\(\int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^5} \, dx\) [460]

Optimal result

Integrand size = 25, antiderivative size = 289 \[ \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^5} \, dx=\frac {5 a^3 (4 c-3 d) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{4 (c-d) (c+d)^4 \sqrt {c^2-d^2} f}+\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{4 d (c+d) f (c+d \sin (e+f x))^4}+\frac {a^3 (c-d) (2 c+9 d) \cos (e+f x)}{12 d^2 (c+d)^2 f (c+d \sin (e+f x))^3}-\frac {a^3 \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{24 d^2 (c+d)^3 f (c+d \sin (e+f x))^2}-\frac {a^3 \left (2 c^3+12 c^2 d+43 c d^2-72 d^3\right ) \cos (e+f x)}{24 (c-d) d^2 (c+d)^4 f (c+d \sin (e+f x))} \] Output:

5/4*a^3*(4*c-3*d)*arctan((d+c*tan(1/2*f*x+1/2*e))/(c^2-d^2)^(1/2))/(c-d)/( 
c+d)^4/(c^2-d^2)^(1/2)/f+1/4*(c-d)*cos(f*x+e)*(a^3+a^3*sin(f*x+e))/d/(c+d) 
/f/(c+d*sin(f*x+e))^4+1/12*a^3*(c-d)*(2*c+9*d)*cos(f*x+e)/d^2/(c+d)^2/f/(c 
+d*sin(f*x+e))^3-1/24*a^3*(2*c^2+12*c*d+45*d^2)*cos(f*x+e)/d^2/(c+d)^3/f/( 
c+d*sin(f*x+e))^2-1/24*a^3*(2*c^3+12*c^2*d+43*c*d^2-72*d^3)*cos(f*x+e)/(c- 
d)/d^2/(c+d)^4/f/(c+d*sin(f*x+e))
 

Mathematica [A] (warning: unable to verify)

Time = 2.27 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.83 \[ \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^5} \, dx=\frac {a^3 \cos (e+f x) \left (-\frac {d (1+\sin (e+f x))^3}{(c+d \sin (e+f x))^4}-\frac {(4 c-3 d) \left (-\frac {5 \text {arctanh}\left (\frac {\sqrt {c-d} \sqrt {1-\sin (e+f x)}}{\sqrt {-c-d} \sqrt {1+\sin (e+f x)}}\right )}{(-c-d)^{7/2} \sqrt {c-d}}-\frac {\sqrt {\cos ^2(e+f x)} \left (22 c^2+9 c d+2 d^2+\left (9 c^2+48 c d+9 d^2\right ) \sin (e+f x)+\left (2 c^2+9 c d+22 d^2\right ) \sin ^2(e+f x)\right )}{6 (c+d)^3 (c+d \sin (e+f x))^3}\right )}{\sqrt {\cos ^2(e+f x)}}\right )}{4 (-c+d) (c+d) f} \] Input:

Integrate[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^5,x]
 

Output:

(a^3*Cos[e + f*x]*(-((d*(1 + Sin[e + f*x])^3)/(c + d*Sin[e + f*x])^4) - (( 
4*c - 3*d)*((-5*ArcTanh[(Sqrt[c - d]*Sqrt[1 - Sin[e + f*x]])/(Sqrt[-c - d] 
*Sqrt[1 + Sin[e + f*x]])])/((-c - d)^(7/2)*Sqrt[c - d]) - (Sqrt[Cos[e + f* 
x]^2]*(22*c^2 + 9*c*d + 2*d^2 + (9*c^2 + 48*c*d + 9*d^2)*Sin[e + f*x] + (2 
*c^2 + 9*c*d + 22*d^2)*Sin[e + f*x]^2))/(6*(c + d)^3*(c + d*Sin[e + f*x])^ 
3)))/Sqrt[Cos[e + f*x]^2]))/(4*(-c + d)*(c + d)*f)
 

Rubi [A] (verified)

Time = 1.48 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.16, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3042, 3241, 3042, 3447, 3042, 3500, 3042, 3233, 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.

\(\displaystyle \int \frac {(a \sin (e+f x)+a)^3}{(c+d \sin (e+f x))^5} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a \sin (e+f x)+a)^3}{(c+d \sin (e+f x))^5}dx\)

\(\Big \downarrow \) 3241

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3447

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3500

\(\displaystyle \frac {(c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{4 d f (c+d) (c+d \sin (e+f x))^4}-\frac {a \left (-\frac {\int \frac {3 (c-d) d (c+15 d) a^2+2 (c-d) \left (c^2+5 d c+18 d^2\right ) \sin (e+f x) a^2}{(c+d \sin (e+f x))^3}dx}{3 d \left (c^2-d^2\right )}-\frac {a^2 (c-d) (2 c+9 d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^3}\right )}{4 d (c+d)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{4 d f (c+d) (c+d \sin (e+f x))^4}-\frac {a \left (-\frac {\int \frac {3 (c-d) d (c+15 d) a^2+2 (c-d) \left (c^2+5 d c+18 d^2\right ) \sin (e+f x) a^2}{(c+d \sin (e+f x))^3}dx}{3 d \left (c^2-d^2\right )}-\frac {a^2 (c-d) (2 c+9 d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^3}\right )}{4 d (c+d)}\)

\(\Big \downarrow \) 3233

\(\displaystyle \frac {(c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{4 d f (c+d) (c+d \sin (e+f x))^4}-\frac {a \left (-\frac {-\frac {\int -\frac {2 a^2 d (c+36 d) (c-d)^2+a^2 \left (2 c^2+12 d c+45 d^2\right ) \sin (e+f x) (c-d)^2}{(c+d \sin (e+f x))^2}dx}{2 \left (c^2-d^2\right )}-\frac {a^2 (c-d) \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{2 f (c+d) (c+d \sin (e+f x))^2}}{3 d \left (c^2-d^2\right )}-\frac {a^2 (c-d) (2 c+9 d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^3}\right )}{4 d (c+d)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{4 d f (c+d) (c+d \sin (e+f x))^4}-\frac {a \left (-\frac {\frac {\int \frac {2 a^2 d (c+36 d) (c-d)^2+a^2 \left (2 c^2+12 d c+45 d^2\right ) \sin (e+f x) (c-d)^2}{(c+d \sin (e+f x))^2}dx}{2 \left (c^2-d^2\right )}-\frac {a^2 (c-d) \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{2 f (c+d) (c+d \sin (e+f x))^2}}{3 d \left (c^2-d^2\right )}-\frac {a^2 (c-d) (2 c+9 d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^3}\right )}{4 d (c+d)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{4 d f (c+d) (c+d \sin (e+f x))^4}-\frac {a \left (-\frac {\frac {\int \frac {2 a^2 d (c+36 d) (c-d)^2+a^2 \left (2 c^2+12 d c+45 d^2\right ) \sin (e+f x) (c-d)^2}{(c+d \sin (e+f x))^2}dx}{2 \left (c^2-d^2\right )}-\frac {a^2 (c-d) \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{2 f (c+d) (c+d \sin (e+f x))^2}}{3 d \left (c^2-d^2\right )}-\frac {a^2 (c-d) (2 c+9 d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^3}\right )}{4 d (c+d)}\)

\(\Big \downarrow \) 3233

\(\displaystyle \frac {(c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{4 d f (c+d) (c+d \sin (e+f x))^4}-\frac {a \left (-\frac {\frac {-\frac {\int -\frac {15 a^2 (4 c-3 d) (c-d)^2 d^2}{c+d \sin (e+f x)}dx}{c^2-d^2}-\frac {a^2 (c-d) \left (2 c^3+12 c^2 d+43 c d^2-72 d^3\right ) \cos (e+f x)}{f (c+d) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {a^2 (c-d) \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{2 f (c+d) (c+d \sin (e+f x))^2}}{3 d \left (c^2-d^2\right )}-\frac {a^2 (c-d) (2 c+9 d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^3}\right )}{4 d (c+d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{4 d f (c+d) (c+d \sin (e+f x))^4}-\frac {a \left (-\frac {\frac {\frac {15 a^2 d^2 (4 c-3 d) (c-d)^2 \int \frac {1}{c+d \sin (e+f x)}dx}{c^2-d^2}-\frac {a^2 (c-d) \left (2 c^3+12 c^2 d+43 c d^2-72 d^3\right ) \cos (e+f x)}{f (c+d) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {a^2 (c-d) \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{2 f (c+d) (c+d \sin (e+f x))^2}}{3 d \left (c^2-d^2\right )}-\frac {a^2 (c-d) (2 c+9 d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^3}\right )}{4 d (c+d)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{4 d f (c+d) (c+d \sin (e+f x))^4}-\frac {a \left (-\frac {\frac {\frac {15 a^2 d^2 (4 c-3 d) (c-d)^2 \int \frac {1}{c+d \sin (e+f x)}dx}{c^2-d^2}-\frac {a^2 (c-d) \left (2 c^3+12 c^2 d+43 c d^2-72 d^3\right ) \cos (e+f x)}{f (c+d) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {a^2 (c-d) \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{2 f (c+d) (c+d \sin (e+f x))^2}}{3 d \left (c^2-d^2\right )}-\frac {a^2 (c-d) (2 c+9 d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^3}\right )}{4 d (c+d)}\)

\(\Big \downarrow \) 3139

\(\displaystyle \frac {(c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{4 d f (c+d) (c+d \sin (e+f x))^4}-\frac {a \left (-\frac {\frac {\frac {30 a^2 d^2 (4 c-3 d) (c-d)^2 \int \frac {1}{c \tan ^2\left (\frac {1}{2} (e+f x)\right )+2 d \tan \left (\frac {1}{2} (e+f x)\right )+c}d\tan \left (\frac {1}{2} (e+f x)\right )}{f \left (c^2-d^2\right )}-\frac {a^2 (c-d) \left (2 c^3+12 c^2 d+43 c d^2-72 d^3\right ) \cos (e+f x)}{f (c+d) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {a^2 (c-d) \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{2 f (c+d) (c+d \sin (e+f x))^2}}{3 d \left (c^2-d^2\right )}-\frac {a^2 (c-d) (2 c+9 d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^3}\right )}{4 d (c+d)}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {(c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{4 d f (c+d) (c+d \sin (e+f x))^4}-\frac {a \left (-\frac {\frac {-\frac {60 a^2 d^2 (4 c-3 d) (c-d)^2 \int \frac {1}{-\left (2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )^2-4 \left (c^2-d^2\right )}d\left (2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )}{f \left (c^2-d^2\right )}-\frac {a^2 (c-d) \left (2 c^3+12 c^2 d+43 c d^2-72 d^3\right ) \cos (e+f x)}{f (c+d) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {a^2 (c-d) \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{2 f (c+d) (c+d \sin (e+f x))^2}}{3 d \left (c^2-d^2\right )}-\frac {a^2 (c-d) (2 c+9 d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^3}\right )}{4 d (c+d)}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {(c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{4 d f (c+d) (c+d \sin (e+f x))^4}-\frac {a \left (-\frac {\frac {\frac {30 a^2 d^2 (4 c-3 d) (c-d)^2 \arctan \left (\frac {2 c \tan \left (\frac {1}{2} (e+f x)\right )+2 d}{2 \sqrt {c^2-d^2}}\right )}{f \left (c^2-d^2\right )^{3/2}}-\frac {a^2 (c-d) \left (2 c^3+12 c^2 d+43 c d^2-72 d^3\right ) \cos (e+f x)}{f (c+d) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {a^2 (c-d) \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{2 f (c+d) (c+d \sin (e+f x))^2}}{3 d \left (c^2-d^2\right )}-\frac {a^2 (c-d) (2 c+9 d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^3}\right )}{4 d (c+d)}\)

Input:

Int[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^5,x]
 

Output:

((c - d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(4*d*(c + d)*f*(c + d*Sin[ 
e + f*x])^4) - (a*(-1/3*(a^2*(c - d)*(2*c + 9*d)*Cos[e + f*x])/(d*(c + d)* 
f*(c + d*Sin[e + f*x])^3) - (-1/2*(a^2*(c - d)*(2*c^2 + 12*c*d + 45*d^2)*C 
os[e + f*x])/((c + d)*f*(c + d*Sin[e + f*x])^2) + ((30*a^2*(4*c - 3*d)*(c 
- d)^2*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) - (a^2*(c - d)*(2*c^3 + 12*c^2*d + 43*c*d^2 - 72*d^3)*Cos 
[e + f*x])/((c + d)*f*(c + d*Sin[e + f*x])))/(2*(c^2 - d^2)))/(3*d*(c^2 - 
d^2))))/(4*d*(c + d))
 

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 3241

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*(b*c - a*d)*Cos[e + f*x]*(a + b 
*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a* 
d))), x] + Simp[b^2/(d*(n + 1)*(b*c + a*d))   Int[(a + b*Sin[e + f*x])^(m - 
 2)*(c + d*Sin[e + f*x])^(n + 1)*Simp[a*c*(m - 2) - b*d*(m - 2*n - 4) - (b* 
c*(m - 1) - a*d*(m + 2*n + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, 
d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] 
 && GtQ[m, 1] && LtQ[n, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || 
 (IntegerQ[m] && EqQ[c, 0]))
 

rule 3447

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

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. \(921\) vs. \(2(274)=548\).

Time = 3.74 (sec) , antiderivative size = 922, normalized size of antiderivative = 3.19

method	result	size
derivativedivides	\(\frac {2 a^{3} \left (\frac {\frac {\left (12 c^{5}-39 c^{4} d -16 c^{3} d^{2}+16 c^{2} d^{3}+24 c \,d^{4}+8 d^{5}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{8 c \left (c^{5}+3 c^{4} d +2 c^{3} d^{2}-2 c^{2} d^{3}-3 c \,d^{4}-d^{5}\right )}-\frac {\left (24 c^{6}-44 c^{5} d +225 c^{4} d^{2}-120 c^{2} d^{4}-96 c \,d^{5}-24 d^{6}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{8 \left (c^{5}+3 c^{4} d +2 c^{3} d^{2}-2 c^{2} d^{3}-3 c \,d^{4}-d^{5}\right ) c^{2}}+\frac {\left (36 c^{7}-813 c^{6} d +288 c^{5} d^{2}-892 c^{4} d^{3}+552 c^{3} d^{4}+664 c^{2} d^{5}+384 c \,d^{6}+96 d^{7}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{24 c^{3} \left (c^{5}+3 c^{4} d +2 c^{3} d^{2}-2 c^{2} d^{3}-3 c \,d^{4}-d^{5}\right )}-\frac {\left (264 c^{8}-108 c^{7} d +2001 c^{6} d^{2}-936 c^{5} d^{3}-202 c^{4} d^{4}-864 c^{3} d^{5}-440 d^{6} c^{2}-192 c \,d^{7}-48 d^{8}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{24 c^{4} \left (c^{5}+3 c^{4} d +2 c^{3} d^{2}-2 c^{2} d^{3}-3 c \,d^{4}-d^{5}\right )}-\frac {\left (36 c^{7}+1299 c^{6} d -576 c^{5} d^{2}+1036 c^{4} d^{3}-1176 c^{3} d^{4}-664 c^{2} d^{5}-384 c \,d^{6}-96 d^{7}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{24 c^{3} \left (c^{5}+3 c^{4} d +2 c^{3} d^{2}-2 c^{2} d^{3}-3 c \,d^{4}-d^{5}\right )}-\frac {\left (280 c^{6}-12 c^{5} d +1289 c^{4} d^{2}-960 c^{3} d^{3}-552 c^{2} d^{4}-288 c \,d^{5}-72 d^{6}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{24 \left (c^{5}+3 c^{4} d +2 c^{3} d^{2}-2 c^{2} d^{3}-3 c \,d^{4}-d^{5}\right ) c^{2}}-\frac {\left (36 c^{5}+587 c^{4} d -336 c^{3} d^{2}-248 c^{2} d^{3}-120 c \,d^{4}-24 d^{5}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{24 c \left (c^{5}+3 c^{4} d +2 c^{3} d^{2}-2 c^{2} d^{3}-3 c \,d^{4}-d^{5}\right )}-\frac {88 c^{4}-36 c^{3} d -37 c^{2} d^{2}-24 c \,d^{3}-6 d^{4}}{24 \left (c^{5}+3 c^{4} d +2 c^{3} d^{2}-2 c^{2} d^{3}-3 c \,d^{4}-d^{5}\right )}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )^{4}}+\frac {5 \left (4 c -3 d \right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{8 \left (c^{5}+3 c^{4} d +2 c^{3} d^{2}-2 c^{2} d^{3}-3 c \,d^{4}-d^{5}\right ) \sqrt {c^{2}-d^{2}}}\right )}{f}\)	\(922\)
default	\(\frac {2 a^{3} \left (\frac {\frac {\left (12 c^{5}-39 c^{4} d -16 c^{3} d^{2}+16 c^{2} d^{3}+24 c \,d^{4}+8 d^{5}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{8 c \left (c^{5}+3 c^{4} d +2 c^{3} d^{2}-2 c^{2} d^{3}-3 c \,d^{4}-d^{5}\right )}-\frac {\left (24 c^{6}-44 c^{5} d +225 c^{4} d^{2}-120 c^{2} d^{4}-96 c \,d^{5}-24 d^{6}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{8 \left (c^{5}+3 c^{4} d +2 c^{3} d^{2}-2 c^{2} d^{3}-3 c \,d^{4}-d^{5}\right ) c^{2}}+\frac {\left (36 c^{7}-813 c^{6} d +288 c^{5} d^{2}-892 c^{4} d^{3}+552 c^{3} d^{4}+664 c^{2} d^{5}+384 c \,d^{6}+96 d^{7}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{24 c^{3} \left (c^{5}+3 c^{4} d +2 c^{3} d^{2}-2 c^{2} d^{3}-3 c \,d^{4}-d^{5}\right )}-\frac {\left (264 c^{8}-108 c^{7} d +2001 c^{6} d^{2}-936 c^{5} d^{3}-202 c^{4} d^{4}-864 c^{3} d^{5}-440 d^{6} c^{2}-192 c \,d^{7}-48 d^{8}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{24 c^{4} \left (c^{5}+3 c^{4} d +2 c^{3} d^{2}-2 c^{2} d^{3}-3 c \,d^{4}-d^{5}\right )}-\frac {\left (36 c^{7}+1299 c^{6} d -576 c^{5} d^{2}+1036 c^{4} d^{3}-1176 c^{3} d^{4}-664 c^{2} d^{5}-384 c \,d^{6}-96 d^{7}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{24 c^{3} \left (c^{5}+3 c^{4} d +2 c^{3} d^{2}-2 c^{2} d^{3}-3 c \,d^{4}-d^{5}\right )}-\frac {\left (280 c^{6}-12 c^{5} d +1289 c^{4} d^{2}-960 c^{3} d^{3}-552 c^{2} d^{4}-288 c \,d^{5}-72 d^{6}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{24 \left (c^{5}+3 c^{4} d +2 c^{3} d^{2}-2 c^{2} d^{3}-3 c \,d^{4}-d^{5}\right ) c^{2}}-\frac {\left (36 c^{5}+587 c^{4} d -336 c^{3} d^{2}-248 c^{2} d^{3}-120 c \,d^{4}-24 d^{5}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{24 c \left (c^{5}+3 c^{4} d +2 c^{3} d^{2}-2 c^{2} d^{3}-3 c \,d^{4}-d^{5}\right )}-\frac {88 c^{4}-36 c^{3} d -37 c^{2} d^{2}-24 c \,d^{3}-6 d^{4}}{24 \left (c^{5}+3 c^{4} d +2 c^{3} d^{2}-2 c^{2} d^{3}-3 c \,d^{4}-d^{5}\right )}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )^{4}}+\frac {5 \left (4 c -3 d \right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{8 \left (c^{5}+3 c^{4} d +2 c^{3} d^{2}-2 c^{2} d^{3}-3 c \,d^{4}-d^{5}\right ) \sqrt {c^{2}-d^{2}}}\right )}{f}\)	\(922\)
risch	\(\text {Expression too large to display}\)	\(1088\)

Input:

int((a+sin(f*x+e)*a)^3/(c+d*sin(f*x+e))^5,x,method=_RETURNVERBOSE)
 

Output:

2/f*a^3*((1/8*(12*c^5-39*c^4*d-16*c^3*d^2+16*c^2*d^3+24*c*d^4+8*d^5)/c/(c^ 
5+3*c^4*d+2*c^3*d^2-2*c^2*d^3-3*c*d^4-d^5)*tan(1/2*f*x+1/2*e)^7-1/8*(24*c^ 
6-44*c^5*d+225*c^4*d^2-120*c^2*d^4-96*c*d^5-24*d^6)/(c^5+3*c^4*d+2*c^3*d^2 
-2*c^2*d^3-3*c*d^4-d^5)/c^2*tan(1/2*f*x+1/2*e)^6+1/24/c^3*(36*c^7-813*c^6* 
d+288*c^5*d^2-892*c^4*d^3+552*c^3*d^4+664*c^2*d^5+384*c*d^6+96*d^7)/(c^5+3 
*c^4*d+2*c^3*d^2-2*c^2*d^3-3*c*d^4-d^5)*tan(1/2*f*x+1/2*e)^5-1/24/c^4*(264 
*c^8-108*c^7*d+2001*c^6*d^2-936*c^5*d^3-202*c^4*d^4-864*c^3*d^5-440*c^2*d^ 
6-192*c*d^7-48*d^8)/(c^5+3*c^4*d+2*c^3*d^2-2*c^2*d^3-3*c*d^4-d^5)*tan(1/2* 
f*x+1/2*e)^4-1/24/c^3*(36*c^7+1299*c^6*d-576*c^5*d^2+1036*c^4*d^3-1176*c^3 
*d^4-664*c^2*d^5-384*c*d^6-96*d^7)/(c^5+3*c^4*d+2*c^3*d^2-2*c^2*d^3-3*c*d^ 
4-d^5)*tan(1/2*f*x+1/2*e)^3-1/24*(280*c^6-12*c^5*d+1289*c^4*d^2-960*c^3*d^ 
3-552*c^2*d^4-288*c*d^5-72*d^6)/(c^5+3*c^4*d+2*c^3*d^2-2*c^2*d^3-3*c*d^4-d 
^5)/c^2*tan(1/2*f*x+1/2*e)^2-1/24*(36*c^5+587*c^4*d-336*c^3*d^2-248*c^2*d^ 
3-120*c*d^4-24*d^5)/c/(c^5+3*c^4*d+2*c^3*d^2-2*c^2*d^3-3*c*d^4-d^5)*tan(1/ 
2*f*x+1/2*e)-1/24*(88*c^4-36*c^3*d-37*c^2*d^2-24*c*d^3-6*d^4)/(c^5+3*c^4*d 
+2*c^3*d^2-2*c^2*d^3-3*c*d^4-d^5))/(tan(1/2*f*x+1/2*e)^2*c+2*d*tan(1/2*f*x 
+1/2*e)+c)^4+5/8*(4*c-3*d)/(c^5+3*c^4*d+2*c^3*d^2-2*c^2*d^3-3*c*d^4-d^5)/( 
c^2-d^2)^(1/2)*arctan(1/2*(2*c*tan(1/2*f*x+1/2*e)+2*d)/(c^2-d^2)^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 962 vs. \(2 (274) = 548\).

Time = 0.17 (sec) , antiderivative size = 2009, normalized size of antiderivative = 6.95 \[ \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^5} \, dx=\text {Too large to display} \] Input:

integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^5,x, algorithm="fricas")
 

Output:

[1/48*(2*(8*a^3*c^6 + 48*a^3*c^5*d + 164*a^3*c^4*d^2 - 276*a^3*c^3*d^3 - 2 
17*a^3*c^2*d^4 + 228*a^3*c*d^5 + 45*a^3*d^6)*cos(f*x + e)^3 - 15*(4*a^3*c^ 
5 - 3*a^3*c^4*d + 24*a^3*c^3*d^2 - 18*a^3*c^2*d^3 + 4*a^3*c*d^4 - 3*a^3*d^ 
5 + (4*a^3*c*d^4 - 3*a^3*d^5)*cos(f*x + e)^4 - 2*(12*a^3*c^3*d^2 - 9*a^3*c 
^2*d^3 + 4*a^3*c*d^4 - 3*a^3*d^5)*cos(f*x + e)^2 + 4*(4*a^3*c^4*d - 3*a^3* 
c^3*d^2 + 4*a^3*c^2*d^3 - 3*a^3*c*d^4 - (4*a^3*c^2*d^3 - 3*a^3*c*d^4)*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*c 
os(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)) - 6*(32*a^3*c^6 + 4*a^3*c^5*d + 13*a^3*c^4*d^2 - 88*a^3*c^3*d^ 
3 - 62*a^3*c^2*d^4 + 84*a^3*c*d^5 + 17*a^3*d^6)*cos(f*x + e) + 2*((2*a^3*c 
^5*d + 12*a^3*c^4*d^2 + 41*a^3*c^3*d^3 - 84*a^3*c^2*d^4 - 43*a^3*c*d^5 + 7 
2*a^3*d^6)*cos(f*x + e)^3 - 3*(12*a^3*c^6 + 79*a^3*c^5*d - 72*a^3*c^4*d^2 
- 98*a^3*c^3*d^3 + 28*a^3*c^2*d^4 + 19*a^3*c*d^5 + 32*a^3*d^6)*cos(f*x + e 
))*sin(f*x + e))/((c^7*d^4 + 3*c^6*d^5 + c^5*d^6 - 5*c^4*d^7 - 5*c^3*d^8 + 
 c^2*d^9 + 3*c*d^10 + d^11)*f*cos(f*x + e)^4 - 2*(3*c^9*d^2 + 9*c^8*d^3 + 
4*c^7*d^4 - 12*c^6*d^5 - 14*c^5*d^6 - 2*c^4*d^7 + 4*c^3*d^8 + 4*c^2*d^9 + 
3*c*d^10 + d^11)*f*cos(f*x + e)^2 + (c^11 + 3*c^10*d + 7*c^9*d^2 + 13*c^8* 
d^3 + 2*c^7*d^4 - 26*c^6*d^5 - 26*c^5*d^6 + 2*c^4*d^7 + 13*c^3*d^8 + 7*c^2 
*d^9 + 3*c*d^10 + d^11)*f - 4*((c^8*d^3 + 3*c^7*d^4 + c^6*d^5 - 5*c^5*d...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^5} \, dx=\text {Timed out} \] Input:

integrate((a+a*sin(f*x+e))**3/(c+d*sin(f*x+e))**5,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^5} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^5,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. 1284 vs. \(2 (274) = 548\).

Time = 0.25 (sec) , antiderivative size = 1284, normalized size of antiderivative = 4.44 \[ \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^5} \, dx=\text {Too large to display} \] Input:

integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^5,x, algorithm="giac")
 

Output:

1/12*(15*(4*a^3*c - 3*a^3*d)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(c) + ar 
ctan((c*tan(1/2*f*x + 1/2*e) + d)/sqrt(c^2 - d^2)))/((c^5 + 3*c^4*d + 2*c^ 
3*d^2 - 2*c^2*d^3 - 3*c*d^4 - d^5)*sqrt(c^2 - d^2)) + (36*a^3*c^8*tan(1/2* 
f*x + 1/2*e)^7 - 117*a^3*c^7*d*tan(1/2*f*x + 1/2*e)^7 - 48*a^3*c^6*d^2*tan 
(1/2*f*x + 1/2*e)^7 + 48*a^3*c^5*d^3*tan(1/2*f*x + 1/2*e)^7 + 72*a^3*c^4*d 
^4*tan(1/2*f*x + 1/2*e)^7 + 24*a^3*c^3*d^5*tan(1/2*f*x + 1/2*e)^7 - 72*a^3 
*c^8*tan(1/2*f*x + 1/2*e)^6 + 132*a^3*c^7*d*tan(1/2*f*x + 1/2*e)^6 - 675*a 
^3*c^6*d^2*tan(1/2*f*x + 1/2*e)^6 + 360*a^3*c^4*d^4*tan(1/2*f*x + 1/2*e)^6 
 + 288*a^3*c^3*d^5*tan(1/2*f*x + 1/2*e)^6 + 72*a^3*c^2*d^6*tan(1/2*f*x + 1 
/2*e)^6 + 36*a^3*c^8*tan(1/2*f*x + 1/2*e)^5 - 813*a^3*c^7*d*tan(1/2*f*x + 
1/2*e)^5 + 288*a^3*c^6*d^2*tan(1/2*f*x + 1/2*e)^5 - 892*a^3*c^5*d^3*tan(1/ 
2*f*x + 1/2*e)^5 + 552*a^3*c^4*d^4*tan(1/2*f*x + 1/2*e)^5 + 664*a^3*c^3*d^ 
5*tan(1/2*f*x + 1/2*e)^5 + 384*a^3*c^2*d^6*tan(1/2*f*x + 1/2*e)^5 + 96*a^3 
*c*d^7*tan(1/2*f*x + 1/2*e)^5 - 264*a^3*c^8*tan(1/2*f*x + 1/2*e)^4 + 108*a 
^3*c^7*d*tan(1/2*f*x + 1/2*e)^4 - 2001*a^3*c^6*d^2*tan(1/2*f*x + 1/2*e)^4 
+ 936*a^3*c^5*d^3*tan(1/2*f*x + 1/2*e)^4 + 202*a^3*c^4*d^4*tan(1/2*f*x + 1 
/2*e)^4 + 864*a^3*c^3*d^5*tan(1/2*f*x + 1/2*e)^4 + 440*a^3*c^2*d^6*tan(1/2 
*f*x + 1/2*e)^4 + 192*a^3*c*d^7*tan(1/2*f*x + 1/2*e)^4 + 48*a^3*d^8*tan(1/ 
2*f*x + 1/2*e)^4 - 36*a^3*c^8*tan(1/2*f*x + 1/2*e)^3 - 1299*a^3*c^7*d*tan( 
1/2*f*x + 1/2*e)^3 + 576*a^3*c^6*d^2*tan(1/2*f*x + 1/2*e)^3 - 1036*a^3*...
 

Mupad [B] (verification not implemented)

Time = 18.86 (sec) , antiderivative size = 1231, normalized size of antiderivative = 4.26 \[ \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^5} \, dx=\text {Too large to display} \] Input:

int((a + a*sin(e + f*x))^3/(c + d*sin(e + f*x))^5,x)
 

Output:

- ((6*a^3*d^4 - 88*a^3*c^4 + 24*a^3*c*d^3 + 36*a^3*c^3*d + 37*a^3*c^2*d^2) 
/(12*(3*c*d^4 - 3*c^4*d - c^5 + d^5 + 2*c^2*d^3 - 2*c^3*d^2)) + (a^3*tan(e 
/2 + (f*x)/2)^7*(24*c*d^4 - 39*c^4*d + 12*c^5 + 8*d^5 + 16*c^2*d^3 - 16*c^ 
3*d^2))/(4*c*(3*c*d^4 - 3*c^4*d - c^5 + d^5 + 2*c^2*d^3 - 2*c^3*d^2)) + (a 
^3*tan(e/2 + (f*x)/2)^6*(96*c*d^5 + 44*c^5*d - 24*c^6 + 24*d^6 + 120*c^2*d 
^4 - 225*c^4*d^2))/(4*c^2*(3*c*d^4 - 3*c^4*d - c^5 + d^5 + 2*c^2*d^3 - 2*c 
^3*d^2)) + (a^3*tan(e/2 + (f*x)/2)*(120*c*d^4 - 587*c^4*d - 36*c^5 + 24*d^ 
5 + 248*c^2*d^3 + 336*c^3*d^2))/(12*c*(3*c*d^4 - 3*c^4*d - c^5 + d^5 + 2*c 
^2*d^3 - 2*c^3*d^2)) + (a^3*tan(e/2 + (f*x)/2)^5*(384*c*d^6 - 813*c^6*d + 
36*c^7 + 96*d^7 + 664*c^2*d^5 + 552*c^3*d^4 - 892*c^4*d^3 + 288*c^5*d^2))/ 
(12*c^3*(3*c*d^4 - 3*c^4*d - c^5 + d^5 + 2*c^2*d^3 - 2*c^3*d^2)) + (a^3*ta 
n(e/2 + (f*x)/2)^3*(384*c*d^6 - 1299*c^6*d - 36*c^7 + 96*d^7 + 664*c^2*d^5 
 + 1176*c^3*d^4 - 1036*c^4*d^3 + 576*c^5*d^2))/(12*c^3*(3*c*d^4 - 3*c^4*d 
- c^5 + d^5 + 2*c^2*d^3 - 2*c^3*d^2)) + (a^3*tan(e/2 + (f*x)/2)^2*(288*c*d 
^5 + 12*c^5*d - 280*c^6 + 72*d^6 + 552*c^2*d^4 + 960*c^3*d^3 - 1289*c^4*d^ 
2))/(12*c^2*(3*c*d^4 - 3*c^4*d - c^5 + d^5 + 2*c^2*d^3 - 2*c^3*d^2)) + (a^ 
3*tan(e/2 + (f*x)/2)^4*(3*c^4 + 8*d^4 + 24*c^2*d^2)*(24*c*d^3 + 36*c^3*d - 
 88*c^4 + 6*d^4 + 37*c^2*d^2))/(12*c^4*(3*c*d^4 - 3*c^4*d - c^5 + d^5 + 2* 
c^2*d^3 - 2*c^3*d^2)))/(f*(tan(e/2 + (f*x)/2)^4*(6*c^4 + 16*d^4 + 48*c^2*d 
^2) + c^4*tan(e/2 + (f*x)/2)^8 + c^4 + tan(e/2 + (f*x)/2)^2*(4*c^4 + 24...
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 2057, normalized size of antiderivative = 7.12 \[ \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^5} \, dx =\text {Too large to display} \] Input:

int((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^5,x)
 

Output:

(a**3*(480*sqrt(c**2 - d**2)*atan((tan((e + f*x)/2)*c + d)/sqrt(c**2 - d** 
2))*sin(e + f*x)**4*c**4*d**5 - 360*sqrt(c**2 - d**2)*atan((tan((e + f*x)/ 
2)*c + d)/sqrt(c**2 - d**2))*sin(e + f*x)**4*c**3*d**6 + 1920*sqrt(c**2 - 
d**2)*atan((tan((e + f*x)/2)*c + d)/sqrt(c**2 - d**2))*sin(e + f*x)**3*c** 
5*d**4 - 1440*sqrt(c**2 - d**2)*atan((tan((e + f*x)/2)*c + d)/sqrt(c**2 - 
d**2))*sin(e + f*x)**3*c**4*d**5 + 2880*sqrt(c**2 - d**2)*atan((tan((e + f 
*x)/2)*c + d)/sqrt(c**2 - d**2))*sin(e + f*x)**2*c**6*d**3 - 2160*sqrt(c** 
2 - d**2)*atan((tan((e + f*x)/2)*c + d)/sqrt(c**2 - d**2))*sin(e + f*x)**2 
*c**5*d**4 + 1920*sqrt(c**2 - d**2)*atan((tan((e + f*x)/2)*c + d)/sqrt(c** 
2 - d**2))*sin(e + f*x)*c**7*d**2 - 1440*sqrt(c**2 - d**2)*atan((tan((e + 
f*x)/2)*c + d)/sqrt(c**2 - d**2))*sin(e + f*x)*c**6*d**3 + 480*sqrt(c**2 - 
 d**2)*atan((tan((e + f*x)/2)*c + d)/sqrt(c**2 - d**2))*c**8*d - 360*sqrt( 
c**2 - d**2)*atan((tan((e + f*x)/2)*c + d)/sqrt(c**2 - d**2))*c**7*d**2 - 
8*cos(e + f*x)*sin(e + f*x)**3*c**8*d**2 - 48*cos(e + f*x)*sin(e + f*x)**3 
*c**7*d**3 - 164*cos(e + f*x)*sin(e + f*x)**3*c**6*d**4 + 336*cos(e + f*x) 
*sin(e + f*x)**3*c**5*d**5 + 172*cos(e + f*x)*sin(e + f*x)**3*c**4*d**6 - 
288*cos(e + f*x)*sin(e + f*x)**3*c**3*d**7 - 32*cos(e + f*x)*sin(e + f*x)* 
*2*c**9*d - 192*cos(e + f*x)*sin(e + f*x)**2*c**8*d**2 - 656*cos(e + f*x)* 
sin(e + f*x)**2*c**7*d**3 + 1104*cos(e + f*x)*sin(e + f*x)**2*c**6*d**4 + 
868*cos(e + f*x)*sin(e + f*x)**2*c**5*d**5 - 912*cos(e + f*x)*sin(e + f...