Integrand size = 27, antiderivative size = 549 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+b \sin (e+f x))^3} \, dx=\frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {3 (b c-a d) \left (2 a b c+a^2 d-3 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 b \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {3 (b c-a d) \left (2 a b c+a^2 d-3 b^2 d\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{4 b^2 \left (a^2-b^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (4 a^3 b c d^2+3 a^4 d^3+a^2 b^2 d \left (7 c^2-5 d^2\right )+b^4 d \left (11 c^2+8 d^2\right )-2 a b^3 c \left (3 c^2+11 d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{4 b^3 \left (a^2-b^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {(b c-a d) \left (4 a^3 b c d-28 a b^3 c d+3 a^4 d^2+2 a^2 b^2 \left (4 c^2-3 d^2\right )+b^4 \left (4 c^2+15 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{4 (a-b)^2 b^3 (a+b)^3 f \sqrt {c+d \sin (e+f x)}} \] Output:
1/2*(-a*d+b*c)^2*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/b/(a^2-b^2)/f/(a+b*sin( f*x+e))^2+3/4*(-a*d+b*c)*(a^2*d+2*a*b*c-3*b^2*d)*cos(f*x+e)*(c+d*sin(f*x+e ))^(1/2)/b/(a^2-b^2)^2/f/(a+b*sin(f*x+e))-3/4*(-a*d+b*c)*(a^2*d+2*a*b*c-3* b^2*d)*EllipticE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*s in(f*x+e))^(1/2)/b^2/(a^2-b^2)^2/f/((c+d*sin(f*x+e))/(c+d))^(1/2)+1/4*(4*a ^3*b*c*d^2+3*a^4*d^3+a^2*b^2*d*(7*c^2-5*d^2)+b^4*d*(11*c^2+8*d^2)-2*a*b^3* c*(3*c^2+11*d^2))*InverseJacobiAM(1/2*e-1/4*Pi+1/2*f*x,2^(1/2)*(d/(c+d))^( 1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/b^3/(a^2-b^2)^2/f/(c+d*sin(f*x+e))^(1 /2)-1/4*(-a*d+b*c)*(4*a^3*b*c*d-28*a*b^3*c*d+3*a^4*d^2+2*a^2*b^2*(4*c^2-3* d^2)+b^4*(4*c^2+15*d^2))*EllipticPi(cos(1/2*e+1/4*Pi+1/2*f*x),2*b/(a+b),2^ (1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/(a-b)^2/b^3/(a+b)^3/ f/(c+d*sin(f*x+e))^(1/2)
Result contains complex when optimal does not.
Time = 11.72 (sec) , antiderivative size = 1149, normalized size of antiderivative = 2.09 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+b \sin (e+f x))^3} \, dx =\text {Too large to display} \] Input:
Integrate[(c + d*Sin[e + f*x])^(5/2)/(a + b*Sin[e + f*x])^3,x]
Output:
(Sqrt[c + d*Sin[e + f*x]]*((-(b^2*c^2*Cos[e + f*x]) + 2*a*b*c*d*Cos[e + f* x] - a^2*d^2*Cos[e + f*x])/(2*b*(-a^2 + b^2)*(a + b*Sin[e + f*x])^2) - (3* (-2*a*b^2*c^2*Cos[e + f*x] + a^2*b*c*d*Cos[e + f*x] + 3*b^3*c*d*Cos[e + f* x] + a^3*d^2*Cos[e + f*x] - 3*a*b^2*d^2*Cos[e + f*x]))/(4*b*(-a^2 + b^2)^2 *(a + b*Sin[e + f*x]))))/f - ((-2*(-16*a^2*b*c^3 - 8*b^3*c^3 + 54*a*b^2*c^ 2*d - 15*a^2*b*c*d^2 - 21*b^3*c*d^2 + a^3*d^3 + 5*a*b^2*d^3)*EllipticPi[(2 *b)/(a + b), (-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x]) /(c + d)])/((a + b)*Sqrt[c + d*Sin[e + f*x]]) - ((2*I)*(-20*a^2*b*c^2*d - 4*b^3*c^2*d + 4*a^3*c*d^2 + 44*a*b^2*c*d^2 - 8*a^2*b*d^3 - 16*b^3*d^3)*Cos [e + f*x]*((b*c - a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d* Sin[e + f*x]]], (c + d)/(c - d)] + a*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] )*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))] *(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b*d^2*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[-((c^2 - d^2 - 2 *c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]) - ((2*I)*(6*a*b^2 *c^2*d - 3*a^2*b*c*d^2 - 9*b^3*c*d^2 - 3*a^3*d^3 + 9*a*b^2*d^3)*Cos[e + f* x]*Cos[2*(e + f*x)]*(2*b*(c - d)*(b*c - a*d)*EllipticE[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + d*(-2*(a + b)*(-( b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + ...
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.
| \(\displaystyle \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+b \sin (e+f x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+b \sin (e+f x))^3}dx\) |
| \(\Big \downarrow \) 3271 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\int -\frac {4 a b c^3-9 b^2 d c^2+6 a b d^2 c-a^2 d^3+d \left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x)+2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{2 (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{2 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3+9 b^2 c^2 d-\left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x) d-a b \left (4 c^3+6 d^2 c\right )-2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\int -\frac {4 a b c^3-9 b^2 d c^2+6 a b d^2 c-a^2 d^3+d \left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x)+2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3+9 b^2 c^2 d-\left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x) d-a b \left (4 c^3+6 d^2 c\right )-2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\int -\frac {4 a b c^3-9 b^2 d c^2+6 a b d^2 c-a^2 d^3+d \left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x)+2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3+9 b^2 c^2 d-\left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x) d-a b \left (4 c^3+6 d^2 c\right )-2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\int -\frac {4 a b c^3-9 b^2 d c^2+6 a b d^2 c-a^2 d^3+d \left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x)+2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3+9 b^2 c^2 d-\left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x) d-a b \left (4 c^3+6 d^2 c\right )-2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\int -\frac {4 a b c^3-9 b^2 d c^2+6 a b d^2 c-a^2 d^3+d \left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x)+2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3+9 b^2 c^2 d-\left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x) d-a b \left (4 c^3+6 d^2 c\right )-2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\int -\frac {4 a b c^3-9 b^2 d c^2+6 a b d^2 c-a^2 d^3+d \left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x)+2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3+9 b^2 c^2 d-\left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x) d-a b \left (4 c^3+6 d^2 c\right )-2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\int -\frac {4 a b c^3-9 b^2 d c^2+6 a b d^2 c-a^2 d^3+d \left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x)+2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3+9 b^2 c^2 d-\left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x) d-a b \left (4 c^3+6 d^2 c\right )-2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\int -\frac {4 a b c^3-9 b^2 d c^2+6 a b d^2 c-a^2 d^3+d \left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x)+2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3+9 b^2 c^2 d-\left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x) d-a b \left (4 c^3+6 d^2 c\right )-2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\int -\frac {4 a b c^3-9 b^2 d c^2+6 a b d^2 c-a^2 d^3+d \left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x)+2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3+9 b^2 c^2 d-\left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x) d-a b \left (4 c^3+6 d^2 c\right )-2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\int -\frac {4 a b c^3-9 b^2 d c^2+6 a b d^2 c-a^2 d^3+d \left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x)+2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3+9 b^2 c^2 d-\left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x) d-a b \left (4 c^3+6 d^2 c\right )-2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\int -\frac {4 a b c^3-9 b^2 d c^2+6 a b d^2 c-a^2 d^3+d \left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x)+2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3+9 b^2 c^2 d-\left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x) d-a b \left (4 c^3+6 d^2 c\right )-2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\int -\frac {4 a b c^3-9 b^2 d c^2+6 a b d^2 c-a^2 d^3+d \left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x)+2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3+9 b^2 c^2 d-\left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x) d-a b \left (4 c^3+6 d^2 c\right )-2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\int -\frac {4 a b c^3-9 b^2 d c^2+6 a b d^2 c-a^2 d^3+d \left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x)+2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3+9 b^2 c^2 d-\left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x) d-a b \left (4 c^3+6 d^2 c\right )-2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\int -\frac {4 a b c^3-9 b^2 d c^2+6 a b d^2 c-a^2 d^3+d \left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x)+2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3+9 b^2 c^2 d-\left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x) d-a b \left (4 c^3+6 d^2 c\right )-2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\int -\frac {4 a b c^3-9 b^2 d c^2+6 a b d^2 c-a^2 d^3+d \left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x)+2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}\) |
Input:
Int[(c + d*Sin[e + f*x])^(5/2)/(a + b*Sin[e + f*x])^3,x]
Output:
$Aborted
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_.)*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])
Leaf count of result is larger than twice the leaf count of optimal. \(1880\) vs. \(2(531)=1062\).
Time = 124.02 (sec) , antiderivative size = 1881, normalized size of antiderivative = 3.43
Input:
int((c+d*sin(f*x+e))^(5/2)/(a+b*sin(f*x+e))^3,x,method=_RETURNVERBOSE)
Output:
(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*(2*d^3/b^3*(c/d-1)*((c+d*sin(f*x+e ))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^( 1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/( c-d))^(1/2),((c-d)/(c+d))^(1/2))-(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3* c^3)/b^3*(-1/2*b^2/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)*(-(-c-d*sin(f*x+e))*cos(f *x+e)^2)^(1/2)/(a+b*sin(f*x+e))^2-3/4*b^2*(3*a^2*d-2*a*b*c-b^2*d)/(a^3*d-a ^2*b*c-a*b^2*d+b^3*c)^2*(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)/(a+b*sin(f *x+e))-1/4*d*(7*a^3*d-4*a^2*b*c-a*b^2*d-2*b^3*c)/(a^3*d-a^2*b*c-a*b^2*d+b^ 3*c)^2*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/ 2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2) *EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-3/4*b*d*(3* a^2*d-2*a*b*c-b^2*d)/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)^2*(c/d-1)*((c+d*sin(f*x +e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d)) ^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*s in(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/( c-d))^(1/2),((c-d)/(c+d))^(1/2)))+1/4*(15*a^4*d^2-20*a^3*b*c*d+8*a^2*b^2*c ^2-6*a^2*b^2*d^2-4*a*b^3*c*d+4*b^4*c^2+3*b^4*d^2)/(a^3*d-a^2*b*c-a*b^2*d+b ^3*c)^2/b*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^ (1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1 /2)/(-c/d+a/b)*EllipticPi(((c+d*sin(f*x+e))/(c-d))^(1/2),(-c/d+1)/(-c/d...
Timed out. \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+b \sin (e+f x))^3} \, dx=\text {Timed out} \] Input:
integrate((c+d*sin(f*x+e))^(5/2)/(a+b*sin(f*x+e))^3,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+b \sin (e+f x))^3} \, dx=\text {Timed out} \] Input:
integrate((c+d*sin(f*x+e))**(5/2)/(a+b*sin(f*x+e))**3,x)
Output:
Timed out
\[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+b \sin (e+f x))^3} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{3}} \,d x } \] Input:
integrate((c+d*sin(f*x+e))^(5/2)/(a+b*sin(f*x+e))^3,x, algorithm="maxima")
Output:
integrate((d*sin(f*x + e) + c)^(5/2)/(b*sin(f*x + e) + a)^3, x)
\[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+b \sin (e+f x))^3} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{3}} \,d x } \] Input:
integrate((c+d*sin(f*x+e))^(5/2)/(a+b*sin(f*x+e))^3,x, algorithm="giac")
Output:
integrate((d*sin(f*x + e) + c)^(5/2)/(b*sin(f*x + e) + a)^3, x)
Timed out. \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+b \sin (e+f x))^3} \, dx=\int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^3} \,d x \] Input:
int((c + d*sin(e + f*x))^(5/2)/(a + b*sin(e + f*x))^3,x)
Output:
int((c + d*sin(e + f*x))^(5/2)/(a + b*sin(e + f*x))^3, x)
\[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+b \sin (e+f x))^3} \, dx=\left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}}{\sin \left (f x +e \right )^{3} b^{3}+3 \sin \left (f x +e \right )^{2} a \,b^{2}+3 \sin \left (f x +e \right ) a^{2} b +a^{3}}d x \right ) c^{2}+\left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{3} b^{3}+3 \sin \left (f x +e \right )^{2} a \,b^{2}+3 \sin \left (f x +e \right ) a^{2} b +a^{3}}d x \right ) d^{2}+2 \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{3} b^{3}+3 \sin \left (f x +e \right )^{2} a \,b^{2}+3 \sin \left (f x +e \right ) a^{2} b +a^{3}}d x \right ) c d \] Input:
int((c+d*sin(f*x+e))^(5/2)/(a+b*sin(f*x+e))^3,x)
Output:
int(sqrt(sin(e + f*x)*d + c)/(sin(e + f*x)**3*b**3 + 3*sin(e + f*x)**2*a*b **2 + 3*sin(e + f*x)*a**2*b + a**3),x)*c**2 + int((sqrt(sin(e + f*x)*d + c )*sin(e + f*x)**2)/(sin(e + f*x)**3*b**3 + 3*sin(e + f*x)**2*a*b**2 + 3*si n(e + f*x)*a**2*b + a**3),x)*d**2 + 2*int((sqrt(sin(e + f*x)*d + c)*sin(e + f*x))/(sin(e + f*x)**3*b**3 + 3*sin(e + f*x)**2*a*b**2 + 3*sin(e + f*x)* a**2*b + a**3),x)*c*d