Integrand size = 25, antiderivative size = 534 \[ \int \frac {(c+d \sin (e+f x))^5}{(a+b \sin (e+f x))^3} \, dx=-\frac {d^3 \left (30 a b c d-12 a^2 d^2-b^2 \left (20 c^2+d^2\right )\right ) x}{2 b^5}+\frac {(b c-a d)^3 \left (6 a^3 b c d-12 a b^3 c d+12 a^4 d^2+a^2 b^2 \left (2 c^2-29 d^2\right )+b^4 \left (c^2+20 d^2\right )\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{b^5 \left (a^2-b^2\right )^{5/2} f}-\frac {d \left (30 a^4 b c d^3-12 a^5 d^4-a^3 b^2 d^2 \left (16 c^2-21 d^2\right )-b^5 c d \left (17 c^2-10 d^2\right )-a^2 b^3 c d \left (4 c^2+55 d^2\right )+a b^4 \left (6 c^4+43 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{2 b^4 \left (a^2-b^2\right )^2 f}+\frac {d^2 \left (7 a^3 b c d^2-6 a^4 d^3+b^4 d \left (8 c^2-d^2\right )+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^3 \left (a^2-b^2\right )^2 f}+\frac {(b c-a d)^2 \left (3 a b c+4 a^2 d-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2} \] Output:
-1/2*d^3*(30*a*b*c*d-12*a^2*d^2-b^2*(20*c^2+d^2))*x/b^5+(-a*d+b*c)^3*(6*a^ 3*b*c*d-12*a*b^3*c*d+12*a^4*d^2+a^2*b^2*(2*c^2-29*d^2)+b^4*(c^2+20*d^2))*a rctan((b+a*tan(1/2*f*x+1/2*e))/(a^2-b^2)^(1/2))/b^5/(a^2-b^2)^(5/2)/f-1/2* d*(30*a^4*b*c*d^3-12*a^5*d^4-a^3*b^2*d^2*(16*c^2-21*d^2)-b^5*c*d*(17*c^2-1 0*d^2)-a^2*b^3*c*d*(4*c^2+55*d^2)+a*b^4*(6*c^4+43*c^2*d^2-6*d^4))*cos(f*x+ e)/b^4/(a^2-b^2)^2/f+1/2*d^2*(7*a^3*b*c*d^2-6*a^4*d^3+b^4*d*(8*c^2-d^2)+a^ 2*b^2*d*(c^2+10*d^2)-a*b^3*c*(3*c^2+16*d^2))*cos(f*x+e)*sin(f*x+e)/b^3/(a^ 2-b^2)^2/f+1/2*(-a*d+b*c)^2*(4*a^2*d+3*a*b*c-7*b^2*d)*cos(f*x+e)*(c+d*sin( f*x+e))^2/b^2/(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))^3/b/(a^2-b^2)/f/(a+b*sin(f*x+e))^2
Result contains complex when optimal does not.
Time = 12.55 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.64 \[ \int \frac {(c+d \sin (e+f x))^5}{(a+b \sin (e+f x))^3} \, dx=\frac {2 d^3 \left (-30 a b c d+12 a^2 d^2+b^2 \left (20 c^2+d^2\right )\right ) (e+f x)+\frac {4 (b c-a d)^3 \left (6 a^3 b c d-12 a b^3 c d+12 a^4 d^2+a^2 b^2 \left (2 c^2-29 d^2\right )+b^4 \left (c^2+20 d^2\right )\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+2 b d^4 (-5 b c+3 a d) (\cos (e+f x)-i \sin (e+f x))+2 b d^4 (-5 b c+3 a d) (\cos (e+f x)+i \sin (e+f x))-\frac {2 b (b c-a d)^5 \cos (e+f x)}{\left (-a^2+b^2\right ) (a+b \sin (e+f x))^2}+\frac {2 b (b c-a d)^4 \left (3 a b c+7 a^2 d-10 b^2 d\right ) \cos (e+f x)}{\left (a^2-b^2\right )^2 (a+b \sin (e+f x))}-b^2 d^5 \sin (2 (e+f x))}{4 b^5 f} \] Input:
Integrate[(c + d*Sin[e + f*x])^5/(a + b*Sin[e + f*x])^3,x]
Output:
(2*d^3*(-30*a*b*c*d + 12*a^2*d^2 + b^2*(20*c^2 + d^2))*(e + f*x) + (4*(b*c - a*d)^3*(6*a^3*b*c*d - 12*a*b^3*c*d + 12*a^4*d^2 + a^2*b^2*(2*c^2 - 29*d ^2) + b^4*(c^2 + 20*d^2))*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]] )/(a^2 - b^2)^(5/2) + 2*b*d^4*(-5*b*c + 3*a*d)*(Cos[e + f*x] - I*Sin[e + f *x]) + 2*b*d^4*(-5*b*c + 3*a*d)*(Cos[e + f*x] + I*Sin[e + f*x]) - (2*b*(b* c - a*d)^5*Cos[e + f*x])/((-a^2 + b^2)*(a + b*Sin[e + f*x])^2) + (2*b*(b*c - a*d)^4*(3*a*b*c + 7*a^2*d - 10*b^2*d)*Cos[e + f*x])/((a^2 - b^2)^2*(a + b*Sin[e + f*x])) - b^2*d^5*Sin[2*(e + f*x)])/(4*b^5*f)
Time = 3.54 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.06, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 3271, 3042, 3526, 3042, 3512, 27, 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.
| \(\displaystyle \int \frac {(c+d \sin (e+f x))^5}{(a+b \sin (e+f x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d \sin (e+f x))^5}{(a+b \sin (e+f x))^3}dx\) |
| \(\Big \downarrow \) 3271 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\int \frac {(c+d \sin (e+f x))^2 \left (3 a^2 d^3+7 b^2 c^2 d+2 \left (-\left (\left (c^2-d^2\right ) b^2\right )+2 a c d b-2 a^2 d^2\right ) \sin ^2(e+f x) d-2 a b c \left (c^2+4 d^2\right )-\left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)\right )}{(a+b \sin (e+f x))^2}dx}{2 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\int \frac {(c+d \sin (e+f x))^2 \left (3 a^2 d^3+7 b^2 c^2 d+2 \left (-\left (\left (c^2-d^2\right ) b^2\right )+2 a c d b-2 a^2 d^2\right ) \sin (e+f x)^2 d-2 a b c \left (c^2+4 d^2\right )-\left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)\right )}{(a+b \sin (e+f x))^2}dx}{2 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {-\frac {\int \frac {(c+d \sin (e+f x)) \left (c^2 \left (c^2+20 d^2\right ) b^4-a c d \left (15 c^2+32 d^2\right ) b^3+a^2 \left (2 c^4+7 d^2 c^2+14 d^4\right ) b^2+11 a^3 c d^3 b-8 a^4 d^4-2 d \left (-6 d^3 a^4+7 b c d^2 a^3+b^2 d \left (c^2+10 d^2\right ) a^2-b^3 c \left (3 c^2+16 d^2\right ) a+b^4 d \left (8 c^2-d^2\right )\right ) \sin ^2(e+f x)+d \left (4 c d^2 a^4-b d \left (4 c^2-d^2\right ) a^3+b^2 c \left (4 c^2-3 d^2\right ) a^2-b^3 d \left (5 c^2+4 d^2\right ) a-b^4 c \left (c^2-8 d^2\right )\right ) \sin (e+f x)\right )}{a+b \sin (e+f x)}dx}{b \left (a^2-b^2\right )}-\frac {(b c-a d)^2 \left (4 a^2 d+3 a b c-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{2 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {-\frac {\int \frac {(c+d \sin (e+f x)) \left (c^2 \left (c^2+20 d^2\right ) b^4-a c d \left (15 c^2+32 d^2\right ) b^3+a^2 \left (2 c^4+7 d^2 c^2+14 d^4\right ) b^2+11 a^3 c d^3 b-8 a^4 d^4-2 d \left (-6 d^3 a^4+7 b c d^2 a^3+b^2 d \left (c^2+10 d^2\right ) a^2-b^3 c \left (3 c^2+16 d^2\right ) a+b^4 d \left (8 c^2-d^2\right )\right ) \sin (e+f x)^2+d \left (4 c d^2 a^4-b d \left (4 c^2-d^2\right ) a^3+b^2 c \left (4 c^2-3 d^2\right ) a^2-b^3 d \left (5 c^2+4 d^2\right ) a-b^4 c \left (c^2-8 d^2\right )\right ) \sin (e+f x)\right )}{a+b \sin (e+f x)}dx}{b \left (a^2-b^2\right )}-\frac {(b c-a d)^2 \left (4 a^2 d+3 a b c-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{2 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {-\frac {\frac {\int -\frac {2 \left (-c^3 \left (c^2+20 d^2\right ) b^5+a d \left (15 c^4+40 d^2 c^2-d^4\right ) b^4-2 a^2 c \left (c^4+5 d^2 c^2+15 d^4\right ) b^3-10 a^3 d^3 \left (c^2-d^2\right ) b^2+15 a^4 c d^4 b-d \left (\left (4 c^2 d^2-2 d^4\right ) a^4-b \left (4 c^3 d-5 c d^3\right ) a^3+b^2 \left (6 c^4+3 d^2 c^2+4 d^4\right ) a^2-b^3 c d \left (17 c^2+20 d^2\right ) a+b^4 d^2 \left (20 c^2+d^2\right )\right ) \sin (e+f x) b-6 a^5 d^5-d \left (-12 d^4 a^5+30 b c d^3 a^4-b^2 d^2 \left (16 c^2-21 d^2\right ) a^3-b^3 c d \left (4 c^2+55 d^2\right ) a^2+b^4 \left (6 c^4+43 d^2 c^2-6 d^4\right ) a-b^5 c d \left (17 c^2-10 d^2\right )\right ) \sin ^2(e+f x)\right )}{a+b \sin (e+f x)}dx}{2 b}+\frac {d^2 \left (-6 a^4 d^3+7 a^3 b c d^2+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )+b^4 d \left (8 c^2-d^2\right )\right ) \sin (e+f x) \cos (e+f x)}{b f}}{b \left (a^2-b^2\right )}-\frac {(b c-a d)^2 \left (4 a^2 d+3 a b c-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{2 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {-\frac {\frac {d^2 \left (-6 a^4 d^3+7 a^3 b c d^2+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )+b^4 d \left (8 c^2-d^2\right )\right ) \sin (e+f x) \cos (e+f x)}{b f}-\frac {\int \frac {-c^3 \left (c^2+20 d^2\right ) b^5+a d \left (15 c^4+40 d^2 c^2-d^4\right ) b^4-2 a^2 c \left (c^4+5 d^2 c^2+15 d^4\right ) b^3-10 a^3 d^3 \left (c^2-d^2\right ) b^2+15 a^4 c d^4 b-d \left (\left (4 c^2 d^2-2 d^4\right ) a^4-b \left (4 c^3 d-5 c d^3\right ) a^3+b^2 \left (6 c^4+3 d^2 c^2+4 d^4\right ) a^2-b^3 c d \left (17 c^2+20 d^2\right ) a+b^4 d^2 \left (20 c^2+d^2\right )\right ) \sin (e+f x) b-6 a^5 d^5-d \left (-12 d^4 a^5+30 b c d^3 a^4-b^2 d^2 \left (16 c^2-21 d^2\right ) a^3-b^3 c d \left (4 c^2+55 d^2\right ) a^2+b^4 \left (6 c^4+43 d^2 c^2-6 d^4\right ) a-b^5 c d \left (17 c^2-10 d^2\right )\right ) \sin ^2(e+f x)}{a+b \sin (e+f x)}dx}{b}}{b \left (a^2-b^2\right )}-\frac {(b c-a d)^2 \left (4 a^2 d+3 a b c-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{2 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {-\frac {\frac {d^2 \left (-6 a^4 d^3+7 a^3 b c d^2+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )+b^4 d \left (8 c^2-d^2\right )\right ) \sin (e+f x) \cos (e+f x)}{b f}-\frac {\int \frac {-c^3 \left (c^2+20 d^2\right ) b^5+a d \left (15 c^4+40 d^2 c^2-d^4\right ) b^4-2 a^2 c \left (c^4+5 d^2 c^2+15 d^4\right ) b^3-10 a^3 d^3 \left (c^2-d^2\right ) b^2+15 a^4 c d^4 b-d \left (\left (4 c^2 d^2-2 d^4\right ) a^4-b \left (4 c^3 d-5 c d^3\right ) a^3+b^2 \left (6 c^4+3 d^2 c^2+4 d^4\right ) a^2-b^3 c d \left (17 c^2+20 d^2\right ) a+b^4 d^2 \left (20 c^2+d^2\right )\right ) \sin (e+f x) b-6 a^5 d^5-d \left (-12 d^4 a^5+30 b c d^3 a^4-b^2 d^2 \left (16 c^2-21 d^2\right ) a^3-b^3 c d \left (4 c^2+55 d^2\right ) a^2+b^4 \left (6 c^4+43 d^2 c^2-6 d^4\right ) a-b^5 c d \left (17 c^2-10 d^2\right )\right ) \sin (e+f x)^2}{a+b \sin (e+f x)}dx}{b}}{b \left (a^2-b^2\right )}-\frac {(b c-a d)^2 \left (4 a^2 d+3 a b c-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{2 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {-\frac {\frac {d^2 \left (-6 a^4 d^3+7 a^3 b c d^2+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )+b^4 d \left (8 c^2-d^2\right )\right ) \sin (e+f x) \cos (e+f x)}{b f}-\frac {\frac {\int \frac {\left (a^2-b^2\right )^2 \left (-\left (\left (20 c^2+d^2\right ) b^2\right )+30 a c d b-12 a^2 d^2\right ) \sin (e+f x) d^3+b \left (-c^3 \left (c^2+20 d^2\right ) b^5+a d \left (15 c^4+40 d^2 c^2-d^4\right ) b^4-2 a^2 c \left (c^4+5 d^2 c^2+15 d^4\right ) b^3-10 a^3 d^3 \left (c^2-d^2\right ) b^2+15 a^4 c d^4 b-6 a^5 d^5\right )}{a+b \sin (e+f x)}dx}{b}+\frac {d \left (-12 a^5 d^4+30 a^4 b c d^3-a^3 b^2 d^2 \left (16 c^2-21 d^2\right )-a^2 b^3 c d \left (4 c^2+55 d^2\right )+a b^4 \left (6 c^4+43 c^2 d^2-6 d^4\right )-b^5 c d \left (17 c^2-10 d^2\right )\right ) \cos (e+f x)}{b f}}{b}}{b \left (a^2-b^2\right )}-\frac {(b c-a d)^2 \left (4 a^2 d+3 a b c-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{2 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {-\frac {\frac {d^2 \left (-6 a^4 d^3+7 a^3 b c d^2+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )+b^4 d \left (8 c^2-d^2\right )\right ) \sin (e+f x) \cos (e+f x)}{b f}-\frac {\frac {\int \frac {\left (a^2-b^2\right )^2 \left (-\left (\left (20 c^2+d^2\right ) b^2\right )+30 a c d b-12 a^2 d^2\right ) \sin (e+f x) d^3+b \left (-c^3 \left (c^2+20 d^2\right ) b^5+a d \left (15 c^4+40 d^2 c^2-d^4\right ) b^4-2 a^2 c \left (c^4+5 d^2 c^2+15 d^4\right ) b^3-10 a^3 d^3 \left (c^2-d^2\right ) b^2+15 a^4 c d^4 b-6 a^5 d^5\right )}{a+b \sin (e+f x)}dx}{b}+\frac {d \left (-12 a^5 d^4+30 a^4 b c d^3-a^3 b^2 d^2 \left (16 c^2-21 d^2\right )-a^2 b^3 c d \left (4 c^2+55 d^2\right )+a b^4 \left (6 c^4+43 c^2 d^2-6 d^4\right )-b^5 c d \left (17 c^2-10 d^2\right )\right ) \cos (e+f x)}{b f}}{b}}{b \left (a^2-b^2\right )}-\frac {(b c-a d)^2 \left (4 a^2 d+3 a b c-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{2 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {-\frac {\frac {d^2 \left (-6 a^4 d^3+7 a^3 b c d^2+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )+b^4 d \left (8 c^2-d^2\right )\right ) \sin (e+f x) \cos (e+f x)}{b f}-\frac {\frac {\frac {d^3 x \left (a^2-b^2\right )^2 \left (-12 a^2 d^2+30 a b c d-\left (b^2 \left (20 c^2+d^2\right )\right )\right )}{b}-\frac {(b c-a d)^3 \left (12 a^4 d^2+6 a^3 b c d+a^2 b^2 \left (2 c^2-29 d^2\right )-12 a b^3 c d+b^4 \left (c^2+20 d^2\right )\right ) \int \frac {1}{a+b \sin (e+f x)}dx}{b}}{b}+\frac {d \left (-12 a^5 d^4+30 a^4 b c d^3-a^3 b^2 d^2 \left (16 c^2-21 d^2\right )-a^2 b^3 c d \left (4 c^2+55 d^2\right )+a b^4 \left (6 c^4+43 c^2 d^2-6 d^4\right )-b^5 c d \left (17 c^2-10 d^2\right )\right ) \cos (e+f x)}{b f}}{b}}{b \left (a^2-b^2\right )}-\frac {(b c-a d)^2 \left (4 a^2 d+3 a b c-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{2 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {-\frac {\frac {d^2 \left (-6 a^4 d^3+7 a^3 b c d^2+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )+b^4 d \left (8 c^2-d^2\right )\right ) \sin (e+f x) \cos (e+f x)}{b f}-\frac {\frac {\frac {d^3 x \left (a^2-b^2\right )^2 \left (-12 a^2 d^2+30 a b c d-\left (b^2 \left (20 c^2+d^2\right )\right )\right )}{b}-\frac {(b c-a d)^3 \left (12 a^4 d^2+6 a^3 b c d+a^2 b^2 \left (2 c^2-29 d^2\right )-12 a b^3 c d+b^4 \left (c^2+20 d^2\right )\right ) \int \frac {1}{a+b \sin (e+f x)}dx}{b}}{b}+\frac {d \left (-12 a^5 d^4+30 a^4 b c d^3-a^3 b^2 d^2 \left (16 c^2-21 d^2\right )-a^2 b^3 c d \left (4 c^2+55 d^2\right )+a b^4 \left (6 c^4+43 c^2 d^2-6 d^4\right )-b^5 c d \left (17 c^2-10 d^2\right )\right ) \cos (e+f x)}{b f}}{b}}{b \left (a^2-b^2\right )}-\frac {(b c-a d)^2 \left (4 a^2 d+3 a b c-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{2 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {-\frac {\frac {d^2 \left (-6 a^4 d^3+7 a^3 b c d^2+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )+b^4 d \left (8 c^2-d^2\right )\right ) \sin (e+f x) \cos (e+f x)}{b f}-\frac {\frac {\frac {d^3 x \left (a^2-b^2\right )^2 \left (-12 a^2 d^2+30 a b c d-\left (b^2 \left (20 c^2+d^2\right )\right )\right )}{b}-\frac {2 (b c-a d)^3 \left (12 a^4 d^2+6 a^3 b c d+a^2 b^2 \left (2 c^2-29 d^2\right )-12 a b^3 c d+b^4 \left (c^2+20 d^2\right )\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (e+f x)\right )+2 b \tan \left (\frac {1}{2} (e+f x)\right )+a}d\tan \left (\frac {1}{2} (e+f x)\right )}{b f}}{b}+\frac {d \left (-12 a^5 d^4+30 a^4 b c d^3-a^3 b^2 d^2 \left (16 c^2-21 d^2\right )-a^2 b^3 c d \left (4 c^2+55 d^2\right )+a b^4 \left (6 c^4+43 c^2 d^2-6 d^4\right )-b^5 c d \left (17 c^2-10 d^2\right )\right ) \cos (e+f x)}{b f}}{b}}{b \left (a^2-b^2\right )}-\frac {(b c-a d)^2 \left (4 a^2 d+3 a b c-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{2 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {-\frac {\frac {d^2 \left (-6 a^4 d^3+7 a^3 b c d^2+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )+b^4 d \left (8 c^2-d^2\right )\right ) \sin (e+f x) \cos (e+f x)}{b f}-\frac {\frac {\frac {4 (b c-a d)^3 \left (12 a^4 d^2+6 a^3 b c d+a^2 b^2 \left (2 c^2-29 d^2\right )-12 a b^3 c d+b^4 \left (c^2+20 d^2\right )\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (e+f x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (e+f x)\right )\right )}{b f}+\frac {d^3 x \left (a^2-b^2\right )^2 \left (-12 a^2 d^2+30 a b c d-\left (b^2 \left (20 c^2+d^2\right )\right )\right )}{b}}{b}+\frac {d \left (-12 a^5 d^4+30 a^4 b c d^3-a^3 b^2 d^2 \left (16 c^2-21 d^2\right )-a^2 b^3 c d \left (4 c^2+55 d^2\right )+a b^4 \left (6 c^4+43 c^2 d^2-6 d^4\right )-b^5 c d \left (17 c^2-10 d^2\right )\right ) \cos (e+f x)}{b f}}{b}}{b \left (a^2-b^2\right )}-\frac {(b c-a d)^2 \left (4 a^2 d+3 a b c-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{2 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {-\frac {(b c-a d)^2 \left (4 a^2 d+3 a b c-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {d^2 \left (-6 a^4 d^3+7 a^3 b c d^2+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )+b^4 d \left (8 c^2-d^2\right )\right ) \sin (e+f x) \cos (e+f x)}{b f}-\frac {\frac {\frac {d^3 x \left (a^2-b^2\right )^2 \left (-12 a^2 d^2+30 a b c d-\left (b^2 \left (20 c^2+d^2\right )\right )\right )}{b}-\frac {2 (b c-a d)^3 \left (12 a^4 d^2+6 a^3 b c d+a^2 b^2 \left (2 c^2-29 d^2\right )-12 a b^3 c d+b^4 \left (c^2+20 d^2\right )\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (e+f x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{b f \sqrt {a^2-b^2}}}{b}+\frac {d \left (-12 a^5 d^4+30 a^4 b c d^3-a^3 b^2 d^2 \left (16 c^2-21 d^2\right )-a^2 b^3 c d \left (4 c^2+55 d^2\right )+a b^4 \left (6 c^4+43 c^2 d^2-6 d^4\right )-b^5 c d \left (17 c^2-10 d^2\right )\right ) \cos (e+f x)}{b f}}{b}}{b \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}\) |
Input:
Int[(c + d*Sin[e + f*x])^5/(a + b*Sin[e + f*x])^3,x]
Output:
((b*c - a*d)^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(2*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2) - (-(((b*c - a*d)^2*(3*a*b*c + 4*a^2*d - 7*b^2*d)*Cos [e + f*x]*(c + d*Sin[e + f*x])^2)/(b*(a^2 - b^2)*f*(a + b*Sin[e + f*x]))) - (-(((((a^2 - b^2)^2*d^3*(30*a*b*c*d - 12*a^2*d^2 - b^2*(20*c^2 + d^2))*x )/b - (2*(b*c - a*d)^3*(6*a^3*b*c*d - 12*a*b^3*c*d + 12*a^4*d^2 + a^2*b^2* (2*c^2 - 29*d^2) + b^4*(c^2 + 20*d^2))*ArcTan[(2*b + 2*a*Tan[(e + f*x)/2]) /(2*Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*f))/b + (d*(30*a^4*b*c*d^3 - 12* a^5*d^4 - a^3*b^2*d^2*(16*c^2 - 21*d^2) - b^5*c*d*(17*c^2 - 10*d^2) - a^2* b^3*c*d*(4*c^2 + 55*d^2) + a*b^4*(6*c^4 + 43*c^2*d^2 - 6*d^4))*Cos[e + f*x ])/(b*f))/b) + (d^2*(7*a^3*b*c*d^2 - 6*a^4*d^3 + b^4*d*(8*c^2 - d^2) + a^2 *b^2*d*(c^2 + 10*d^2) - a*b^3*c*(3*c^2 + 16*d^2))*Cos[e + f*x]*Sin[e + f*x ])/(b*f))/(b*(a^2 - b^2)))/(2*b*(a^2 - b^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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]
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]
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])
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]
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[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si n[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3)) Int[(a + b*Si n[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e + f*x]^2 , x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1]
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[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((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 - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1122\) vs. \(2(519)=1038\).
Time = 183.45 (sec) , antiderivative size = 1123, normalized size of antiderivative = 2.10
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1123\) |
| default | \(\text {Expression too large to display}\) | \(1123\) |
| risch | \(\text {Expression too large to display}\) | \(3634\) |
Input:
int((c+d*sin(f*x+e))^5/(a+b*sin(f*x+e))^3,x,method=_RETURNVERBOSE)
Output:
1/f*(2*d^3/b^5*((1/2*d^2*b^2*tan(1/2*f*x+1/2*e)^3+(3*a*b*d^2-5*b^2*c*d)*ta n(1/2*f*x+1/2*e)^2-1/2*d^2*b^2*tan(1/2*f*x+1/2*e)+3*a*b*d^2-5*b^2*c*d)/(1+ tan(1/2*f*x+1/2*e)^2)^2+1/2*(12*a^2*d^2-30*a*b*c*d+20*b^2*c^2+b^2*d^2)*arc tan(tan(1/2*f*x+1/2*e)))-2/b^5*((-1/2*b^2*(5*a^7*d^5-15*a^6*b*c*d^4+10*a^5 *b^2*c^2*d^3-8*a^5*b^2*d^5+10*a^4*b^3*c^3*d^2+30*a^4*b^3*c*d^4-15*a^3*b^4* c^4*d-40*a^3*b^4*c^2*d^3+5*a^2*b^5*c^5+20*a^2*b^5*c^3*d^2-2*b^7*c^5)/(a^4- 2*a^2*b^2+b^4)/a*tan(1/2*f*x+1/2*e)^3-1/2*b*(6*a^9*d^5-20*a^8*b*c*d^4+20*a ^7*b^2*c^2*d^3+3*a^7*b^2*d^5-5*a^6*b^3*c*d^4-10*a^5*b^4*c^4*d-10*a^5*b^4*c ^2*d^3-18*a^5*b^4*d^5+4*a^4*b^5*c^5+30*a^4*b^5*c^3*d^2+70*a^4*b^5*c*d^4-25 *a^3*b^6*c^4*d-100*a^3*b^6*c^2*d^3+7*a^2*b^7*c^5+60*a^2*b^7*c^3*d^2-10*a*b ^8*c^4*d-2*b^9*c^5)/(a^4-2*a^2*b^2+b^4)/a^2*tan(1/2*f*x+1/2*e)^2-1/2*b^2*( 19*a^7*d^5-65*a^6*b*c*d^4+70*a^5*b^2*c^2*d^3-28*a^5*b^2*d^5-10*a^4*b^3*c^3 *d^2+110*a^4*b^3*c*d^4-25*a^3*b^4*c^4*d-160*a^3*b^4*c^2*d^3+11*a^2*b^5*c^5 +100*a^2*b^5*c^3*d^2-20*a*b^6*c^4*d-2*b^7*c^5)/a/(a^4-2*a^2*b^2+b^4)*tan(1 /2*f*x+1/2*e)-1/2*b*(6*a^7*d^5-20*a^6*b*c*d^4+20*a^5*b^2*c^2*d^3-9*a^5*b^2 *d^5+35*a^4*b^3*c*d^4-10*a^3*b^4*c^4*d-50*a^3*b^4*c^2*d^3+4*a^2*b^5*c^5+30 *a^2*b^5*c^3*d^2-5*a*b^6*c^4*d-b^7*c^5)/(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*(12*a^7*d^5-30*a^6*b*c*d^4+20*a ^5*b^2*c^2*d^3-29*a^5*b^2*d^5+75*a^4*b^3*c*d^4-50*a^3*b^4*c^2*d^3+20*a^3*b ^4*d^5-2*a^2*b^5*c^5-10*a^2*b^5*c^3*d^2-60*a^2*b^5*c*d^4+15*a*b^6*c^4*d...
Leaf count of result is larger than twice the leaf count of optimal. 1544 vs. \(2 (519) = 1038\).
Time = 0.26 (sec) , antiderivative size = 3174, normalized size of antiderivative = 5.94 \[ \int \frac {(c+d \sin (e+f x))^5}{(a+b \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:
integrate((c+d*sin(f*x+e))^5/(a+b*sin(f*x+e))^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {(c+d \sin (e+f x))^5}{(a+b \sin (e+f x))^3} \, dx=\text {Timed out} \] Input:
integrate((c+d*sin(f*x+e))**5/(a+b*sin(f*x+e))**3,x)
Output:
Timed out
Exception generated. \[ \int \frac {(c+d \sin (e+f x))^5}{(a+b \sin (e+f x))^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((c+d*sin(f*x+e))^5/(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
Leaf count of result is larger than twice the leaf count of optimal. 3043 vs. \(2 (519) = 1038\).
Time = 0.44 (sec) , antiderivative size = 3043, normalized size of antiderivative = 5.70 \[ \int \frac {(c+d \sin (e+f x))^5}{(a+b \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:
integrate((c+d*sin(f*x+e))^5/(a+b*sin(f*x+e))^3,x, algorithm="giac")
Output:
1/2*(2*(2*a^2*b^5*c^5 + b^7*c^5 - 15*a*b^6*c^4*d + 10*a^2*b^5*c^3*d^2 + 20 *b^7*c^3*d^2 - 20*a^5*b^2*c^2*d^3 + 50*a^3*b^4*c^2*d^3 - 60*a*b^6*c^2*d^3 + 30*a^6*b*c*d^4 - 75*a^4*b^3*c*d^4 + 60*a^2*b^5*c*d^4 - 12*a^7*d^5 + 29*a ^5*b^2*d^5 - 20*a^3*b^4*d^5)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(a) + ar ctan((a*tan(1/2*f*x + 1/2*e) + b)/sqrt(a^2 - b^2)))/((a^4*b^5 - 2*a^2*b^7 + b^9)*sqrt(a^2 - b^2)) + 2*(5*a^3*b^6*c^5*tan(1/2*f*x + 1/2*e)^7 - 2*a*b^ 8*c^5*tan(1/2*f*x + 1/2*e)^7 - 15*a^4*b^5*c^4*d*tan(1/2*f*x + 1/2*e)^7 + 1 0*a^5*b^4*c^3*d^2*tan(1/2*f*x + 1/2*e)^7 + 20*a^3*b^6*c^3*d^2*tan(1/2*f*x + 1/2*e)^7 + 10*a^6*b^3*c^2*d^3*tan(1/2*f*x + 1/2*e)^7 - 40*a^4*b^5*c^2*d^ 3*tan(1/2*f*x + 1/2*e)^7 - 15*a^7*b^2*c*d^4*tan(1/2*f*x + 1/2*e)^7 + 30*a^ 5*b^4*c*d^4*tan(1/2*f*x + 1/2*e)^7 + 6*a^8*b*d^5*tan(1/2*f*x + 1/2*e)^7 - 10*a^6*b^3*d^5*tan(1/2*f*x + 1/2*e)^7 + a^4*b^5*d^5*tan(1/2*f*x + 1/2*e)^7 + 4*a^4*b^5*c^5*tan(1/2*f*x + 1/2*e)^6 + 7*a^2*b^7*c^5*tan(1/2*f*x + 1/2* e)^6 - 2*b^9*c^5*tan(1/2*f*x + 1/2*e)^6 - 10*a^5*b^4*c^4*d*tan(1/2*f*x + 1 /2*e)^6 - 25*a^3*b^6*c^4*d*tan(1/2*f*x + 1/2*e)^6 - 10*a*b^8*c^4*d*tan(1/2 *f*x + 1/2*e)^6 + 30*a^4*b^5*c^3*d^2*tan(1/2*f*x + 1/2*e)^6 + 60*a^2*b^7*c ^3*d^2*tan(1/2*f*x + 1/2*e)^6 + 20*a^7*b^2*c^2*d^3*tan(1/2*f*x + 1/2*e)^6 - 10*a^5*b^4*c^2*d^3*tan(1/2*f*x + 1/2*e)^6 - 100*a^3*b^6*c^2*d^3*tan(1/2* f*x + 1/2*e)^6 - 30*a^8*b*c*d^4*tan(1/2*f*x + 1/2*e)^6 + 15*a^6*b^3*c*d^4* tan(1/2*f*x + 1/2*e)^6 + 60*a^4*b^5*c*d^4*tan(1/2*f*x + 1/2*e)^6 + 12*a...
Time = 31.76 (sec) , antiderivative size = 23910, normalized size of antiderivative = 44.78 \[ \int \frac {(c+d \sin (e+f x))^5}{(a+b \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:
int((c + d*sin(e + f*x))^5/(a + b*sin(e + f*x))^3,x)
Output:
- ((b^7*c^5 - 12*a^7*d^5 - 4*a^2*b^5*c^5 - 6*a^3*b^4*d^5 + 21*a^5*b^2*d^5 + 10*a^2*b^5*c*d^4 + 10*a^3*b^4*c^4*d - 55*a^4*b^3*c*d^4 - 30*a^2*b^5*c^3* d^2 + 50*a^3*b^4*c^2*d^3 - 20*a^5*b^2*c^2*d^3 + 5*a*b^6*c^4*d + 30*a^6*b*c *d^4)/(b^4*(a^4 + b^4 - 2*a^2*b^2)) - (tan(e/2 + (f*x)/2)^6*(12*a^9*d^5 - 2*b^9*c^5 + 7*a^2*b^7*c^5 + 4*a^4*b^5*c^5 + 4*a^3*b^6*d^5 - 20*a^5*b^4*d^5 - 5*a^7*b^2*d^5 - 25*a^3*b^6*c^4*d + 60*a^4*b^5*c*d^4 - 10*a^5*b^4*c^4*d + 15*a^6*b^3*c*d^4 + 60*a^2*b^7*c^3*d^2 - 100*a^3*b^6*c^2*d^3 + 30*a^4*b^5 *c^3*d^2 - 10*a^5*b^4*c^2*d^3 + 20*a^7*b^2*c^2*d^3 - 10*a*b^8*c^4*d - 30*a ^8*b*c*d^4))/(a^2*b^4*(a^4 + b^4 - 2*a^2*b^2)) + (tan(e/2 + (f*x)/2)^2*(2* b^9*c^5 - 36*a^9*d^5 - 5*a^2*b^7*c^5 - 12*a^4*b^5*c^5 - 20*a^3*b^6*d^5 + 4 0*a^5*b^4*d^5 + 31*a^7*b^2*d^5 + 40*a^2*b^7*c*d^4 + 35*a^3*b^6*c^4*d - 120 *a^4*b^5*c*d^4 + 30*a^5*b^4*c^4*d - 85*a^6*b^3*c*d^4 - 60*a^2*b^7*c^3*d^2 + 100*a^3*b^6*c^2*d^3 - 90*a^4*b^5*c^3*d^2 + 110*a^5*b^4*c^2*d^3 - 60*a^7* b^2*c^2*d^3 + 10*a*b^8*c^4*d + 90*a^8*b*c*d^4))/(a^2*b^4*(a^4 + b^4 - 2*a^ 2*b^2)) - (tan(e/2 + (f*x)/2)^5*(54*a^7*d^5 - 6*b^7*c^5 + 4*a*b^6*d^5 + 21 *a^2*b^5*c^5 + 17*a^3*b^4*d^5 - 90*a^5*b^2*d^5 - 40*a^2*b^5*c*d^4 - 55*a^3 *b^4*c^4*d + 250*a^4*b^3*c*d^4 + 140*a^2*b^5*c^3*d^2 - 240*a^3*b^4*c^2*d^3 + 10*a^4*b^3*c^3*d^2 + 90*a^5*b^2*c^2*d^3 - 20*a*b^6*c^4*d - 135*a^6*b*c* d^4))/(a*b^3*(a^4 + b^4 - 2*a^2*b^2)) + (tan(e/2 + (f*x)/2)^3*(6*b^7*c^5 - 90*a^7*d^5 + 4*a*b^6*d^5 - 27*a^2*b^5*c^5 - 55*a^3*b^4*d^5 + 162*a^5*b...
\[ \int \frac {(c+d \sin (e+f x))^5}{(a+b \sin (e+f x))^3} \, dx=\int \frac {\left (\sin \left (f x +e \right ) d +c \right )^{5}}{\left (\sin \left (f x +e \right ) b +a \right )^{3}}d x \] Input:
int((c+d*sin(f*x+e))^5/(a+b*sin(f*x+e))^3,x)
Output:
int((c+d*sin(f*x+e))^5/(a+b*sin(f*x+e))^3,x)