\(\int \frac {(c+d \sin (e+f x))^4}{(a+b \sin (e+f x))^3} \, dx\) [722]

Optimal result

Integrand size = 25, antiderivative size = 318 \[ \int \frac {(c+d \sin (e+f x))^4}{(a+b \sin (e+f x))^3} \, dx=\frac {d^3 (4 b c-3 a d) x}{b^4}+\frac {(b c-a d)^2 \left (4 a^3 b c d-10 a b^3 c d+6 a^4 d^2+a^2 b^2 \left (2 c^2-15 d^2\right )+b^4 \left (c^2+12 d^2\right )\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{b^4 \left (a^2-b^2\right )^{5/2} f}+\frac {d^2 \left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{2 b^3 \left (a^2-b^2\right ) f}+\frac {3 (b c-a d)^3 \left (a b c+a^2 d-2 b^2 d\right ) \cos (e+f x)}{2 b^3 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2} \] Output:

d^3*(-3*a*d+4*b*c)*x/b^4+(-a*d+b*c)^2*(4*a^3*b*c*d-10*a*b^3*c*d+6*a^4*d^2+ 
a^2*b^2*(2*c^2-15*d^2)+b^4*(c^2+12*d^2))*arctan((b+a*tan(1/2*f*x+1/2*e))/( 
a^2-b^2)^(1/2))/b^4/(a^2-b^2)^(5/2)/f+1/2*d^2*(2*a*b*c*d-3*a^2*d^2-b^2*(c^ 
2-2*d^2))*cos(f*x+e)/b^3/(a^2-b^2)/f+3/2*(-a*d+b*c)^3*(a^2*d+a*b*c-2*b^2*d 
)*cos(f*x+e)/b^3/(a^2-b^2)^2/f/(a+b*sin(f*x+e))+1/2*(-a*d+b*c)^2*cos(f*x+e 
)*(c+d*sin(f*x+e))^2/b/(a^2-b^2)/f/(a+b*sin(f*x+e))^2
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(894\) vs. \(2(318)=636\).

Time = 13.59 (sec) , antiderivative size = 894, normalized size of antiderivative = 2.81 \[ \int \frac {(c+d \sin (e+f x))^4}{(a+b \sin (e+f x))^3} \, dx=\frac {\frac {4 (b c-a d)^2 \left (4 a^3 b c d-10 a b^3 c d+6 a^4 d^2+a^2 b^2 \left (2 c^2-15 d^2\right )+b^4 \left (c^2+12 d^2\right )\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac {16 a^6 b c d^3 e-24 a^4 b^3 c d^3 e+8 b^7 c d^3 e-12 a^7 d^4 e+18 a^5 b^2 d^4 e-6 a b^6 d^4 e+16 a^6 b c d^3 f x-24 a^4 b^3 c d^3 f x+8 b^7 c d^3 f x-12 a^7 d^4 f x+18 a^5 b^2 d^4 f x-6 a b^6 d^4 f x-b \left (8 a b^5 c^3 d-16 a^5 b c d^3+12 a^6 d^4-21 a^4 b^2 d^4+8 a^3 b^3 c d \left (2 c^2+5 d^2\right )+b^6 \left (2 c^4+d^4\right )+2 a^2 b^4 \left (-4 c^4-18 c^2 d^2+d^4\right )\right ) \cos (e+f x)+2 b^2 \left (a^2-b^2\right )^2 d^3 (-4 b c+3 a d) (e+f x) \cos (2 (e+f x))+a^4 b^3 d^4 \cos (3 (e+f x))-2 a^2 b^5 d^4 \cos (3 (e+f x))+b^7 d^4 \cos (3 (e+f x))+32 a^5 b^2 c d^3 e \sin (e+f x)-64 a^3 b^4 c d^3 e \sin (e+f x)+32 a b^6 c d^3 e \sin (e+f x)-24 a^6 b d^4 e \sin (e+f x)+48 a^4 b^3 d^4 e \sin (e+f x)-24 a^2 b^5 d^4 e \sin (e+f x)+32 a^5 b^2 c d^3 f x \sin (e+f x)-64 a^3 b^4 c d^3 f x \sin (e+f x)+32 a b^6 c d^3 f x \sin (e+f x)-24 a^6 b d^4 f x \sin (e+f x)+48 a^4 b^3 d^4 f x \sin (e+f x)-24 a^2 b^5 d^4 f x \sin (e+f x)+3 a b^6 c^4 \sin (2 (e+f x))-4 a^2 b^5 c^3 d \sin (2 (e+f x))-8 b^7 c^3 d \sin (2 (e+f x))-6 a^3 b^4 c^2 d^2 \sin (2 (e+f x))+24 a b^6 c^2 d^2 \sin (2 (e+f x))+12 a^4 b^3 c d^3 \sin (2 (e+f x))-24 a^2 b^5 c d^3 \sin (2 (e+f x))-9 a^5 b^2 d^4 \sin (2 (e+f x))+16 a^3 b^4 d^4 \sin (2 (e+f x))-4 a b^6 d^4 \sin (2 (e+f x))}{\left (a^2-b^2\right )^2 (a+b \sin (e+f x))^2}}{4 b^4 f} \] Input:

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

Output:

((4*(b*c - a*d)^2*(4*a^3*b*c*d - 10*a*b^3*c*d + 6*a^4*d^2 + a^2*b^2*(2*c^2 
 - 15*d^2) + b^4*(c^2 + 12*d^2))*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 
- b^2]])/(a^2 - b^2)^(5/2) + (16*a^6*b*c*d^3*e - 24*a^4*b^3*c*d^3*e + 8*b^ 
7*c*d^3*e - 12*a^7*d^4*e + 18*a^5*b^2*d^4*e - 6*a*b^6*d^4*e + 16*a^6*b*c*d 
^3*f*x - 24*a^4*b^3*c*d^3*f*x + 8*b^7*c*d^3*f*x - 12*a^7*d^4*f*x + 18*a^5* 
b^2*d^4*f*x - 6*a*b^6*d^4*f*x - b*(8*a*b^5*c^3*d - 16*a^5*b*c*d^3 + 12*a^6 
*d^4 - 21*a^4*b^2*d^4 + 8*a^3*b^3*c*d*(2*c^2 + 5*d^2) + b^6*(2*c^4 + d^4) 
+ 2*a^2*b^4*(-4*c^4 - 18*c^2*d^2 + d^4))*Cos[e + f*x] + 2*b^2*(a^2 - b^2)^ 
2*d^3*(-4*b*c + 3*a*d)*(e + f*x)*Cos[2*(e + f*x)] + a^4*b^3*d^4*Cos[3*(e + 
 f*x)] - 2*a^2*b^5*d^4*Cos[3*(e + f*x)] + b^7*d^4*Cos[3*(e + f*x)] + 32*a^ 
5*b^2*c*d^3*e*Sin[e + f*x] - 64*a^3*b^4*c*d^3*e*Sin[e + f*x] + 32*a*b^6*c* 
d^3*e*Sin[e + f*x] - 24*a^6*b*d^4*e*Sin[e + f*x] + 48*a^4*b^3*d^4*e*Sin[e 
+ f*x] - 24*a^2*b^5*d^4*e*Sin[e + f*x] + 32*a^5*b^2*c*d^3*f*x*Sin[e + f*x] 
 - 64*a^3*b^4*c*d^3*f*x*Sin[e + f*x] + 32*a*b^6*c*d^3*f*x*Sin[e + f*x] - 2 
4*a^6*b*d^4*f*x*Sin[e + f*x] + 48*a^4*b^3*d^4*f*x*Sin[e + f*x] - 24*a^2*b^ 
5*d^4*f*x*Sin[e + f*x] + 3*a*b^6*c^4*Sin[2*(e + f*x)] - 4*a^2*b^5*c^3*d*Si 
n[2*(e + f*x)] - 8*b^7*c^3*d*Sin[2*(e + f*x)] - 6*a^3*b^4*c^2*d^2*Sin[2*(e 
 + f*x)] + 24*a*b^6*c^2*d^2*Sin[2*(e + f*x)] + 12*a^4*b^3*c*d^3*Sin[2*(e + 
 f*x)] - 24*a^2*b^5*c*d^3*Sin[2*(e + f*x)] - 9*a^5*b^2*d^4*Sin[2*(e + f*x) 
] + 16*a^3*b^4*d^4*Sin[2*(e + f*x)] - 4*a*b^6*d^4*Sin[2*(e + f*x)])/((a...
 

Rubi [A] (verified)

Time = 1.91 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.14, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3042, 3271, 3042, 3510, 25, 3042, 3502, 3042, 3214, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3271

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3510

\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\frac {\int -\frac {-b \left (a^2-b^2\right ) \left (-\left (\left (c^2-2 d^2\right ) b^2\right )+2 a c d b-3 a^2 d^2\right ) \sin ^2(e+f x) d^2+\left (a^2-b^2\right ) \left (-3 d^2 a^3+6 b c d a^2+b^2 \left (c^2+4 d^2\right ) a-8 b^3 c d\right ) \sin (e+f x) d^2+b \left (c^2 \left (c^2+12 d^2\right ) b^4-4 a c d \left (3 c^2+4 d^2\right ) b^3+2 a^2 \left (c^4+3 d^2 c^2+3 d^4\right ) b^2+4 a^3 c d^3 b-3 a^4 d^4\right )}{a+b \sin (e+f x)}dx}{b^2 \left (a^2-b^2\right )}-\frac {3 (b c-a d)^3 \left (a^2 d+a b c-2 b^2 d\right ) \cos (e+f x)}{b^2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {-\frac {\int \frac {-b \left (a^2-b^2\right ) \left (-\left (\left (c^2-2 d^2\right ) b^2\right )+2 a c d b-3 a^2 d^2\right ) \sin ^2(e+f x) d^2+\left (a^2-b^2\right ) \left (-3 d^2 a^3+6 b c d a^2+b^2 \left (c^2+4 d^2\right ) a-8 b^3 c d\right ) \sin (e+f x) d^2+b \left (c^2 \left (c^2+12 d^2\right ) b^4-4 a c d \left (3 c^2+4 d^2\right ) b^3+2 a^2 \left (c^4+3 d^2 c^2+3 d^4\right ) b^2+4 a^3 c d^3 b-3 a^4 d^4\right )}{a+b \sin (e+f x)}dx}{b^2 \left (a^2-b^2\right )}-\frac {3 \left (a^2 d+a b c-2 b^2 d\right ) (b c-a d)^3 \cos (e+f x)}{b^2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {-\frac {\int \frac {-b \left (a^2-b^2\right ) \left (-\left (\left (c^2-2 d^2\right ) b^2\right )+2 a c d b-3 a^2 d^2\right ) \sin (e+f x)^2 d^2+\left (a^2-b^2\right ) \left (-3 d^2 a^3+6 b c d a^2+b^2 \left (c^2+4 d^2\right ) a-8 b^3 c d\right ) \sin (e+f x) d^2+b \left (c^2 \left (c^2+12 d^2\right ) b^4-4 a c d \left (3 c^2+4 d^2\right ) b^3+2 a^2 \left (c^4+3 d^2 c^2+3 d^4\right ) b^2+4 a^3 c d^3 b-3 a^4 d^4\right )}{a+b \sin (e+f x)}dx}{b^2 \left (a^2-b^2\right )}-\frac {3 \left (a^2 d+a b c-2 b^2 d\right ) (b c-a d)^3 \cos (e+f x)}{b^2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3502

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3214

\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {-\frac {\frac {(b c-a d)^2 \left (6 a^4 d^2+4 a^3 b c d+a^2 b^2 \left (2 c^2-15 d^2\right )-10 a b^3 c d+b^4 \left (c^2+12 d^2\right )\right ) \int \frac {1}{a+b \sin (e+f x)}dx+2 d^3 x \left (a^2-b^2\right )^2 (4 b c-3 a d)}{b}+\frac {d^2 \left (a^2-b^2\right ) \left (-3 a^2 d^2+2 a b c d-\left (b^2 \left (c^2-2 d^2\right )\right )\right ) \cos (e+f x)}{f}}{b^2 \left (a^2-b^2\right )}-\frac {3 \left (a^2 d+a b c-2 b^2 d\right ) (b c-a d)^3 \cos (e+f x)}{b^2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {-\frac {\frac {(b c-a d)^2 \left (6 a^4 d^2+4 a^3 b c d+a^2 b^2 \left (2 c^2-15 d^2\right )-10 a b^3 c d+b^4 \left (c^2+12 d^2\right )\right ) \int \frac {1}{a+b \sin (e+f x)}dx+2 d^3 x \left (a^2-b^2\right )^2 (4 b c-3 a d)}{b}+\frac {d^2 \left (a^2-b^2\right ) \left (-3 a^2 d^2+2 a b c d-\left (b^2 \left (c^2-2 d^2\right )\right )\right ) \cos (e+f x)}{f}}{b^2 \left (a^2-b^2\right )}-\frac {3 \left (a^2 d+a b c-2 b^2 d\right ) (b c-a d)^3 \cos (e+f x)}{b^2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3139

\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {-\frac {\frac {\frac {2 (b c-a d)^2 \left (6 a^4 d^2+4 a^3 b c d+a^2 b^2 \left (2 c^2-15 d^2\right )-10 a b^3 c d+b^4 \left (c^2+12 d^2\right )\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (e+f x)\right )+2 b \tan \left (\frac {1}{2} (e+f x)\right )+a}d\tan \left (\frac {1}{2} (e+f x)\right )}{f}+2 d^3 x \left (a^2-b^2\right )^2 (4 b c-3 a d)}{b}+\frac {d^2 \left (a^2-b^2\right ) \left (-3 a^2 d^2+2 a b c d-\left (b^2 \left (c^2-2 d^2\right )\right )\right ) \cos (e+f x)}{f}}{b^2 \left (a^2-b^2\right )}-\frac {3 \left (a^2 d+a b c-2 b^2 d\right ) (b c-a d)^3 \cos (e+f x)}{b^2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {-\frac {\frac {2 d^3 x \left (a^2-b^2\right )^2 (4 b c-3 a d)-\frac {4 (b c-a d)^2 \left (6 a^4 d^2+4 a^3 b c d+a^2 b^2 \left (2 c^2-15 d^2\right )-10 a b^3 c d+b^4 \left (c^2+12 d^2\right )\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (e+f x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (e+f x)\right )\right )}{f}}{b}+\frac {d^2 \left (a^2-b^2\right ) \left (-3 a^2 d^2+2 a b c d-\left (b^2 \left (c^2-2 d^2\right )\right )\right ) \cos (e+f x)}{f}}{b^2 \left (a^2-b^2\right )}-\frac {3 \left (a^2 d+a b c-2 b^2 d\right ) (b c-a d)^3 \cos (e+f x)}{b^2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {-\frac {3 \left (a^2 d+a b c-2 b^2 d\right ) (b c-a d)^3 \cos (e+f x)}{b^2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {d^2 \left (a^2-b^2\right ) \left (-3 a^2 d^2+2 a b c d-\left (b^2 \left (c^2-2 d^2\right )\right )\right ) \cos (e+f x)}{f}+\frac {2 d^3 x \left (a^2-b^2\right )^2 (4 b c-3 a d)+\frac {2 (b c-a d)^2 \left (6 a^4 d^2+4 a^3 b c d+a^2 b^2 \left (2 c^2-15 d^2\right )-10 a b^3 c d+b^4 \left (c^2+12 d^2\right )\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (e+f x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{f \sqrt {a^2-b^2}}}{b}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}\)

Input:

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

Output:

((b*c - a*d)^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(2*b*(a^2 - b^2)*f*(a 
+ b*Sin[e + f*x])^2) - (-(((2*(a^2 - b^2)^2*d^3*(4*b*c - 3*a*d)*x + (2*(b* 
c - a*d)^2*(4*a^3*b*c*d - 10*a*b^3*c*d + 6*a^4*d^2 + a^2*b^2*(2*c^2 - 15*d 
^2) + b^4*(c^2 + 12*d^2))*ArcTan[(2*b + 2*a*Tan[(e + f*x)/2])/(2*Sqrt[a^2 
- b^2])])/(Sqrt[a^2 - b^2]*f))/b + ((a^2 - b^2)*d^2*(2*a*b*c*d - 3*a^2*d^2 
 - b^2*(c^2 - 2*d^2))*Cos[e + f*x])/f)/(b^2*(a^2 - b^2))) - (3*(b*c - a*d) 
^3*(a*b*c + a^2*d - 2*b^2*d)*Cos[e + f*x])/(b^2*(a^2 - b^2)*f*(a + b*Sin[e 
 + f*x])))/(2*b*(a^2 - b^2))
 

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 3214

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

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 3502

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

rule 3510

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f 
_.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)*Cos[ 
e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - S 
imp[1/(b^2*(m + 1)*(a^2 - b^2))   Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*( 
m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b^2*d*(m 
 + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)) 
))*Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; F 
reeQ[{a, b, c, d, e, f, A, B, C}, 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. \(853\) vs. \(2(307)=614\).

Time = 22.52 (sec) , antiderivative size = 854, normalized size of antiderivative = 2.69

method	result	size
derivativedivides	\(\frac {\frac {\frac {2 \left (-\frac {b^{2} \left (3 a^{6} d^{4}-4 a^{5} b c \,d^{3}-6 b^{2} a^{4} c^{2} d^{2}-6 a^{4} b^{2} d^{4}+12 a^{3} b^{3} c^{3} d +16 a^{3} b^{3} c \,d^{3}-5 b^{4} a^{2} c^{4}-12 a^{2} b^{4} c^{2} d^{2}+2 b^{6} c^{4}\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 (4 a^{8} d^{4}-8 a^{7} b c \,d^{3}+a^{6} b^{2} d^{4}+8 a^{5} b^{3} c^{3} d +4 a^{5} b^{3} c \,d^{3}-4 a^{4} b^{4} c^{4}-18 a^{4} b^{4} c^{2} d^{2}-14 a^{4} b^{4} d^{4}+20 a^{3} b^{5} c^{3} d +40 a^{3} b^{5} c \,d^{3}-7 a^{2} b^{6} c^{4}-36 a^{2} b^{6} c^{2} d^{2}+8 a \,b^{7} c^{3} d +2 b^{8} c^{4}\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 (13 a^{6} d^{4}-28 a^{5} b c \,d^{3}+6 b^{2} a^{4} c^{2} d^{2}-22 a^{4} b^{2} d^{4}+20 a^{3} b^{3} c^{3} d +64 a^{3} b^{3} c \,d^{3}-11 b^{4} a^{2} c^{4}-60 a^{2} b^{4} c^{2} d^{2}+16 b^{5} a \,c^{3} d +2 b^{6} c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a}-\frac {b \left (4 a^{6} d^{4}-8 a^{5} b c \,d^{3}-7 a^{4} b^{2} d^{4}+8 a^{3} b^{3} c^{3} d +20 a^{3} b^{3} c \,d^{3}-4 b^{4} a^{2} c^{4}-18 a^{2} b^{4} c^{2} d^{2}+4 b^{5} a \,c^{3} d +b^{6} c^{4}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}\right )}{\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^{6} d^{4}-8 a^{5} b c \,d^{3}-15 a^{4} b^{2} d^{4}+20 a^{3} b^{3} c \,d^{3}+2 b^{4} a^{2} c^{4}+6 a^{2} b^{4} c^{2} d^{2}+12 a^{2} b^{4} d^{4}-12 b^{5} a \,c^{3} d -24 a \,b^{5} c \,d^{3}+b^{6} c^{4}+12 b^{6} c^{2} 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 )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}-b^{2}}}}{b^{4}}-\frac {2 d^{3} \left (\frac {b d}{1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}+\left (3 a d -4 b c \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{b^{4}}}{f}\)	\(854\)
default	\(\frac {\frac {\frac {2 \left (-\frac {b^{2} \left (3 a^{6} d^{4}-4 a^{5} b c \,d^{3}-6 b^{2} a^{4} c^{2} d^{2}-6 a^{4} b^{2} d^{4}+12 a^{3} b^{3} c^{3} d +16 a^{3} b^{3} c \,d^{3}-5 b^{4} a^{2} c^{4}-12 a^{2} b^{4} c^{2} d^{2}+2 b^{6} c^{4}\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 (4 a^{8} d^{4}-8 a^{7} b c \,d^{3}+a^{6} b^{2} d^{4}+8 a^{5} b^{3} c^{3} d +4 a^{5} b^{3} c \,d^{3}-4 a^{4} b^{4} c^{4}-18 a^{4} b^{4} c^{2} d^{2}-14 a^{4} b^{4} d^{4}+20 a^{3} b^{5} c^{3} d +40 a^{3} b^{5} c \,d^{3}-7 a^{2} b^{6} c^{4}-36 a^{2} b^{6} c^{2} d^{2}+8 a \,b^{7} c^{3} d +2 b^{8} c^{4}\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 (13 a^{6} d^{4}-28 a^{5} b c \,d^{3}+6 b^{2} a^{4} c^{2} d^{2}-22 a^{4} b^{2} d^{4}+20 a^{3} b^{3} c^{3} d +64 a^{3} b^{3} c \,d^{3}-11 b^{4} a^{2} c^{4}-60 a^{2} b^{4} c^{2} d^{2}+16 b^{5} a \,c^{3} d +2 b^{6} c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a}-\frac {b \left (4 a^{6} d^{4}-8 a^{5} b c \,d^{3}-7 a^{4} b^{2} d^{4}+8 a^{3} b^{3} c^{3} d +20 a^{3} b^{3} c \,d^{3}-4 b^{4} a^{2} c^{4}-18 a^{2} b^{4} c^{2} d^{2}+4 b^{5} a \,c^{3} d +b^{6} c^{4}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}\right )}{\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^{6} d^{4}-8 a^{5} b c \,d^{3}-15 a^{4} b^{2} d^{4}+20 a^{3} b^{3} c \,d^{3}+2 b^{4} a^{2} c^{4}+6 a^{2} b^{4} c^{2} d^{2}+12 a^{2} b^{4} d^{4}-12 b^{5} a \,c^{3} d -24 a \,b^{5} c \,d^{3}+b^{6} c^{4}+12 b^{6} c^{2} 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 )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}-b^{2}}}}{b^{4}}-\frac {2 d^{3} \left (\frac {b d}{1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}+\left (3 a d -4 b c \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{b^{4}}}{f}\)	\(854\)
risch	\(\text {Expression too large to display}\)	\(2791\)

Input:

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

Output:

1/f*(2/b^4*((-1/2*b^2*(3*a^6*d^4-4*a^5*b*c*d^3-6*a^4*b^2*c^2*d^2-6*a^4*b^2 
*d^4+12*a^3*b^3*c^3*d+16*a^3*b^3*c*d^3-5*a^2*b^4*c^4-12*a^2*b^4*c^2*d^2+2* 
b^6*c^4)/(a^4-2*a^2*b^2+b^4)/a*tan(1/2*f*x+1/2*e)^3-1/2*b*(4*a^8*d^4-8*a^7 
*b*c*d^3+a^6*b^2*d^4+8*a^5*b^3*c^3*d+4*a^5*b^3*c*d^3-4*a^4*b^4*c^4-18*a^4* 
b^4*c^2*d^2-14*a^4*b^4*d^4+20*a^3*b^5*c^3*d+40*a^3*b^5*c*d^3-7*a^2*b^6*c^4 
-36*a^2*b^6*c^2*d^2+8*a*b^7*c^3*d+2*b^8*c^4)/(a^4-2*a^2*b^2+b^4)/a^2*tan(1 
/2*f*x+1/2*e)^2-1/2*b^2*(13*a^6*d^4-28*a^5*b*c*d^3+6*a^4*b^2*c^2*d^2-22*a^ 
4*b^2*d^4+20*a^3*b^3*c^3*d+64*a^3*b^3*c*d^3-11*a^2*b^4*c^4-60*a^2*b^4*c^2* 
d^2+16*a*b^5*c^3*d+2*b^6*c^4)/(a^4-2*a^2*b^2+b^4)/a*tan(1/2*f*x+1/2*e)-1/2 
*b*(4*a^6*d^4-8*a^5*b*c*d^3-7*a^4*b^2*d^4+8*a^3*b^3*c^3*d+20*a^3*b^3*c*d^3 
-4*a^2*b^4*c^4-18*a^2*b^4*c^2*d^2+4*a*b^5*c^3*d+b^6*c^4)/(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^6*d^4-8*a 
^5*b*c*d^3-15*a^4*b^2*d^4+20*a^3*b^3*c*d^3+2*a^2*b^4*c^4+6*a^2*b^4*c^2*d^2 
+12*a^2*b^4*d^4-12*a*b^5*c^3*d-24*a*b^5*c*d^3+b^6*c^4+12*b^6*c^2*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)))-2*d^3/b^4*(b*d/(1+tan(1/2*f*x+1/2*e)^2)+(3*a*d-4*b*c)*arct 
an(tan(1/2*f*x+1/2*e))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1125 vs. \(2 (307) = 614\).

Time = 0.20 (sec) , antiderivative size = 2335, normalized size of antiderivative = 7.34 \[ \int \frac {(c+d \sin (e+f x))^4}{(a+b \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/4*(4*(a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*d^4*cos(f*x + e)^3 - 4*(4 
*(a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*c*d^3 - 3*(a^7*b^2 - 3*a^5*b^4 + 
3*a^3*b^6 - a*b^8)*d^4)*f*x*cos(f*x + e)^2 + 4*(4*(a^8*b - 2*a^6*b^3 + 2*a 
^2*b^7 - b^9)*c*d^3 - 3*(a^9 - 2*a^7*b^2 + 2*a^3*b^6 - a*b^8)*d^4)*f*x - ( 
(2*a^4*b^4 + 3*a^2*b^6 + b^8)*c^4 - 12*(a^3*b^5 + a*b^7)*c^3*d + 6*(a^4*b^ 
4 + 3*a^2*b^6 + 2*b^8)*c^2*d^2 - 4*(2*a^7*b - 3*a^5*b^3 + a^3*b^5 + 6*a*b^ 
7)*c*d^3 + 3*(2*a^8 - 3*a^6*b^2 - a^4*b^4 + 4*a^2*b^6)*d^4 + (12*a*b^7*c^3 
*d - (2*a^2*b^6 + b^8)*c^4 - 6*(a^2*b^6 + 2*b^8)*c^2*d^2 + 4*(2*a^5*b^3 - 
5*a^3*b^5 + 6*a*b^7)*c*d^3 - 3*(2*a^6*b^2 - 5*a^4*b^4 + 4*a^2*b^6)*d^4)*co 
s(f*x + e)^2 - 2*(12*a^2*b^6*c^3*d - (2*a^3*b^5 + a*b^7)*c^4 - 6*(a^3*b^5 
+ 2*a*b^7)*c^2*d^2 + 4*(2*a^6*b^2 - 5*a^4*b^4 + 6*a^2*b^6)*c*d^3 - 3*(2*a^ 
7*b - 5*a^5*b^3 + 4*a^3*b^5)*d^4)*sin(f*x + e))*sqrt(-a^2 + b^2)*log(((2*a 
^2 - b^2)*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2 + 2*(a*cos(f*x + 
 e)*sin(f*x + e) + b*cos(f*x + e))*sqrt(-a^2 + b^2))/(b^2*cos(f*x + e)^2 - 
 2*a*b*sin(f*x + e) - a^2 - b^2)) + 2*((4*a^4*b^5 - 5*a^2*b^7 + b^9)*c^4 - 
 4*(2*a^5*b^4 - a^3*b^6 - a*b^8)*c^3*d + 18*(a^4*b^5 - a^2*b^7)*c^2*d^2 + 
4*(2*a^7*b^2 - 7*a^5*b^4 + 5*a^3*b^6)*c*d^3 - (6*a^8*b - 15*a^6*b^3 + 7*a^ 
4*b^5 + 4*a^2*b^7 - 2*b^9)*d^4)*cos(f*x + e) + 2*(4*(4*(a^7*b^2 - 3*a^5*b^ 
4 + 3*a^3*b^6 - a*b^8)*c*d^3 - 3*(a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7) 
*d^4)*f*x + (3*(a^3*b^6 - a*b^8)*c^4 - 4*(a^4*b^5 + a^2*b^7 - 2*b^9)*c^...
 

Sympy [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F(-2)]

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

integrate((c+d*sin(f*x+e))^4/(a+b*sin(f*x+e))^3,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*b^2-4*a^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. 1120 vs. \(2 (307) = 614\).

Time = 0.41 (sec) , antiderivative size = 1120, normalized size of antiderivative = 3.52 \[ \int \frac {(c+d \sin (e+f x))^4}{(a+b \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:

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

Output:

((2*a^2*b^4*c^4 + b^6*c^4 - 12*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 + 12*b^6*c^ 
2*d^2 - 8*a^5*b*c*d^3 + 20*a^3*b^3*c*d^3 - 24*a*b^5*c*d^3 + 6*a^6*d^4 - 15 
*a^4*b^2*d^4 + 12*a^2*b^4*d^4)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(a) + 
arctan((a*tan(1/2*f*x + 1/2*e) + b)/sqrt(a^2 - b^2)))/((a^4*b^4 - 2*a^2*b^ 
6 + b^8)*sqrt(a^2 - b^2)) - 2*d^4/((tan(1/2*f*x + 1/2*e)^2 + 1)*b^3) + (5* 
a^3*b^5*c^4*tan(1/2*f*x + 1/2*e)^3 - 2*a*b^7*c^4*tan(1/2*f*x + 1/2*e)^3 - 
12*a^4*b^4*c^3*d*tan(1/2*f*x + 1/2*e)^3 + 6*a^5*b^3*c^2*d^2*tan(1/2*f*x + 
1/2*e)^3 + 12*a^3*b^5*c^2*d^2*tan(1/2*f*x + 1/2*e)^3 + 4*a^6*b^2*c*d^3*tan 
(1/2*f*x + 1/2*e)^3 - 16*a^4*b^4*c*d^3*tan(1/2*f*x + 1/2*e)^3 - 3*a^7*b*d^ 
4*tan(1/2*f*x + 1/2*e)^3 + 6*a^5*b^3*d^4*tan(1/2*f*x + 1/2*e)^3 + 4*a^4*b^ 
4*c^4*tan(1/2*f*x + 1/2*e)^2 + 7*a^2*b^6*c^4*tan(1/2*f*x + 1/2*e)^2 - 2*b^ 
8*c^4*tan(1/2*f*x + 1/2*e)^2 - 8*a^5*b^3*c^3*d*tan(1/2*f*x + 1/2*e)^2 - 20 
*a^3*b^5*c^3*d*tan(1/2*f*x + 1/2*e)^2 - 8*a*b^7*c^3*d*tan(1/2*f*x + 1/2*e) 
^2 + 18*a^4*b^4*c^2*d^2*tan(1/2*f*x + 1/2*e)^2 + 36*a^2*b^6*c^2*d^2*tan(1/ 
2*f*x + 1/2*e)^2 + 8*a^7*b*c*d^3*tan(1/2*f*x + 1/2*e)^2 - 4*a^5*b^3*c*d^3* 
tan(1/2*f*x + 1/2*e)^2 - 40*a^3*b^5*c*d^3*tan(1/2*f*x + 1/2*e)^2 - 4*a^8*d 
^4*tan(1/2*f*x + 1/2*e)^2 - a^6*b^2*d^4*tan(1/2*f*x + 1/2*e)^2 + 14*a^4*b^ 
4*d^4*tan(1/2*f*x + 1/2*e)^2 + 11*a^3*b^5*c^4*tan(1/2*f*x + 1/2*e) - 2*a*b 
^7*c^4*tan(1/2*f*x + 1/2*e) - 20*a^4*b^4*c^3*d*tan(1/2*f*x + 1/2*e) - 16*a 
^2*b^6*c^3*d*tan(1/2*f*x + 1/2*e) - 6*a^5*b^3*c^2*d^2*tan(1/2*f*x + 1/2...
 

Mupad [B] (verification not implemented)

Time = 27.75 (sec) , antiderivative size = 16958, normalized size of antiderivative = 53.33 \[ \int \frac {(c+d \sin (e+f x))^4}{(a+b \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:

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

Output:

(2*d^3*atan(((d^3*(3*a*d - 4*b*c)*((8*tan(e/2 + (f*x)/2)*(a*b^15*c^8 + 4*a 
^3*b^13*c^8 + 4*a^5*b^11*c^8 - 72*a^3*b^13*d^8 + 468*a^5*b^11*d^8 - 936*a^ 
7*b^9*d^8 + 873*a^9*b^7*d^8 - 396*a^11*b^5*d^8 + 72*a^13*b^3*d^8 - 128*a*b 
^15*c^2*d^6 + 144*a*b^15*c^4*d^4 + 24*a*b^15*c^6*d^2 + 192*a^2*b^14*c*d^7 
- 24*a^2*b^14*c^7*d - 1440*a^4*b^12*c*d^7 - 48*a^4*b^12*c^7*d + 2736*a^6*b 
^10*c*d^7 - 2424*a^8*b^8*c*d^7 + 1056*a^10*b^6*c*d^7 - 192*a^12*b^4*c*d^7 
- 576*a^2*b^14*c^3*d^5 - 336*a^2*b^14*c^5*d^3 + 1440*a^3*b^13*c^2*d^6 + 74 
4*a^3*b^13*c^4*d^4 + 204*a^3*b^13*c^6*d^2 - 96*a^4*b^12*c^3*d^5 - 200*a^4* 
b^12*c^5*d^3 - 2200*a^5*b^11*c^2*d^6 - 426*a^5*b^11*c^4*d^4 + 24*a^5*b^11* 
c^6*d^2 + 408*a^6*b^10*c^3*d^5 + 64*a^6*b^10*c^5*d^3 + 1644*a^7*b^9*c^2*d^ 
6 + 144*a^7*b^9*c^4*d^4 - 240*a^8*b^8*c^3*d^5 - 32*a^8*b^8*c^5*d^3 - 632*a 
^9*b^7*c^2*d^6 + 24*a^9*b^7*c^4*d^4 + 128*a^11*b^5*c^2*d^6))/(b^17 - 4*a^2 
*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9) - (8*(36*a^4*b^11*d^8 - 144*a^6 
*b^9*d^8 + 216*a^8*b^7*d^8 - 144*a^10*b^5*d^8 + 36*a^12*b^3*d^8 - 96*a^3*b 
^12*c*d^7 + 384*a^5*b^10*c*d^7 - 576*a^7*b^8*c*d^7 + 384*a^9*b^6*c*d^7 - 9 
6*a^11*b^4*c*d^7 + 64*a^2*b^13*c^2*d^6 - 256*a^4*b^11*c^2*d^6 + 384*a^6*b^ 
9*c^2*d^6 - 256*a^8*b^7*c^2*d^6 + 64*a^10*b^5*c^2*d^6))/(b^16 - 4*a^2*b^14 
 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + (d^3*(3*a*d - 4*b*c)*((8*(2*a^2*b^ 
16*c^4 - 6*a^6*b^12*c^4 + 4*a^8*b^10*c^4 + 12*a^2*b^16*d^4 - 36*a^4*b^14*d 
^4 + 42*a^6*b^12*d^4 - 24*a^8*b^10*d^4 + 6*a^10*b^8*d^4 + 32*a^3*b^15*c...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 3794, normalized size of antiderivative = 11.93 \[ \int \frac {(c+d \sin (e+f x))^4}{(a+b \sin (e+f x))^3} \, dx =\text {Too large to display} \] Input:

int((c+d*sin(f*x+e))^4/(a+b*sin(f*x+e))^3,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**7*b**2*d**4 - 32*sqrt(a**2 - b**2)*atan((tan((e + f*x)/2)* 
a + b)/sqrt(a**2 - b**2))*sin(e + f*x)**2*a**6*b**3*c*d**3 - 60*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**4*d**4 + 80*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**5*c*d**3 + 8*sqrt(a**2 - b**2)*atan((t 
an((e + f*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(e + f*x)**2*a**3*b**6*c**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**3*b**6*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)**2*a**3*b**6*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)** 
2*a**2*b**7*c**3*d - 96*sqrt(a**2 - b**2)*atan((tan((e + f*x)/2)*a + b)/sq 
rt(a**2 - b**2))*sin(e + f*x)**2*a**2*b**7*c*d**3 + 4*sqrt(a**2 - b**2)*at 
an((tan((e + f*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(e + f*x)**2*a*b**8*c**4 
 + 48*sqrt(a**2 - b**2)*atan((tan((e + f*x)/2)*a + b)/sqrt(a**2 - b**2))*s 
in(e + f*x)**2*a*b**8*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**8*b*d**4 - 64*sqrt(a**2 - b* 
*2)*atan((tan((e + f*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(e + f*x)*a**7*b** 
2*c*d**3 - 120*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**3*d**4 + 160*sqrt(a**2 - b**2)*atan((tan((...