\(\int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx\) [692]

Optimal result

Integrand size = 25, antiderivative size = 305 \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx=-\frac {\left (2 a b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )-a^2 \left (2 c^3+3 c d^2\right )\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\left (c^2-d^2\right )^{7/2} f}+\frac {(b c-a d)^2 \cos (e+f x)}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}-\frac {(b c-a d) \left (5 a c d+b \left (c^2-6 d^2\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}+\frac {\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )-b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))} \] Output:

-(2*a*b*d*(4*c^2+d^2)-b^2*c*(c^2+4*d^2)-a^2*(2*c^3+3*c*d^2))*arctan((d+c*t 
an(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)/d/(c^2-d^2)/f/(c+d*sin(f*x+e))^3-1/6*(-a*d+b*c)*(5*a*c*d+b*(c^2-6* 
d^2))*cos(f*x+e)/d/(c^2-d^2)^2/f/(c+d*sin(f*x+e))^2+1/6*(a^2*d^2*(11*c^2+4 
*d^2)-a*b*(4*c^3*d+26*c*d^3)-b^2*(c^4-10*c^2*d^2-6*d^4))*cos(f*x+e)/d/(c^2 
-d^2)^3/f/(c+d*sin(f*x+e))
 

Mathematica [A] (verified)

Time = 1.81 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.13 \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx=\frac {\frac {12 \left (-2 a b d \left (4 c^2+d^2\right )+b^2 c \left (c^2+4 d^2\right )+a^2 \left (2 c^3+3 c d^2\right )\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\left (c^2-d^2\right )^{7/2}}+\frac {\cos (e+f x) \left (-24 a b c^5+36 a^2 c^4 d+25 b^2 c^4 d-44 a b c^3 d^2+a^2 c^2 d^3+14 b^2 c^2 d^3-22 a b c d^4+8 a^2 d^5+6 b^2 d^5+d \left (-a^2 d^2 \left (11 c^2+4 d^2\right )+a b \left (4 c^3 d+26 c d^3\right )+b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right ) \cos (2 (e+f x))-6 \left (-a^2 c d^2 \left (9 c^2+d^2\right )-2 a b d \left (-2 c^4-9 c^2 d^2+d^4\right )+b^2 \left (c^5-9 c^3 d^2-2 c d^4\right )\right ) \sin (e+f x)\right )}{\left (c^2-d^2\right )^3 (c+d \sin (e+f x))^3}}{12 f} \] Input:

Integrate[(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^4,x]
 

Output:

((12*(-2*a*b*d*(4*c^2 + d^2) + b^2*c*(c^2 + 4*d^2) + a^2*(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) + (C 
os[e + f*x]*(-24*a*b*c^5 + 36*a^2*c^4*d + 25*b^2*c^4*d - 44*a*b*c^3*d^2 + 
a^2*c^2*d^3 + 14*b^2*c^2*d^3 - 22*a*b*c*d^4 + 8*a^2*d^5 + 6*b^2*d^5 + d*(- 
(a^2*d^2*(11*c^2 + 4*d^2)) + a*b*(4*c^3*d + 26*c*d^3) + b^2*(c^4 - 10*c^2* 
d^2 - 6*d^4))*Cos[2*(e + f*x)] - 6*(-(a^2*c*d^2*(9*c^2 + d^2)) - 2*a*b*d*( 
-2*c^4 - 9*c^2*d^2 + d^4) + b^2*(c^5 - 9*c^3*d^2 - 2*c*d^4))*Sin[e + f*x]) 
)/((c^2 - d^2)^3*(c + d*Sin[e + f*x])^3))/(12*f)
 

Rubi [A] (verified)

Time = 1.16 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.11, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3042, 3269, 3042, 3233, 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.

\(\displaystyle \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^4}dx\)

\(\Big \downarrow \) 3269

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3233

\(\displaystyle \frac {-\frac {\int \frac {2 d \left (-\left (\left (3 c^2+2 d^2\right ) a^2\right )+10 b c d a-b^2 \left (2 c^2+3 d^2\right )\right )-(b c-a d) \left (5 a c d+b \left (c^2-6 d^2\right )\right ) \sin (e+f x)}{(c+d \sin (e+f x))^2}dx}{2 \left (c^2-d^2\right )}-\frac {(b c-a d) \left (5 a c d+b c^2-6 b d^2\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{3 d \left (c^2-d^2\right )}+\frac {(b c-a d)^2 \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\int \frac {2 d \left (-\left (\left (3 c^2+2 d^2\right ) a^2\right )+10 b c d a-b^2 \left (2 c^2+3 d^2\right )\right )-(b c-a d) \left (5 a c d+b \left (c^2-6 d^2\right )\right ) \sin (e+f x)}{(c+d \sin (e+f x))^2}dx}{2 \left (c^2-d^2\right )}-\frac {(b c-a d) \left (5 a c d+b c^2-6 b d^2\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{3 d \left (c^2-d^2\right )}+\frac {(b c-a d)^2 \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^3}\)

\(\Big \downarrow \) 3233

\(\displaystyle \frac {-\frac {-\frac {\int -\frac {3 d \left (-\left (\left (2 c^3+3 d^2 c\right ) a^2\right )+2 b d \left (4 c^2+d^2\right ) a-b^2 c \left (c^2+4 d^2\right )\right )}{c+d \sin (e+f x)}dx}{c^2-d^2}-\frac {\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )-\left (b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {(b c-a d) \left (5 a c d+b c^2-6 b d^2\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{3 d \left (c^2-d^2\right )}+\frac {(b c-a d)^2 \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\frac {3 d \left (-\left (a^2 \left (2 c^3+3 c d^2\right )\right )+2 a b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )\right ) \int \frac {1}{c+d \sin (e+f x)}dx}{c^2-d^2}-\frac {\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )-\left (b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {(b c-a d) \left (5 a c d+b c^2-6 b d^2\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{3 d \left (c^2-d^2\right )}+\frac {(b c-a d)^2 \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\frac {3 d \left (-\left (a^2 \left (2 c^3+3 c d^2\right )\right )+2 a b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )\right ) \int \frac {1}{c+d \sin (e+f x)}dx}{c^2-d^2}-\frac {\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )-\left (b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {(b c-a d) \left (5 a c d+b c^2-6 b d^2\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{3 d \left (c^2-d^2\right )}+\frac {(b c-a d)^2 \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^3}\)

\(\Big \downarrow \) 3139

\(\displaystyle \frac {-\frac {\frac {6 d \left (-\left (a^2 \left (2 c^3+3 c d^2\right )\right )+2 a b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )\right ) \int \frac {1}{c \tan ^2\left (\frac {1}{2} (e+f x)\right )+2 d \tan \left (\frac {1}{2} (e+f x)\right )+c}d\tan \left (\frac {1}{2} (e+f x)\right )}{f \left (c^2-d^2\right )}-\frac {\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )-\left (b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {(b c-a d) \left (5 a c d+b c^2-6 b d^2\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{3 d \left (c^2-d^2\right )}+\frac {(b c-a d)^2 \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^3}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {-\frac {-\frac {12 d \left (-\left (a^2 \left (2 c^3+3 c d^2\right )\right )+2 a b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )\right ) \int \frac {1}{-\left (2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )^2-4 \left (c^2-d^2\right )}d\left (2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )}{f \left (c^2-d^2\right )}-\frac {\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )-\left (b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {(b c-a d) \left (5 a c d+b c^2-6 b d^2\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{3 d \left (c^2-d^2\right )}+\frac {(b c-a d)^2 \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^3}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {-\frac {\frac {6 d \left (-\left (a^2 \left (2 c^3+3 c d^2\right )\right )+2 a b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )\right ) \arctan \left (\frac {2 c \tan \left (\frac {1}{2} (e+f x)\right )+2 d}{2 \sqrt {c^2-d^2}}\right )}{f \left (c^2-d^2\right )^{3/2}}-\frac {\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )-\left (b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right )}-\frac {(b c-a d) \left (5 a c d+b c^2-6 b d^2\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}}{3 d \left (c^2-d^2\right )}+\frac {(b c-a d)^2 \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^3}\)

Input:

Int[(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^4,x]
 

Output:

((b*c - a*d)^2*Cos[e + f*x])/(3*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^3) + 
(-1/2*((b*c - a*d)*(b*c^2 + 5*a*c*d - 6*b*d^2)*Cos[e + f*x])/((c^2 - d^2)* 
f*(c + d*Sin[e + f*x])^2) - ((6*d*(2*a*b*d*(4*c^2 + d^2) - b^2*c*(c^2 + 4* 
d^2) - a^2*(2*c^3 + 3*c*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*d^2*(11*c^2 + 4*d^2) - a*b*(4* 
c^3*d + 26*c*d^3) - b^2*(c^4 - 10*c^2*d^2 - 6*d^4))*Cos[e + f*x])/((c^2 - 
d^2)*f*(c + d*Sin[e + f*x])))/(2*(c^2 - d^2)))/(3*d*(c^2 - d^2))
 

Defintions of rubi rules used

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 3269

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)])^2, x_Symbol] :> Simp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*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*(m + 1) 
*(2*b*c*d - a*(c^2 + d^2)) + (a^2*d^2 - 2*a*b*c*d*(m + 2) + b^2*(d^2*(m + 1 
) + c^2*(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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(932\) vs. \(2(294)=588\).

Time = 2.71 (sec) , antiderivative size = 933, normalized size of antiderivative = 3.06

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (9 a^{2} c^{4} d^{2}-6 a^{2} c^{2} d^{4}+2 a^{2} d^{6}-8 a b \,c^{5} d -2 a b \,c^{3} d^{3}+b^{2} c^{6}+4 b^{2} c^{4} d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{c \left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right )}+\frac {\left (6 a^{2} c^{6} d +27 a^{2} c^{4} d^{3}-12 a^{2} c^{2} d^{5}+4 a^{2} d^{7}-4 a b \,c^{7}-28 a b \,c^{5} d^{2}-22 a b \,c^{3} d^{4}+4 a b c \,d^{6}+5 b^{2} c^{6} d +20 b^{2} c^{4} d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{\left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right ) c^{2}}+\frac {2 d \left (54 a^{2} c^{6} d +21 a^{2} c^{4} d^{3}-4 a^{2} c^{2} d^{5}+4 a^{2} d^{7}-36 a b \,c^{7}-84 a b \,c^{5} d^{2}-34 a b \,c^{3} d^{4}+4 a b c \,d^{6}+39 b^{2} c^{6} d +32 b^{2} c^{4} d^{3}+4 b^{2} c^{2} d^{5}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3 c^{3} \left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right )}+\frac {2 \left (6 a^{2} c^{6} d +20 a^{2} c^{4} d^{3}-3 a^{2} c^{2} d^{5}+2 a^{2} d^{7}-4 a b \,c^{7}-20 a b \,c^{5} d^{2}-28 a b \,c^{3} d^{4}+2 a b c \,d^{6}+4 b^{2} c^{6} d +17 b^{2} c^{4} d^{3}+4 b^{2} c^{2} d^{5}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{c^{2} \left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right )}+\frac {\left (27 a^{2} c^{4} d^{2}-4 a^{2} c^{2} d^{4}+2 a^{2} d^{6}-16 a b \,c^{5} d -38 a b \,c^{3} d^{3}+4 a b c \,d^{5}-b^{2} c^{6}+22 b^{2} c^{4} d^{2}+4 b^{2} c^{2} d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{c \left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right )}+\frac {2 \left (18 a^{2} c^{4} d -5 a^{2} c^{2} d^{3}+2 a^{2} d^{5}-12 a b \,c^{5}-20 a b \,c^{3} d^{2}+2 a b c \,d^{4}+13 b^{2} c^{4} d +2 b^{2} c^{2} d^{3}\right )}{6 c^{6}-18 c^{4} d^{2}+18 c^{2} d^{4}-6 d^{6}}}{\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 )^{3}}+\frac {\left (2 a^{2} c^{3}+3 a^{2} c \,d^{2}-8 a b d \,c^{2}-2 a b \,d^{3}+b^{2} c^{3}+4 b^{2} c \,d^{2}\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 )}{\left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right ) \sqrt {c^{2}-d^{2}}}}{f}\)	\(933\)
default	\(\frac {\frac {\frac {\left (9 a^{2} c^{4} d^{2}-6 a^{2} c^{2} d^{4}+2 a^{2} d^{6}-8 a b \,c^{5} d -2 a b \,c^{3} d^{3}+b^{2} c^{6}+4 b^{2} c^{4} d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{c \left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right )}+\frac {\left (6 a^{2} c^{6} d +27 a^{2} c^{4} d^{3}-12 a^{2} c^{2} d^{5}+4 a^{2} d^{7}-4 a b \,c^{7}-28 a b \,c^{5} d^{2}-22 a b \,c^{3} d^{4}+4 a b c \,d^{6}+5 b^{2} c^{6} d +20 b^{2} c^{4} d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{\left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right ) c^{2}}+\frac {2 d \left (54 a^{2} c^{6} d +21 a^{2} c^{4} d^{3}-4 a^{2} c^{2} d^{5}+4 a^{2} d^{7}-36 a b \,c^{7}-84 a b \,c^{5} d^{2}-34 a b \,c^{3} d^{4}+4 a b c \,d^{6}+39 b^{2} c^{6} d +32 b^{2} c^{4} d^{3}+4 b^{2} c^{2} d^{5}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3 c^{3} \left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right )}+\frac {2 \left (6 a^{2} c^{6} d +20 a^{2} c^{4} d^{3}-3 a^{2} c^{2} d^{5}+2 a^{2} d^{7}-4 a b \,c^{7}-20 a b \,c^{5} d^{2}-28 a b \,c^{3} d^{4}+2 a b c \,d^{6}+4 b^{2} c^{6} d +17 b^{2} c^{4} d^{3}+4 b^{2} c^{2} d^{5}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{c^{2} \left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right )}+\frac {\left (27 a^{2} c^{4} d^{2}-4 a^{2} c^{2} d^{4}+2 a^{2} d^{6}-16 a b \,c^{5} d -38 a b \,c^{3} d^{3}+4 a b c \,d^{5}-b^{2} c^{6}+22 b^{2} c^{4} d^{2}+4 b^{2} c^{2} d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{c \left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right )}+\frac {2 \left (18 a^{2} c^{4} d -5 a^{2} c^{2} d^{3}+2 a^{2} d^{5}-12 a b \,c^{5}-20 a b \,c^{3} d^{2}+2 a b c \,d^{4}+13 b^{2} c^{4} d +2 b^{2} c^{2} d^{3}\right )}{6 c^{6}-18 c^{4} d^{2}+18 c^{2} d^{4}-6 d^{6}}}{\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 )^{3}}+\frac {\left (2 a^{2} c^{3}+3 a^{2} c \,d^{2}-8 a b d \,c^{2}-2 a b \,d^{3}+b^{2} c^{3}+4 b^{2} c \,d^{2}\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 )}{\left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right ) \sqrt {c^{2}-d^{2}}}}{f}\)	\(933\)
risch	\(\text {Expression too large to display}\)	\(1942\)

Input:

int((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^4,x,method=_RETURNVERBOSE)
 

Output:

1/f*(2*(1/2*(9*a^2*c^4*d^2-6*a^2*c^2*d^4+2*a^2*d^6-8*a*b*c^5*d-2*a*b*c^3*d 
^3+b^2*c^6+4*b^2*c^4*d^2)/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^2*c^6*d+27*a^2*c^4*d^3-12*a^2*c^2*d^5+4*a^2*d^7-4*a*b*c^7-28 
*a*b*c^5*d^2-22*a*b*c^3*d^4+4*a*b*c*d^6+5*b^2*c^6*d+20*b^2*c^4*d^3)/(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^2*c^6*d+2 
1*a^2*c^4*d^3-4*a^2*c^2*d^5+4*a^2*d^7-36*a*b*c^7-84*a*b*c^5*d^2-34*a*b*c^3 
*d^4+4*a*b*c*d^6+39*b^2*c^6*d+32*b^2*c^4*d^3+4*b^2*c^2*d^5)/(c^6-3*c^4*d^2 
+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^3+1/c^2*(6*a^2*c^6*d+20*a^2*c^4*d^3-3*a 
^2*c^2*d^5+2*a^2*d^7-4*a*b*c^7-20*a*b*c^5*d^2-28*a*b*c^3*d^4+2*a*b*c*d^6+4 
*b^2*c^6*d+17*b^2*c^4*d^3+4*b^2*c^2*d^5)/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan 
(1/2*f*x+1/2*e)^2+1/2*(27*a^2*c^4*d^2-4*a^2*c^2*d^4+2*a^2*d^6-16*a*b*c^5*d 
-38*a*b*c^3*d^3+4*a*b*c*d^5-b^2*c^6+22*b^2*c^4*d^2+4*b^2*c^2*d^4)/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^2*c^4*d-5*a^2*c^2*d^3 
+2*a^2*d^5-12*a*b*c^5-20*a*b*c^3*d^2+2*a*b*c*d^4+13*b^2*c^4*d+2*b^2*c^2*d^ 
3)/(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^2*c^3+3*a^2*c*d^2-8*a*b*c^2*d-2*a*b*d^3+b^2*c^3+4*b^2*c*d 
^2)/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)/(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. 820 vs. \(2 (291) = 582\).

Time = 0.17 (sec) , antiderivative size = 1724, normalized size of antiderivative = 5.65 \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx=\text {Too large to display} \] Input:

integrate((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^4,x, algorithm="fricas")
 

Output:

[-1/12*(2*(b^2*c^6*d + 4*a*b*c^5*d^2 + 22*a*b*c^3*d^4 - 26*a*b*c*d^6 - 11* 
(a^2 + b^2)*c^4*d^3 + (7*a^2 + 4*b^2)*c^2*d^5 + 2*(2*a^2 + 3*b^2)*d^7)*cos 
(f*x + e)^3 - 6*(b^2*c^7 + 4*a*b*c^6*d + 14*a*b*c^4*d^3 - 20*a*b*c^2*d^5 + 
 2*a*b*d^7 - (9*a^2 + 10*b^2)*c^5*d^2 + (8*a^2 + 7*b^2)*c^3*d^4 + (a^2 + 2 
*b^2)*c*d^6)*cos(f*x + e)*sin(f*x + e) + 3*(8*a*b*c^5*d + 26*a*b*c^3*d^3 + 
 6*a*b*c*d^5 - (2*a^2 + b^2)*c^6 - (9*a^2 + 7*b^2)*c^4*d^2 - 3*(3*a^2 + 4* 
b^2)*c^2*d^4 - 3*(8*a*b*c^3*d^3 + 2*a*b*c*d^5 - (2*a^2 + b^2)*c^4*d^2 - (3 
*a^2 + 4*b^2)*c^2*d^4)*cos(f*x + e)^2 + (24*a*b*c^4*d^2 + 14*a*b*c^2*d^4 + 
 2*a*b*d^6 - 3*(2*a^2 + b^2)*c^5*d - (11*a^2 + 13*b^2)*c^3*d^3 - (3*a^2 + 
4*b^2)*c*d^5 - (8*a*b*c^2*d^4 + 2*a*b*d^6 - (2*a^2 + b^2)*c^3*d^3 - (3*a^2 
 + 4*b^2)*c*d^5)*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*(2*a*b*c^7 + 2*a*b*c^5*d^2 + 2*a^2*c 
^4*d^3 + b^2*c^2*d^5 - 4*a*b*c*d^6 - (3*a^2 + 2*b^2)*c^6*d + (a^2 + b^2)*d 
^7)*cos(f*x + e))/(3*(c^9*d^2 - 4*c^7*d^4 + 6*c^5*d^6 - 4*c^3*d^8 + c*d^10 
)*f*cos(f*x + e)^2 - (c^11 - c^9*d^2 - 6*c^7*d^4 + 14*c^5*d^6 - 11*c^3*d^8 
 + 3*c*d^10)*f + ((c^8*d^3 - 4*c^6*d^5 + 6*c^4*d^7 - 4*c^2*d^9 + d^11)*f*c 
os(f*x + e)^2 - (3*c^10*d - 11*c^8*d^3 + 14*c^6*d^5 - 6*c^4*d^7 - c^2*d^9 
+ d^11)*f)*sin(f*x + e)), -1/6*((b^2*c^6*d + 4*a*b*c^5*d^2 + 22*a*b*c^3...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx=\text {Timed out} \] Input:

integrate((a+b*sin(f*x+e))**2/(c+d*sin(f*x+e))**4,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a+b*sin(f*x+e))^2/(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. 1264 vs. \(2 (291) = 582\).

Time = 0.18 (sec) , antiderivative size = 1264, normalized size of antiderivative = 4.14 \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx=\text {Too large to display} \] Input:

integrate((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^4,x, algorithm="giac")
 

Output:

1/3*(3*(2*a^2*c^3 + b^2*c^3 - 8*a*b*c^2*d + 3*a^2*c*d^2 + 4*b^2*c*d^2 - 2* 
a*b*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)) + (3*b^2*c^8*tan(1/2*f*x + 1/2*e)^5 - 24*a*b*c^7*d*tan(1/2*f*x 
 + 1/2*e)^5 + 27*a^2*c^6*d^2*tan(1/2*f*x + 1/2*e)^5 + 12*b^2*c^6*d^2*tan(1 
/2*f*x + 1/2*e)^5 - 6*a*b*c^5*d^3*tan(1/2*f*x + 1/2*e)^5 - 18*a^2*c^4*d^4* 
tan(1/2*f*x + 1/2*e)^5 + 6*a^2*c^2*d^6*tan(1/2*f*x + 1/2*e)^5 - 12*a*b*c^8 
*tan(1/2*f*x + 1/2*e)^4 + 18*a^2*c^7*d*tan(1/2*f*x + 1/2*e)^4 + 15*b^2*c^7 
*d*tan(1/2*f*x + 1/2*e)^4 - 84*a*b*c^6*d^2*tan(1/2*f*x + 1/2*e)^4 + 81*a^2 
*c^5*d^3*tan(1/2*f*x + 1/2*e)^4 + 60*b^2*c^5*d^3*tan(1/2*f*x + 1/2*e)^4 - 
66*a*b*c^4*d^4*tan(1/2*f*x + 1/2*e)^4 - 36*a^2*c^3*d^5*tan(1/2*f*x + 1/2*e 
)^4 + 12*a*b*c^2*d^6*tan(1/2*f*x + 1/2*e)^4 + 12*a^2*c*d^7*tan(1/2*f*x + 1 
/2*e)^4 - 72*a*b*c^7*d*tan(1/2*f*x + 1/2*e)^3 + 108*a^2*c^6*d^2*tan(1/2*f* 
x + 1/2*e)^3 + 78*b^2*c^6*d^2*tan(1/2*f*x + 1/2*e)^3 - 168*a*b*c^5*d^3*tan 
(1/2*f*x + 1/2*e)^3 + 42*a^2*c^4*d^4*tan(1/2*f*x + 1/2*e)^3 + 64*b^2*c^4*d 
^4*tan(1/2*f*x + 1/2*e)^3 - 68*a*b*c^3*d^5*tan(1/2*f*x + 1/2*e)^3 - 8*a^2* 
c^2*d^6*tan(1/2*f*x + 1/2*e)^3 + 8*b^2*c^2*d^6*tan(1/2*f*x + 1/2*e)^3 + 8* 
a*b*c*d^7*tan(1/2*f*x + 1/2*e)^3 + 8*a^2*d^8*tan(1/2*f*x + 1/2*e)^3 - 24*a 
*b*c^8*tan(1/2*f*x + 1/2*e)^2 + 36*a^2*c^7*d*tan(1/2*f*x + 1/2*e)^2 + 24*b 
^2*c^7*d*tan(1/2*f*x + 1/2*e)^2 - 120*a*b*c^6*d^2*tan(1/2*f*x + 1/2*e)^...
 

Mupad [B] (verification not implemented)

Time = 20.12 (sec) , antiderivative size = 1220, normalized size of antiderivative = 4.00 \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx =\text {Too large to display} \] Input:

int((a + b*sin(e + f*x))^2/(c + d*sin(e + f*x))^4,x)
 

Output:

((2*a^2*d^5 + 18*a^2*c^4*d + 13*b^2*c^4*d - 5*a^2*c^2*d^3 + 2*b^2*c^2*d^3 
- 12*a*b*c^5 + 2*a*b*c*d^4 - 20*a*b*c^3*d^2)/(3*(c^6 - d^6 + 3*c^2*d^4 - 3 
*c^4*d^2)) + (tan(e/2 + (f*x)/2)^4*(4*a^2*d^7 + 6*a^2*c^6*d + 5*b^2*c^6*d 
- 12*a^2*c^2*d^5 + 27*a^2*c^4*d^3 + 20*b^2*c^4*d^3 - 4*a*b*c^7 + 4*a*b*c*d 
^6 - 22*a*b*c^3*d^4 - 28*a*b*c^5*d^2))/(c^2*(c^6 - d^6 + 3*c^2*d^4 - 3*c^4 
*d^2)) + (tan(e/2 + (f*x)/2)*(2*a^2*d^6 - b^2*c^6 - 4*a^2*c^2*d^4 + 27*a^2 
*c^4*d^2 + 4*b^2*c^2*d^4 + 22*b^2*c^4*d^2 + 4*a*b*c*d^5 - 16*a*b*c^5*d - 3 
8*a*b*c^3*d^3))/(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^2*d^7 + 6*a^2*c^6*d + 4*b^2*c^6*d - 3*a^2*c^2*d^5 + 20*a^2*c^ 
4*d^3 + 4*b^2*c^2*d^5 + 17*b^2*c^4*d^3 - 4*a*b*c^7 + 2*a*b*c*d^6 - 28*a*b* 
c^3*d^4 - 20*a*b*c^5*d^2))/(c^2*(c^6 - d^6 + 3*c^2*d^4 - 3*c^4*d^2)) + (ta 
n(e/2 + (f*x)/2)^5*(2*a^2*d^6 + b^2*c^6 - 6*a^2*c^2*d^4 + 9*a^2*c^4*d^2 + 
4*b^2*c^4*d^2 - 8*a*b*c^5*d - 2*a*b*c^3*d^3))/(c*(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^2*d^5 + 18*a^ 
2*c^4*d + 13*b^2*c^4*d - 5*a^2*c^2*d^3 + 2*b^2*c^2*d^3 - 12*a*b*c^5 + 2*a* 
b*c*d^4 - 20*a*b*c^3*d^2))/(3*c^3*(c^6 - d^6 + 3*c^2*d^4 - 3*c^4*d^2)))/(f 
*(c^3*tan(e/2 + (f*x)/2)^6 + tan(e/2 + (f*x)/2)^2*(12*c*d^2 + 3*c^3) + tan 
(e/2 + (f*x)/2)^4*(12*c*d^2 + 3*c^3) + tan(e/2 + (f*x)/2)^3*(12*c^2*d + 8* 
d^3) + c^3 + 6*c^2*d*tan(e/2 + (f*x)/2) + 6*c^2*d*tan(e/2 + (f*x)/2)^5)) + 
 (atan((((c*tan(e/2 + (f*x)/2)*(2*a^2*c^3 + b^2*c^3 + 3*a^2*c*d^2 + 4*b...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 2911, normalized size of antiderivative = 9.54 \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx =\text {Too large to display} \] Input:

int((a+b*sin(f*x+e))^2/(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**2*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**2*c**3*d**6 - 48*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 
*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*a*b*c**2*d**7 + 6*sqrt(c**2 - d**2)*atan((tan((e 
+ f*x)/2)*c + d)/sqrt(c**2 - d**2))*sin(e + f*x)**3*b**2*c**5*d**4 + 24*sq 
rt(c**2 - d**2)*atan((tan((e + f*x)/2)*c + d)/sqrt(c**2 - d**2))*sin(e + f 
*x)**3*b**2*c**3*d**6 + 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**2*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**2*c**4 
*d**5 - 144*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*c**5*d**4 - 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*b*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)* 
*2*b**2*c**6*d**3 + 72*sqrt(c**2 - d**2)*atan((tan((e + f*x)/2)*c + d)/sqr 
t(c**2 - d**2))*sin(e + f*x)**2*b**2*c**4*d**5 + 36*sqrt(c**2 - d**2)*atan 
((tan((e + f*x)/2)*c + d)/sqrt(c**2 - d**2))*sin(e + f*x)*a**2*c**7*d**2 + 
 54*sqrt(c**2 - d**2)*atan((tan((e + f*x)/2)*c + d)/sqrt(c**2 - d**2))*sin 
(e + f*x)*a**2*c**5*d**4 - 144*sqrt(c**2 - d**2)*atan((tan((e + f*x)/2)...