\(\int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))} \, dx\) [727]

Optimal result

Integrand size = 25, antiderivative size = 285 \[ \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))} \, dx=-\frac {b \left (6 a^3 b c d-6 a^4 d^2-a^2 b^2 \left (2 c^2-5 d^2\right )-b^4 \left (c^2+2 d^2\right )\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2} (b c-a d)^3 f}-\frac {2 d^3 \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(b c-a d)^3 \sqrt {c^2-d^2} f}+\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2}+\frac {b^2 \left (3 a b c-5 a^2 d+2 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x))} \] Output:

-b*(6*a^3*b*c*d-6*a^4*d^2-a^2*b^2*(2*c^2-5*d^2)-b^4*(c^2+2*d^2))*arctan((b 
+a*tan(1/2*f*x+1/2*e))/(a^2-b^2)^(1/2))/(a^2-b^2)^(5/2)/(-a*d+b*c)^3/f-2*d 
^3*arctan((d+c*tan(1/2*f*x+1/2*e))/(c^2-d^2)^(1/2))/(-a*d+b*c)^3/(c^2-d^2) 
^(1/2)/f+1/2*b^2*cos(f*x+e)/(a^2-b^2)/(-a*d+b*c)/f/(a+b*sin(f*x+e))^2+1/2* 
b^2*(-5*a^2*d+3*a*b*c+2*b^2*d)*cos(f*x+e)/(a^2-b^2)^2/(-a*d+b*c)^2/f/(a+b* 
sin(f*x+e))
 

Mathematica [A] (verified)

Time = 2.07 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.96 \[ \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))} \, dx=\frac {-\frac {2 b \left (-6 a^3 b c d+6 a^4 d^2+a^2 b^2 \left (2 c^2-5 d^2\right )+b^4 \left (c^2+2 d^2\right )\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2} (-b c+a d)^3}+\frac {4 d^3 \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(-b c+a d)^3 \sqrt {c^2-d^2}}-\frac {b^2 \cos (e+f x)}{(a-b) (a+b) (-b c+a d) (a+b \sin (e+f x))^2}+\frac {b^2 \left (3 a b c-5 a^2 d+2 b^2 d\right ) \cos (e+f x)}{(a-b)^2 (a+b)^2 (b c-a d)^2 (a+b \sin (e+f x))}}{2 f} \] Input:

Integrate[1/((a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])),x]
 

Output:

((-2*b*(-6*a^3*b*c*d + 6*a^4*d^2 + a^2*b^2*(2*c^2 - 5*d^2) + b^4*(c^2 + 2* 
d^2))*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2) 
*(-(b*c) + a*d)^3) + (4*d^3*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2 
]])/((-(b*c) + a*d)^3*Sqrt[c^2 - d^2]) - (b^2*Cos[e + f*x])/((a - b)*(a + 
b)*(-(b*c) + a*d)*(a + b*Sin[e + f*x])^2) + (b^2*(3*a*b*c - 5*a^2*d + 2*b^ 
2*d)*Cos[e + f*x])/((a - b)^2*(a + b)^2*(b*c - a*d)^2*(a + b*Sin[e + f*x]) 
))/(2*f)
 

Rubi [A] (verified)

Time = 1.66 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.25, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3042, 3281, 25, 3042, 3534, 3042, 3480, 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.

\(\displaystyle \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))}dx\)

\(\Big \downarrow \) 3281

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3534

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3480

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3139

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

\(\Big \downarrow \) 1083

\(\displaystyle \frac {\frac {b^2 \left (-5 a^2 d+3 a b c+2 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))}-\frac {-\frac {8 d^3 \left (a^2-b^2\right )^2 \int \frac {1}{-\left (2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )^2-4 \left (c^2-d^2\right )}d\left (2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )}{f (b c-a d)}-\frac {4 b \left (-6 a^4 d^2+6 a^3 b c d-a^2 b^2 \left (2 c^2-5 d^2\right )-b^4 \left (c^2+2 d^2\right )\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (e+f x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (e+f x)\right )\right )}{f (b c-a d)}}{\left (a^2-b^2\right ) (b c-a d)}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2}\)

\(\Big \downarrow \) 217

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

Input:

Int[1/((a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])),x]
 

Output:

(b^2*Cos[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^2) + 
(-(((2*b*(6*a^3*b*c*d - 6*a^4*d^2 - a^2*b^2*(2*c^2 - 5*d^2) - b^4*(c^2 + 2 
*d^2))*ArcTan[(2*b + 2*a*Tan[(e + f*x)/2])/(2*Sqrt[a^2 - b^2])])/(Sqrt[a^2 
 - b^2]*(b*c - a*d)*f) + (4*(a^2 - b^2)^2*d^3*ArcTan[(2*d + 2*c*Tan[(e + f 
*x)/2])/(2*Sqrt[c^2 - d^2])])/((b*c - a*d)*Sqrt[c^2 - d^2]*f))/((a^2 - b^2 
)*(b*c - a*d))) + (b^2*(3*a*b*c - 5*a^2*d + 2*b^2*d)*Cos[e + f*x])/((a^2 - 
 b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])))/(2*(a^2 - b^2)*(b*c - a*d))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 3281

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

rule 3480

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

rule 3534

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_)*((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)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* 
c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2))   Int 
[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* 
d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - 
a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A 
*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && 
NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ 
[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) | 
| EqQ[a, 0])))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(624\) vs. \(2(271)=542\).

Time = 4.16 (sec) , antiderivative size = 625, normalized size of antiderivative = 2.19

method	result	size
derivativedivides	\(\frac {\frac {2 d^{3} \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} d \,c^{2}-b^{3} c^{3}\right ) \sqrt {c^{2}-d^{2}}}-\frac {2 b \left (\frac {\frac {b^{2} \left (7 a^{4} d^{2}-12 a^{3} b c d +5 a^{2} b^{2} c^{2}-4 a^{2} b^{2} d^{2}+6 a \,b^{3} c d -2 b^{4} c^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a}+\frac {b \left (6 a^{6} d^{2}-10 a^{5} b c d +4 a^{4} b^{2} c^{2}+9 a^{4} b^{2} d^{2}-16 a^{3} b^{3} c d +7 a^{2} b^{4} c^{2}-6 a^{2} b^{4} d^{2}+8 a \,b^{5} c d -2 b^{6} c^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a^{2}}+\frac {b^{2} \left (17 a^{4} d^{2}-28 a^{3} b c d +11 a^{2} b^{2} c^{2}-8 a^{2} b^{2} d^{2}+10 a \,b^{3} c d -2 b^{4} c^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {b \left (6 a^{4} d^{2}-10 a^{3} b c d +4 a^{2} b^{2} c^{2}-3 a^{2} b^{2} d^{2}+4 a \,b^{3} c d -b^{4} c^{2}\right )}{2 a^{4}-4 a^{2} b^{2}+2 b^{4}}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} a +2 b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+a \right )^{2}}+\frac {\left (6 a^{4} d^{2}-6 a^{3} b c d +2 a^{2} b^{2} c^{2}-5 a^{2} b^{2} d^{2}+b^{4} c^{2}+2 b^{4} d^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}-b^{2}}}\right )}{\left (a d -b c \right )^{3}}}{f}\)	\(625\)
default	\(\frac {\frac {2 d^{3} \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} d \,c^{2}-b^{3} c^{3}\right ) \sqrt {c^{2}-d^{2}}}-\frac {2 b \left (\frac {\frac {b^{2} \left (7 a^{4} d^{2}-12 a^{3} b c d +5 a^{2} b^{2} c^{2}-4 a^{2} b^{2} d^{2}+6 a \,b^{3} c d -2 b^{4} c^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a}+\frac {b \left (6 a^{6} d^{2}-10 a^{5} b c d +4 a^{4} b^{2} c^{2}+9 a^{4} b^{2} d^{2}-16 a^{3} b^{3} c d +7 a^{2} b^{4} c^{2}-6 a^{2} b^{4} d^{2}+8 a \,b^{5} c d -2 b^{6} c^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a^{2}}+\frac {b^{2} \left (17 a^{4} d^{2}-28 a^{3} b c d +11 a^{2} b^{2} c^{2}-8 a^{2} b^{2} d^{2}+10 a \,b^{3} c d -2 b^{4} c^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {b \left (6 a^{4} d^{2}-10 a^{3} b c d +4 a^{2} b^{2} c^{2}-3 a^{2} b^{2} d^{2}+4 a \,b^{3} c d -b^{4} c^{2}\right )}{2 a^{4}-4 a^{2} b^{2}+2 b^{4}}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} a +2 b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+a \right )^{2}}+\frac {\left (6 a^{4} d^{2}-6 a^{3} b c d +2 a^{2} b^{2} c^{2}-5 a^{2} b^{2} d^{2}+b^{4} c^{2}+2 b^{4} d^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}-b^{2}}}\right )}{\left (a d -b c \right )^{3}}}{f}\)	\(625\)
risch	\(\text {Expression too large to display}\)	\(1603\)

Input:

int(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e)),x,method=_RETURNVERBOSE)
 

Output:

1/f*(2*d^3/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(c^2-d^2)^(1/2)*a 
rctan(1/2*(2*c*tan(1/2*f*x+1/2*e)+2*d)/(c^2-d^2)^(1/2))-2*b/(a*d-b*c)^3*(( 
1/2*b^2*(7*a^4*d^2-12*a^3*b*c*d+5*a^2*b^2*c^2-4*a^2*b^2*d^2+6*a*b^3*c*d-2* 
b^4*c^2)/(a^4-2*a^2*b^2+b^4)/a*tan(1/2*f*x+1/2*e)^3+1/2*b*(6*a^6*d^2-10*a^ 
5*b*c*d+4*a^4*b^2*c^2+9*a^4*b^2*d^2-16*a^3*b^3*c*d+7*a^2*b^4*c^2-6*a^2*b^4 
*d^2+8*a*b^5*c*d-2*b^6*c^2)/(a^4-2*a^2*b^2+b^4)/a^2*tan(1/2*f*x+1/2*e)^2+1 
/2*b^2*(17*a^4*d^2-28*a^3*b*c*d+11*a^2*b^2*c^2-8*a^2*b^2*d^2+10*a*b^3*c*d- 
2*b^4*c^2)/a/(a^4-2*a^2*b^2+b^4)*tan(1/2*f*x+1/2*e)+1/2*b*(6*a^4*d^2-10*a^ 
3*b*c*d+4*a^2*b^2*c^2-3*a^2*b^2*d^2+4*a*b^3*c*d-b^4*c^2)/(a^4-2*a^2*b^2+b^ 
4))/(tan(1/2*f*x+1/2*e)^2*a+2*b*tan(1/2*f*x+1/2*e)+a)^2+1/2*(6*a^4*d^2-6*a 
^3*b*c*d+2*a^2*b^2*c^2-5*a^2*b^2*d^2+b^4*c^2+2*b^4*d^2)/(a^4-2*a^2*b^2+b^4 
)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*f*x+1/2*e)+2*b)/(a^2-b^2)^(1/2)) 
))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))} \, dx=\text {Timed out} \] Input:

integrate(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e)),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))} \, dx=\text {Timed out} \] Input:

integrate(1/(a+b*sin(f*x+e))**3/(c+d*sin(f*x+e)),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e)),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. 768 vs. \(2 (271) = 542\).

Time = 0.43 (sec) , antiderivative size = 768, normalized size of antiderivative = 2.69 \[ \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))} \, dx =\text {Too large to display} \] Input:

integrate(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e)),x, algorithm="giac")
 

Output:

-(2*(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)))*d^3/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - 
 a^3*d^3)*sqrt(c^2 - d^2)) - (2*a^2*b^3*c^2 + b^5*c^2 - 6*a^3*b^2*c*d + 6* 
a^4*b*d^2 - 5*a^2*b^3*d^2 + 2*b^5*d^2)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*s 
gn(a) + arctan((a*tan(1/2*f*x + 1/2*e) + b)/sqrt(a^2 - b^2)))/((a^4*b^3*c^ 
3 - 2*a^2*b^5*c^3 + b^7*c^3 - 3*a^5*b^2*c^2*d + 6*a^3*b^4*c^2*d - 3*a*b^6* 
c^2*d + 3*a^6*b*c*d^2 - 6*a^4*b^3*c*d^2 + 3*a^2*b^5*c*d^2 - a^7*d^3 + 2*a^ 
5*b^2*d^3 - a^3*b^4*d^3)*sqrt(a^2 - b^2)) - (5*a^3*b^4*c*tan(1/2*f*x + 1/2 
*e)^3 - 2*a*b^6*c*tan(1/2*f*x + 1/2*e)^3 - 7*a^4*b^3*d*tan(1/2*f*x + 1/2*e 
)^3 + 4*a^2*b^5*d*tan(1/2*f*x + 1/2*e)^3 + 4*a^4*b^3*c*tan(1/2*f*x + 1/2*e 
)^2 + 7*a^2*b^5*c*tan(1/2*f*x + 1/2*e)^2 - 2*b^7*c*tan(1/2*f*x + 1/2*e)^2 
- 6*a^5*b^2*d*tan(1/2*f*x + 1/2*e)^2 - 9*a^3*b^4*d*tan(1/2*f*x + 1/2*e)^2 
+ 6*a*b^6*d*tan(1/2*f*x + 1/2*e)^2 + 11*a^3*b^4*c*tan(1/2*f*x + 1/2*e) - 2 
*a*b^6*c*tan(1/2*f*x + 1/2*e) - 17*a^4*b^3*d*tan(1/2*f*x + 1/2*e) + 8*a^2* 
b^5*d*tan(1/2*f*x + 1/2*e) + 4*a^4*b^3*c - a^2*b^5*c - 6*a^5*b^2*d + 3*a^3 
*b^4*d)/((a^6*b^2*c^2 - 2*a^4*b^4*c^2 + a^2*b^6*c^2 - 2*a^7*b*c*d + 4*a^5* 
b^3*c*d - 2*a^3*b^5*c*d + a^8*d^2 - 2*a^6*b^2*d^2 + a^4*b^4*d^2)*(a*tan(1/ 
2*f*x + 1/2*e)^2 + 2*b*tan(1/2*f*x + 1/2*e) + a)^2))/f
 

Mupad [B] (verification not implemented)

Time = 32.21 (sec) , antiderivative size = 62873, normalized size of antiderivative = 220.61 \[ \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))} \, dx=\text {Too large to display} \] Input:

int(1/((a + b*sin(e + f*x))^3*(c + d*sin(e + f*x))),x)
 

Output:

(d^3*atan(((d^3*(d^2 - c^2)^(1/2)*((8*(4*a*b^12*c^4*d^5 + 4*a*b^12*c^6*d^3 
 + 4*a^3*b^10*c^8*d + 4*a^4*b^9*c*d^8 + 4*a^5*b^8*c^8*d - 16*a^6*b^7*c*d^8 
 + 24*a^8*b^5*c*d^8 - 16*a^10*b^3*c*d^8 - 4*a^2*b^11*c^3*d^6 - 8*a^2*b^11* 
c^5*d^4 - 2*a^2*b^11*c^7*d^2 - 4*a^3*b^10*c^2*d^7 - 16*a^3*b^10*c^4*d^5 - 
a^3*b^10*c^6*d^3 + 24*a^4*b^9*c^3*d^6 - 20*a^4*b^9*c^5*d^4 - 20*a^4*b^9*c^ 
7*d^2 + 12*a^5*b^8*c^2*d^7 + 95*a^5*b^8*c^4*d^5 + 20*a^5*b^8*c^6*d^3 - 98* 
a^6*b^7*c^3*d^6 + 64*a^6*b^7*c^5*d^4 - 32*a^6*b^7*c^7*d^2 + a^7*b^6*c^2*d^ 
7 - 188*a^7*b^6*c^4*d^5 + 112*a^7*b^6*c^6*d^3 + 164*a^8*b^5*c^3*d^6 - 216* 
a^8*b^5*c^5*d^4 - 28*a^9*b^4*c^2*d^7 + 240*a^9*b^4*c^4*d^5 - 140*a^10*b^3* 
c^3*d^6 + 28*a^11*b^2*c^2*d^7 + a*b^12*c^8*d + 4*a^12*b*c*d^8))/(a^14*d^6 
+ b^14*c^6 - 4*a^2*b^12*c^6 + 6*a^4*b^10*c^6 - 4*a^6*b^8*c^6 + a^8*b^6*c^6 
 + a^6*b^8*d^6 - 4*a^8*b^6*d^6 + 6*a^10*b^4*d^6 - 4*a^12*b^2*d^6 + 24*a^3* 
b^11*c^5*d - 6*a^5*b^9*c*d^5 - 36*a^5*b^9*c^5*d + 24*a^7*b^7*c*d^5 + 24*a^ 
7*b^7*c^5*d - 36*a^9*b^5*c*d^5 - 6*a^9*b^5*c^5*d + 24*a^11*b^3*c*d^5 + 15* 
a^2*b^12*c^4*d^2 - 20*a^3*b^11*c^3*d^3 + 15*a^4*b^10*c^2*d^4 - 60*a^4*b^10 
*c^4*d^2 + 80*a^5*b^9*c^3*d^3 - 60*a^6*b^8*c^2*d^4 + 90*a^6*b^8*c^4*d^2 - 
120*a^7*b^7*c^3*d^3 + 90*a^8*b^6*c^2*d^4 - 60*a^8*b^6*c^4*d^2 + 80*a^9*b^5 
*c^3*d^3 - 60*a^10*b^4*c^2*d^4 + 15*a^10*b^4*c^4*d^2 - 20*a^11*b^3*c^3*d^3 
 + 15*a^12*b^2*c^2*d^4 - 6*a*b^13*c^5*d - 6*a^13*b*c*d^5) - (8*tan(e/2 + ( 
f*x)/2)*(a*b^12*c^9 + 4*a^13*c*d^8 + 4*a^3*b^10*c^9 + 4*a^5*b^8*c^9 - 1...
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 4848, normalized size of antiderivative = 17.01 \[ \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))} \, dx =\text {Too large to display} \] Input:

int(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e)),x)
 

Output:

( - 24*sqrt(a**2 - b**2)*atan((tan((e + f*x)/2)*a + b)/sqrt(a**2 - b**2))* 
sin(e + f*x)**2*a**5*b**3*c**2*d**2 + 24*sqrt(a**2 - b**2)*atan((tan((e + 
f*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(e + f*x)**2*a**5*b**3*d**4 + 24*sqrt 
(a**2 - b**2)*atan((tan((e + f*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(e + f*x 
)**2*a**4*b**4*c**3*d - 24*sqrt(a**2 - b**2)*atan((tan((e + f*x)/2)*a + b) 
/sqrt(a**2 - b**2))*sin(e + f*x)**2*a**4*b**4*c*d**3 - 8*sqrt(a**2 - b**2) 
*atan((tan((e + f*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(e + f*x)**2*a**3*b** 
5*c**4 + 28*sqrt(a**2 - b**2)*atan((tan((e + f*x)/2)*a + b)/sqrt(a**2 - b* 
*2))*sin(e + f*x)**2*a**3*b**5*c**2*d**2 - 20*sqrt(a**2 - b**2)*atan((tan( 
(e + f*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(e + f*x)**2*a**3*b**5*d**4 - 4* 
sqrt(a**2 - b**2)*atan((tan((e + f*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(e + 
 f*x)**2*a*b**7*c**4 - 4*sqrt(a**2 - b**2)*atan((tan((e + f*x)/2)*a + b)/s 
qrt(a**2 - b**2))*sin(e + f*x)**2*a*b**7*c**2*d**2 + 8*sqrt(a**2 - b**2)*a 
tan((tan((e + f*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(e + f*x)**2*a*b**7*d** 
4 - 48*sqrt(a**2 - b**2)*atan((tan((e + f*x)/2)*a + b)/sqrt(a**2 - b**2))* 
sin(e + f*x)*a**6*b**2*c**2*d**2 + 48*sqrt(a**2 - b**2)*atan((tan((e + f*x 
)/2)*a + b)/sqrt(a**2 - b**2))*sin(e + f*x)*a**6*b**2*d**4 + 48*sqrt(a**2 
- b**2)*atan((tan((e + f*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(e + f*x)*a**5 
*b**3*c**3*d - 48*sqrt(a**2 - b**2)*atan((tan((e + f*x)/2)*a + b)/sqrt(a** 
2 - b**2))*sin(e + f*x)*a**5*b**3*c*d**3 - 16*sqrt(a**2 - b**2)*atan((t...