Integrand size = 27, antiderivative size = 512 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+b \sin (e+f x))^3} \, dx=\frac {(b c-3 d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b \left (9-b^2\right ) f (3+b \sin (e+f x))^2}+\frac {3 (b c-3 d) \left (6 b c+9 d-3 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 b \left (9-b^2\right )^2 f (3+b \sin (e+f x))}+\frac {3 (b c-3 d) \left (6 b c+9 d-3 b^2 d\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{4 b^2 \left (9-b^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (108 b c d^2+243 d^3+9 b^2 d \left (7 c^2-5 d^2\right )+b^4 d \left (11 c^2+8 d^2\right )-6 b^3 c \left (3 c^2+11 d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{4 b^3 \left (9-b^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {(b c-3 d) \left (108 b c d-84 b^3 c d+243 d^2+18 b^2 \left (4 c^2-3 d^2\right )+b^4 \left (4 c^2+15 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {2 b}{3+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{4 (3-b)^2 b^3 (3+b)^3 f \sqrt {c+d \sin (e+f x)}} \]
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)*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*Ellip ticE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin(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))*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*Ellipt icF(cos(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))* (sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticPi(c os(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 = 10.52 (sec) , antiderivative size = 1088, normalized size of antiderivative = 2.12 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+b \sin (e+f x))^3} \, dx=\frac {\sqrt {c+d \sin (e+f x)} \left (\frac {-b^2 c^2 \cos (e+f x)+6 b c d \cos (e+f x)-9 d^2 \cos (e+f x)}{2 b \left (-9+b^2\right ) (3+b \sin (e+f x))^2}-\frac {9 \left (-2 b^2 c^2 \cos (e+f x)+3 b c d \cos (e+f x)+b^3 c d \cos (e+f x)+9 d^2 \cos (e+f x)-3 b^2 d^2 \cos (e+f x)\right )}{4 b \left (-9+b^2\right )^2 (3+b \sin (e+f x))}\right )}{f}+\frac {-\frac {2 \left (144 b c^3+8 b^3 c^3-162 b^2 c^2 d+135 b c d^2+21 b^3 c d^2-27 d^3-15 b^2 d^3\right ) \operatorname {EllipticPi}\left (\frac {2 b}{3+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}}}{(3+b) \sqrt {c+d \sin (e+f x)}}-\frac {2 i \left (180 b c^2 d+4 b^3 c^2 d-108 c d^2-132 b^2 c d^2+72 b d^3+16 b^3 d^3\right ) \cos (e+f x) \left ((b c-3 d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )+3 d \operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-3 d},i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )\right ) \sqrt {\frac {d-d \sin (e+f x)}{c+d}} \sqrt {-\frac {d+d \sin (e+f x)}{c-d}} (-b c+3 d+b (c+d \sin (e+f x)))}{b (b c-3 d) d^2 \sqrt {-\frac {1}{c+d}} (3+b \sin (e+f x)) \sqrt {1-\sin ^2(e+f x)} \sqrt {-\frac {c^2-d^2-2 c (c+d \sin (e+f x))+(c+d \sin (e+f x))^2}{d^2}}}-\frac {2 i \left (-18 b^2 c^2 d+27 b c d^2+9 b^3 c d^2+81 d^3-27 b^2 d^3\right ) \cos (e+f x) \cos (2 (e+f x)) \left (2 b (b c-3 d) (c-d) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )+d \left (2 (3+b) (b c-3 d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )-\left (-18+b^2\right ) d \operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-3 d},i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )\right )\right ) \sqrt {\frac {d-d \sin (e+f x)}{c+d}} \sqrt {-\frac {d+d \sin (e+f x)}{c-d}} (-b c+3 d+b (c+d \sin (e+f x)))}{b^2 (b c-3 d) d \sqrt {-\frac {1}{c+d}} (3+b \sin (e+f x)) \sqrt {1-\sin ^2(e+f x)} \left (-2 c^2+d^2+4 c (c+d \sin (e+f x))-2 (c+d \sin (e+f x))^2\right ) \sqrt {-\frac {c^2-d^2-2 c (c+d \sin (e+f x))+(c+d \sin (e+f x))^2}{d^2}}}}{16 (-3+b)^2 b (3+b)^2 f} \]
(Sqrt[c + d*Sin[e + f*x]]*((-(b^2*c^2*Cos[e + f*x]) + 6*b*c*d*Cos[e + f*x] - 9*d^2*Cos[e + f*x])/(2*b*(-9 + b^2)*(3 + b*Sin[e + f*x])^2) - (9*(-2*b^ 2*c^2*Cos[e + f*x] + 3*b*c*d*Cos[e + f*x] + b^3*c*d*Cos[e + f*x] + 9*d^2*C os[e + f*x] - 3*b^2*d^2*Cos[e + f*x]))/(4*b*(-9 + b^2)^2*(3 + b*Sin[e + f* x]))))/f + ((-2*(144*b*c^3 + 8*b^3*c^3 - 162*b^2*c^2*d + 135*b*c*d^2 + 21* b^3*c*d^2 - 27*d^3 - 15*b^2*d^3)*EllipticPi[(2*b)/(3 + b), (-e + Pi/2 - f* x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((3 + b)*Sqrt[c + d*Sin[e + f*x]]) - ((2*I)*(180*b*c^2*d + 4*b^3*c^2*d - 108*c*d^2 - 132*b^ 2*c*d^2 + 72*b*d^3 + 16*b^3*d^3)*Cos[e + f*x]*((b*c - 3*d)*EllipticF[I*Arc Sinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + 3*d *EllipticPi[(b*(c + d))/(b*c - 3*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)]*S qrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + 3*d + b*(c + d*Sin[e + f*x] )))/(b*(b*c - 3*d)*d^2*Sqrt[-(c + d)^(-1)]*(3 + b*Sin[e + f*x])*Sqrt[1 - S in[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)*(-18*b^2*c^2*d + 27*b*c*d^2 + 9*b^3*c*d^2 + 81 *d^3 - 27*b^2*d^3)*Cos[e + f*x]*Cos[2*(e + f*x)]*(2*b*(b*c - 3*d)*(c - d)* EllipticE[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d) /(c - d)] + d*(2*(3 + b)*(b*c - 3*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1 )]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] - (-18 + b^2)*d*Elliptic...
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 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\int -\frac {4 a b c^3-9 b^2 d c^2+6 a b d^2 c-a^2 d^3+d \left (-\left (\left (c^2+4 d^2\right ) b^2\right )+2 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x)+2 \left (-\left (\left (c^3+6 d^2 c\right ) b^2\right )+2 a d \left (2 c^2+d^2\right ) b+a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}dx}{4 b \left (a^2-b^2\right )}\) |
3.8.60.3.1 Defintions of rubi rules used
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. \(1887\) vs. \(2(622)=1244\).
Time = 79.61 (sec) , antiderivative size = 1888, normalized size of antiderivative = 3.69
(-(-d*sin(f*x+e)-c)*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)*(1/(c-d)*(-sin(f*x+e)-1)*d) ^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e)) /(c-d))^(1/2),((c-d)/(c+d))^(1/2))-6*d^2/b^4*(a*d-b*c)*(c/d-1)*((c+d*sin(f *x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1 )*d)^(1/2)/(-(-d*sin(f*x+e)-c)*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+a/b),((c-d)/(c+d))^(1/2))+3/b ^3*d*(a^2*d^2-2*a*b*c*d+b^2*c^2)*(-b^2/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)*(-(-d *sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(a+b*sin(f*x+e))-a*d/(a^3*d-a^2*b*c-a*b ^2*d+b^3*c)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d) )^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2 )^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-b*d/ (a^3*d-a^2*b*c-a*b^2*d+b^3*c)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1 -sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+ e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(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)))+(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)/b*(c/d- 1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)* (-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*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+a/b),((c-d)/...
Timed out. \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+b \sin (e+f x))^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+b \sin (e+f x))^3} \, dx=\text {Timed out} \]
\[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+b \sin (e+f x))^3} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{3}} \,d x } \]
\[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+b \sin (e+f x))^3} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+b \sin (e+f x))^3} \, dx=\int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^3} \,d x \]