right ) \cos (e+f x)}{2 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{3 d \left (c^2-d^2\right )}\)

Input:

Int[(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^4,x]
 

Output:

((b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(3*d*(c^2 - d^2)*f*(c + 
d*Sin[e + f*x])^3) - (-1/2*((b*c - a*d)^2*(2*b*c^2 + 5*a*c*d - 7*b*d^2)*Co 
s[e + f*x])/(d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^2) + ((6*d^2*(a*c - b*d) 
*(10*a*b*c*d - b^2*(3*c^2 + 2*d^2) - a^2*(2*c^2 + 3*d^2))*ArcTan[(2*d + 2* 
c*Tan[(e + f*x)/2])/(2*Sqrt[c^2 - d^2])])/((c^2 - d^2)^(3/2)*f) + ((b*c - 
a*d)*(2*b^2*c^4 + 5*a*b*c^3*d + 11*a^2*c^2*d^2 - 5*b^2*c^2*d^2 - 35*a*b*c* 
d^3 + 4*a^2*d^4 + 18*b^2*d^4)*Cos[e + f*x])/((c^2 - d^2)*f*(c + d*Sin[e + 
f*x])))/(2*d*(c^2 - d^2)))/(3*d*(c^2 - d^2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 3042

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

rule 3139

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre 
eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d)   Subst[Int[1/(a + 2*b*e*x + a 
*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ 
[a^2 - b^2, 0]
 

rule 3233

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

rule 3271

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Co 
s[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f* 
(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2))   Int[(a + b*Sin 
[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^ 
2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 + c*d*(a^2 
+ b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - 
 d^2) - m*(b*c - a*d)^2 + d*n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x] 
, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - 
b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || 
IntegersQ[2*m, 2*n])
 

rule 3500

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + 
 (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 
 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)* 
(a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2))   Int[(a + b*Sin[e + f*x 
])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A 
*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, 
B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1153\) vs. \(2(314)=628\).

Time = 4.72 (sec) , antiderivative size = 1154, normalized size of antiderivative = 3.55

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1154\)
default	\(\text {Expression too large to display}\)	\(1154\)
risch	\(\text {Expression too large to display}\)	\(2740\)

Input:

int((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^4,x,method=_RETURNVERBOSE)
 

Output:

1/f*(2*(1/2*(9*a^3*c^4*d^2-6*a^3*c^2*d^4+2*a^3*d^6-12*a^2*b*c^5*d-3*a^2*b* 
c^3*d^3+3*a*b^2*c^6+12*a*b^2*c^4*d^2-3*b^3*c^5*d-2*b^3*c^3*d^3)/c/(c^6-3*c 
^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^5+1/2*(6*a^3*c^6*d+27*a^3*c^4*d^3 
-12*a^3*c^2*d^5+4*a^3*d^7-6*a^2*b*c^7-42*a^2*b*c^5*d^2-33*a^2*b*c^3*d^4+6* 
a^2*b*c*d^6+15*a*b^2*c^6*d+60*a*b^2*c^4*d^3-15*b^3*c^5*d^2-10*b^3*c^3*d^4) 
/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)/c^2*tan(1/2*f*x+1/2*e)^4+1/3/c^3*d*(54*a^3* 
c^6*d+21*a^3*c^4*d^3-4*a^3*c^2*d^5+4*a^3*d^7-54*a^2*b*c^7-126*a^2*b*c^5*d^ 
2-51*a^2*b*c^3*d^4+6*a^2*b*c*d^6+117*a*b^2*c^6*d+96*a*b^2*c^4*d^3+12*a*b^2 
*c^2*d^5-12*b^3*c^7-41*b^3*c^5*d^2-22*b^3*c^3*d^4)/(c^6-3*c^4*d^2+3*c^2*d^ 
4-d^6)*tan(1/2*f*x+1/2*e)^3+(6*a^3*c^6*d+20*a^3*c^4*d^3-3*a^3*c^2*d^5+2*a^ 
3*d^7-6*a^2*b*c^7-30*a^2*b*c^5*d^2-42*a^2*b*c^3*d^4+3*a^2*b*c*d^6+12*a*b^2 
*c^6*d+51*a*b^2*c^4*d^3+12*a*b^2*c^2*d^5-2*b^3*c^7-6*b^3*c^5*d^2-17*b^3*c^ 
3*d^4)/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)/c^2*tan(1/2*f*x+1/2*e)^2+1/2*(27*a^3* 
c^4*d^2-4*a^3*c^2*d^4+2*a^3*d^6-24*a^2*b*c^5*d-57*a^2*b*c^3*d^3+6*a^2*b*c* 
d^5-3*a*b^2*c^6+66*a*b^2*c^4*d^2+12*a*b^2*c^2*d^4-5*b^3*c^5*d-20*b^3*c^3*d 
^3)/c/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)+1/6*(18*a^3*c^4*d-5 
*a^3*c^2*d^3+2*a^3*d^5-18*a^2*b*c^5-30*a^2*b*c^3*d^2+3*a^2*b*c*d^4+39*a*b^ 
2*c^4*d+6*a*b^2*c^2*d^3-4*b^3*c^5-11*b^3*c^3*d^2)/(c^6-3*c^4*d^2+3*c^2*d^4 
-d^6))/(tan(1/2*f*x+1/2*e)^2*c+2*d*tan(1/2*f*x+1/2*e)+c)^3+(2*a^3*c^3+3*a^ 
3*c*d^2-12*a^2*b*c^2*d-3*a^2*b*d^3+3*a*b^2*c^3+12*a*b^2*c*d^2-3*b^3*c^2...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1026 vs. \(2 (314) = 628\).

Time = 0.21 (sec) , antiderivative size = 2136, normalized size of antiderivative = 6.57 \[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^4} \, dx=\text {Too large to display} \] Input:

integrate((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^4,x, algorithm="fricas")
 

Output:

[-1/12*(2*(2*b^3*c^7 + 3*a*b^2*c^6*d + (6*a^2*b - 7*b^3)*c^5*d^2 - 11*(a^3 
 + 3*a*b^2)*c^4*d^3 + (33*a^2*b + 23*b^3)*c^3*d^4 + (7*a^3 + 12*a*b^2)*c^2 
*d^5 - 3*(13*a^2*b + 6*b^3)*c*d^6 + 2*(2*a^3 + 9*a*b^2)*d^7)*cos(f*x + e)^ 
3 - 6*(3*a*b^2*c^7 + 3*a^2*b*d^7 + (6*a^2*b + b^3)*c^6*d - 3*(3*a^3 + 10*a 
*b^2)*c^5*d^2 + (21*a^2*b + 8*b^3)*c^4*d^3 + (8*a^3 + 21*a*b^2)*c^3*d^4 - 
3*(10*a^2*b + 3*b^3)*c^2*d^5 + (a^3 + 6*a*b^2)*c*d^6)*cos(f*x + e)*sin(f*x 
 + e) - 3*((2*a^3 + 3*a*b^2)*c^6 - 3*(4*a^2*b + b^3)*c^5*d + 3*(3*a^3 + 7* 
a*b^2)*c^4*d^2 - (39*a^2*b + 11*b^3)*c^3*d^3 + 9*(a^3 + 4*a*b^2)*c^2*d^4 - 
 3*(3*a^2*b + 2*b^3)*c*d^5 - 3*((2*a^3 + 3*a*b^2)*c^4*d^2 - 3*(4*a^2*b + b 
^3)*c^3*d^3 + 3*(a^3 + 4*a*b^2)*c^2*d^4 - (3*a^2*b + 2*b^3)*c*d^5)*cos(f*x 
 + e)^2 + (3*(2*a^3 + 3*a*b^2)*c^5*d - 9*(4*a^2*b + b^3)*c^4*d^2 + (11*a^3 
 + 39*a*b^2)*c^3*d^3 - 3*(7*a^2*b + 3*b^3)*c^2*d^4 + 3*(a^3 + 4*a*b^2)*c*d 
^5 - (3*a^2*b + 2*b^3)*d^6 - ((2*a^3 + 3*a*b^2)*c^3*d^3 - 3*(4*a^2*b + b^3 
)*c^2*d^4 + 3*(a^3 + 4*a*b^2)*c*d^5 - (3*a^2*b + 2*b^3)*d^6)*cos(f*x + e)^ 
2)*sin(f*x + e))*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c* 
d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + 
e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 
)) - 12*(3*a^2*b*c^5*d^2 + 2*a^3*c^4*d^3 + 2*b^3*c^3*d^4 + 3*a*b^2*c^2*d^5 
 + (3*a^2*b + b^3)*c^7 - 3*(a^3 + 2*a*b^2)*c^6*d - 3*(2*a^2*b + b^3)*c*d^6 
 + (a^3 + 3*a*b^2)*d^7)*cos(f*x + e))/(3*(c^9*d^2 - 4*c^7*d^4 + 6*c^5*d...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^4} \, dx=\text {Timed out} \] Input:

integrate((a+b*sin(f*x+e))**3/(c+d*sin(f*x+e))**4,x)
 

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^4} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^4,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*d^2-4*c^2>0)', see `assume?` f 
or more de
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1613 vs. \(2 (314) = 628\).

Time = 0.21 (sec) , antiderivative size = 1613, normalized size of antiderivative = 4.96 \[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^4} \, dx=\text {Too large to display} \] Input:

integrate((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^4,x, algorithm="giac")
 

Output:

1/3*(3*(2*a^3*c^3 + 3*a*b^2*c^3 - 12*a^2*b*c^2*d - 3*b^3*c^2*d + 3*a^3*c*d 
^2 + 12*a*b^2*c*d^2 - 3*a^2*b*d^3 - 2*b^3*d^3)*(pi*floor(1/2*(f*x + e)/pi 
+ 1/2)*sgn(c) + arctan((c*tan(1/2*f*x + 1/2*e) + d)/sqrt(c^2 - d^2)))/((c^ 
6 - 3*c^4*d^2 + 3*c^2*d^4 - d^6)*sqrt(c^2 - d^2)) + (9*a*b^2*c^8*tan(1/2*f 
*x + 1/2*e)^5 - 36*a^2*b*c^7*d*tan(1/2*f*x + 1/2*e)^5 - 9*b^3*c^7*d*tan(1/ 
2*f*x + 1/2*e)^5 + 27*a^3*c^6*d^2*tan(1/2*f*x + 1/2*e)^5 + 36*a*b^2*c^6*d^ 
2*tan(1/2*f*x + 1/2*e)^5 - 9*a^2*b*c^5*d^3*tan(1/2*f*x + 1/2*e)^5 - 6*b^3* 
c^5*d^3*tan(1/2*f*x + 1/2*e)^5 - 18*a^3*c^4*d^4*tan(1/2*f*x + 1/2*e)^5 + 6 
*a^3*c^2*d^6*tan(1/2*f*x + 1/2*e)^5 - 18*a^2*b*c^8*tan(1/2*f*x + 1/2*e)^4 
+ 18*a^3*c^7*d*tan(1/2*f*x + 1/2*e)^4 + 45*a*b^2*c^7*d*tan(1/2*f*x + 1/2*e 
)^4 - 126*a^2*b*c^6*d^2*tan(1/2*f*x + 1/2*e)^4 - 45*b^3*c^6*d^2*tan(1/2*f* 
x + 1/2*e)^4 + 81*a^3*c^5*d^3*tan(1/2*f*x + 1/2*e)^4 + 180*a*b^2*c^5*d^3*t 
an(1/2*f*x + 1/2*e)^4 - 99*a^2*b*c^4*d^4*tan(1/2*f*x + 1/2*e)^4 - 30*b^3*c 
^4*d^4*tan(1/2*f*x + 1/2*e)^4 - 36*a^3*c^3*d^5*tan(1/2*f*x + 1/2*e)^4 + 18 
*a^2*b*c^2*d^6*tan(1/2*f*x + 1/2*e)^4 + 12*a^3*c*d^7*tan(1/2*f*x + 1/2*e)^ 
4 - 108*a^2*b*c^7*d*tan(1/2*f*x + 1/2*e)^3 - 24*b^3*c^7*d*tan(1/2*f*x + 1/ 
2*e)^3 + 108*a^3*c^6*d^2*tan(1/2*f*x + 1/2*e)^3 + 234*a*b^2*c^6*d^2*tan(1/ 
2*f*x + 1/2*e)^3 - 252*a^2*b*c^5*d^3*tan(1/2*f*x + 1/2*e)^3 - 82*b^3*c^5*d 
^3*tan(1/2*f*x + 1/2*e)^3 + 42*a^3*c^4*d^4*tan(1/2*f*x + 1/2*e)^3 + 192*a* 
b^2*c^4*d^4*tan(1/2*f*x + 1/2*e)^3 - 102*a^2*b*c^3*d^5*tan(1/2*f*x + 1/...
 

Mupad [B] (verification not implemented)

Time = 20.56 (sec) , antiderivative size = 1423, normalized size of antiderivative = 4.38 \[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^4} \, dx=\text {Too large to display} \] Input:

int((a + b*sin(e + f*x))^3/(c + d*sin(e + f*x))^4,x)
 

Output:

((2*a^3*d^5 - 4*b^3*c^5 - 18*a^2*b*c^5 + 18*a^3*c^4*d - 5*a^3*c^2*d^3 - 11 
*b^3*c^3*d^2 + 6*a*b^2*c^2*d^3 - 30*a^2*b*c^3*d^2 + 39*a*b^2*c^4*d + 3*a^2 
*b*c*d^4)/(3*(c^6 - d^6 + 3*c^2*d^4 - 3*c^4*d^2)) - (tan(e/2 + (f*x)/2)^5* 
(3*b^3*c^5*d - 3*a*b^2*c^6 - 2*a^3*d^6 + 6*a^3*c^2*d^4 - 9*a^3*c^4*d^2 + 2 
*b^3*c^3*d^3 - 12*a*b^2*c^4*d^2 + 3*a^2*b*c^3*d^3 + 12*a^2*b*c^5*d))/(c*(c 
^6 - d^6 + 3*c^2*d^4 - 3*c^4*d^2)) + (2*tan(e/2 + (f*x)/2)^2*(2*a^3*d^7 - 
2*b^3*c^7 - 6*a^2*b*c^7 + 6*a^3*c^6*d - 3*a^3*c^2*d^5 + 20*a^3*c^4*d^3 - 1 
7*b^3*c^3*d^4 - 6*b^3*c^5*d^2 + 12*a*b^2*c^2*d^5 + 51*a*b^2*c^4*d^3 - 42*a 
^2*b*c^3*d^4 - 30*a^2*b*c^5*d^2 + 12*a*b^2*c^6*d + 3*a^2*b*c*d^6))/(c^2*(c 
^6 - d^6 + 3*c^2*d^4 - 3*c^4*d^2)) - (tan(e/2 + (f*x)/2)*(3*a*b^2*c^6 - 2* 
a^3*d^6 + 5*b^3*c^5*d + 4*a^3*c^2*d^4 - 27*a^3*c^4*d^2 + 20*b^3*c^3*d^3 - 
12*a*b^2*c^2*d^4 - 66*a*b^2*c^4*d^2 + 57*a^2*b*c^3*d^3 - 6*a^2*b*c*d^5 + 2 
4*a^2*b*c^5*d))/(c*(c^6 - d^6 + 3*c^2*d^4 - 3*c^4*d^2)) + (tan(e/2 + (f*x) 
/2)^4*(4*a^3*d^7 - 6*a^2*b*c^7 + 6*a^3*c^6*d - 12*a^3*c^2*d^5 + 27*a^3*c^4 
*d^3 - 10*b^3*c^3*d^4 - 15*b^3*c^5*d^2 + 60*a*b^2*c^4*d^3 - 33*a^2*b*c^3*d 
^4 - 42*a^2*b*c^5*d^2 + 15*a*b^2*c^6*d + 6*a^2*b*c*d^6))/(c^2*(c^6 - d^6 + 
 3*c^2*d^4 - 3*c^4*d^2)) + (2*d*tan(e/2 + (f*x)/2)^3*(3*c^2 + 2*d^2)*(2*a^ 
3*d^5 - 4*b^3*c^5 - 18*a^2*b*c^5 + 18*a^3*c^4*d - 5*a^3*c^2*d^3 - 11*b^3*c 
^3*d^2 + 6*a*b^2*c^2*d^3 - 30*a^2*b*c^3*d^2 + 39*a*b^2*c^4*d + 3*a^2*b*c*d 
^4))/(3*c^3*(c^6 - d^6 + 3*c^2*d^4 - 3*c^4*d^2)))/(f*(c^3*tan(e/2 + (f*...
 

Reduce [B] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 3856, normalized size of antiderivative = 11.86 \[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^4} \, dx =\text {Too large to display} \] Input:

int((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^4,x)
 

Output:

(12*sqrt(c**2 - d**2)*atan((tan((e + f*x)/2)*c + d)/sqrt(c**2 - d**2))*sin 
(e + f*x)**3*a**3*c**5*d**4 + 18*sqrt(c**2 - d**2)*atan((tan((e + f*x)/2)* 
c + d)/sqrt(c**2 - d**2))*sin(e + f*x)**3*a**3*c**3*d**6 - 72*sqrt(c**2 - 
d**2)*atan((tan((e + f*x)/2)*c + d)/sqrt(c**2 - d**2))*sin(e + f*x)**3*a** 
2*b*c**4*d**5 - 18*sqrt(c**2 - d**2)*atan((tan((e + f*x)/2)*c + d)/sqrt(c* 
*2 - d**2))*sin(e + f*x)**3*a**2*b*c**2*d**7 + 18*sqrt(c**2 - d**2)*atan(( 
tan((e + f*x)/2)*c + d)/sqrt(c**2 - d**2))*sin(e + f*x)**3*a*b**2*c**5*d** 
4 + 72*sqrt(c**2 - d**2)*atan((tan((e + f*x)/2)*c + d)/sqrt(c**2 - d**2))* 
sin(e + f*x)**3*a*b**2*c**3*d**6 - 18*sqrt(c**2 - d**2)*atan((tan((e + f*x 
)/2)*c + d)/sqrt(c**2 - d**2))*sin(e + f*x)**3*b**3*c**4*d**5 - 12*sqrt(c* 
*2 - d**2)*atan((tan((e + f*x)/2)*c + d)/sqrt(c**2 - d**2))*sin(e + f*x)** 
3*b**3*c**2*d**7 + 36*sqrt(c**2 - d**2)*atan((tan((e + f*x)/2)*c + d)/sqrt 
(c**2 - d**2))*sin(e + f*x)**2*a**3*c**6*d**3 + 54*sqrt(c**2 - d**2)*atan( 
(tan((e + f*x)/2)*c + d)/sqrt(c**2 - d**2))*sin(e + f*x)**2*a**3*c**4*d**5 
 - 216*sqrt(c**2 - d**2)*atan((tan((e + f*x)/2)*c + d)/sqrt(c**2 - d**2))* 
sin(e + f*x)**2*a**2*b*c**5*d**4 - 54*sqrt(c**2 - d**2)*atan((tan((e + f*x 
)/2)*c + d)/sqrt(c**2 - d**2))*sin(e + f*x)**2*a**2*b*c**3*d**6 + 54*sqrt( 
c**2 - d**2)*atan((tan((e + f*x)/2)*c + d)/sqrt(c**2 - d**2))*sin(e + f*x) 
**2*a*b**2*c**6*d**3 + 216*sqrt(c**2 - d**2)*atan((tan((e + f*x)/2)*c + d) 
/sqrt(c**2 - d**2))*sin(e + f*x)**2*a*b**2*c**4*d**5 - 54*sqrt(c**2 - d...