Integrand size = 27, antiderivative size = 661 \[ \int \frac {1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx=\frac {d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 \left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac {\left (8 a^3 c d^4-8 a b^2 c d^4-4 a^2 b d^3 \left (5 c^2-3 d^2\right )-b^3 \left (3 c^4 d-26 c^2 d^3+15 d^5\right )\right ) \cos (e+f x)}{3 \left (a^2-b^2\right ) (b c-a d)^3 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}-\frac {\left (8 a^3 c d^3-8 a b^2 c d^3-4 a^2 b d^2 \left (5 c^2-3 d^2\right )-b^3 \left (3 c^4-26 c^2 d^2+15 d^4\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 \left (a^2-b^2\right ) (b c-a d)^3 \left (c^2-d^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 \left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {b^2 \left (2 a b c-7 a^2 d+5 b^2 d\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(a-b) (a+b)^2 (b c-a d)^3 f \sqrt {c+d \sin (e+f x)}} \] Output:
1/3*d*(2*a^2*d^2+b^2*(3*c^2-5*d^2))*cos(f*x+e)/(a^2-b^2)/(-a*d+b*c)^2/(c^2 -d^2)/f/(c+d*sin(f*x+e))^(3/2)+b^2*cos(f*x+e)/(a^2-b^2)/(-a*d+b*c)/f/(a+b* sin(f*x+e))/(c+d*sin(f*x+e))^(3/2)-1/3*(8*a^3*c*d^4-8*a*b^2*c*d^4-4*a^2*b* d^3*(5*c^2-3*d^2)-b^3*(3*c^4*d-26*c^2*d^3+15*d^5))*cos(f*x+e)/(a^2-b^2)/(- a*d+b*c)^3/(c^2-d^2)^2/f/(c+d*sin(f*x+e))^(1/2)+1/3*(8*a^3*c*d^3-8*a*b^2*c *d^3-4*a^2*b*d^2*(5*c^2-3*d^2)-b^3*(3*c^4-26*c^2*d^2+15*d^4))*EllipticE(co s(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)/(a ^2-b^2)/(-a*d+b*c)^3/(c^2-d^2)^2/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-1/3*(2*a ^2*d^2+b^2*(3*c^2-5*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)/(a^2-b^2)/(-a*d+b*c)^2/(c^2-d ^2)/f/(c+d*sin(f*x+e))^(1/2)-b^2*(-7*a^2*d+2*a*b*c+5*b^2*d)*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)/(a+b)^2/(-a*d+b*c)^3/f/(c+d*sin(f*x+e))^(1/2)
Result contains complex when optimal does not.
Time = 8.22 (sec) , antiderivative size = 1319, normalized size of antiderivative = 2.00 \[ \int \frac {1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx =\text {Too large to display} \] Input:
Integrate[1/((a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(5/2)),x]
Output:
(Sqrt[c + d*Sin[e + f*x]]*(-((b^4*Cos[e + f*x])/((a^2 - b^2)*(-(b*c) + a*d )^3*(a + b*Sin[e + f*x]))) + (2*d^3*Cos[e + f*x])/(3*(b*c - a*d)^2*(c^2 - d^2)*(c + d*Sin[e + f*x])^2) - (4*(-5*b*c^2*d^3*Cos[e + f*x] + 2*a*c*d^4*C os[e + f*x] + 3*b*d^5*Cos[e + f*x]))/(3*(b*c - a*d)^3*(c^2 - d^2)^2*(c + d *Sin[e + f*x]))))/f + ((-2*(-12*a*b^3*c^5 + 36*a^2*b^2*c^4*d - 33*b^4*c^4* d - 36*a^3*b*c^3*d^2 + 60*a*b^3*c^3*d^2 + 12*a^4*c^2*d^3 - 104*a^2*b^2*c^2 *d^3 + 86*b^4*c^2*d^3 + 28*a^3*b*c*d^4 - 40*a*b^3*c*d^4 + 4*a^4*d^5 + 44*a ^2*b^2*d^5 - 45*b^4*d^5)*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)*(-12*a*b^3*c^4*d - 36*a^2*b^2*c^3*d^2 + 36*b^4*c^3*d^2 - 28*a^3*b*c^2*d^3 + 52*a*b^3*c^2*d^3 + 16*a^4*c*d^4 + 4*a^2*b^2*c*d^4 - 2 0*b^4*c*d^4 + 28*a^3*b*d^5 - 40*a*b^3*d^5)*Cos[e + f*x]*((b*c - a*d)*Ellip ticF[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*Si n[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)*(3*b^4*c^4*d + 20*a^2*b^2*c^2*d^3 - 26*b^4*c^2*d^3 - 8*a^3*b*c*d^4 + 8*a*b^3*c*d^4 - 12*a^2*b^2*d^5 + 15*b^...
Time = 6.26 (sec) , antiderivative size = 668, normalized size of antiderivative = 1.01, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.852, Rules used = {3042, 3281, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3538, 3042, 3134, 3042, 3132, 3481, 3042, 3142, 3042, 3140, 3286, 3042, 3284}
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.
| \(\displaystyle \int \frac {1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3281 |
| \(\displaystyle \frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac {\int -\frac {-2 d a^2+2 b c a+2 b d \sin (e+f x) a-3 b^2 d \sin ^2(e+f x)+5 b^2 d}{2 (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}dx}{\left (a^2-b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {-2 d a^2+2 b c a+2 b d \sin (e+f x) a-3 b^2 d \sin ^2(e+f x)+5 b^2 d}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}dx}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {-2 d a^2+2 b c a+2 b d \sin (e+f x) a-3 b^2 d \sin (e+f x)^2+5 b^2 d}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}dx}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {2 \int \frac {-b d \left (\left (3 c^2-5 d^2\right ) b^2+2 a^2 d^2\right ) \sin ^2(e+f x)+2 d \left (-d^2 a^3+3 b c d a^2+b^2 \left (3 c^2-2 d^2\right ) a-3 b^3 c d\right ) \sin (e+f x)+3 \left (2 c d^2 a^3-4 b d \left (c^2-d^2\right ) a^2+2 b^2 c \left (c^2-2 d^2\right ) a+5 b^3 d \left (c^2-d^2\right )\right )}{2 (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {-b d \left (\left (3 c^2-5 d^2\right ) b^2+2 a^2 d^2\right ) \sin ^2(e+f x)+2 d \left (-d^2 a^3+3 b c d a^2+b^2 \left (3 c^2-2 d^2\right ) a-3 b^3 c d\right ) \sin (e+f x)+3 \left (2 c d^2 a^3-4 b d \left (c^2-d^2\right ) a^2+2 b^2 c \left (c^2-2 d^2\right ) a+5 b^3 d \left (c^2-d^2\right )\right )}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {-b d \left (\left (3 c^2-5 d^2\right ) b^2+2 a^2 d^2\right ) \sin (e+f x)^2+2 d \left (-d^2 a^3+3 b c d a^2+b^2 \left (3 c^2-2 d^2\right ) a-3 b^3 c d\right ) \sin (e+f x)+3 \left (2 c d^2 a^3-4 b d \left (c^2-d^2\right ) a^2+2 b^2 c \left (c^2-2 d^2\right ) a+5 b^3 d \left (c^2-d^2\right )\right )}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {-2 \left (d^5+3 c^2 d^3\right ) a^4+2 b c d^2 \left (9 c^2-5 d^2\right ) a^3-2 b^2 d \left (9 c^4-21 d^2 c^2+8 d^4\right ) a^2+2 b^3 c \left (3 c^4-15 d^2 c^2+8 d^4\right ) a+15 b^4 d \left (c^2-d^2\right )^2-b \left (8 a^3 c d^4-8 a b^2 c d^4-4 a^2 b \left (5 c^2-3 d^2\right ) d^3-b^3 \left (15 d^5-26 c^2 d^3+3 c^4 d\right )\right ) \sin ^2(e+f x)-2 d \left (4 c d^3 a^4-7 b d^2 \left (c^2-d^2\right ) a^3-b^2 c d \left (9 c^2-d^2\right ) a^2-b^3 \left (3 c^4-13 d^2 c^2+10 d^4\right ) a+b^4 c d \left (9 c^2-5 d^2\right )\right ) \sin (e+f x)}{2 (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{\left (c^2-d^2\right ) (b c-a d)}-\frac {2 \left (8 a^3 c d^4-4 a^2 b d^3 \left (5 c^2-3 d^2\right )-8 a b^2 c d^4-b^3 \left (3 c^4 d-26 c^2 d^3+15 d^5\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {-2 \left (d^5+3 c^2 d^3\right ) a^4+2 b c d^2 \left (9 c^2-5 d^2\right ) a^3-2 b^2 d \left (9 c^4-21 d^2 c^2+8 d^4\right ) a^2+2 b^3 c \left (3 c^4-15 d^2 c^2+8 d^4\right ) a+15 b^4 d \left (c^2-d^2\right )^2-b \left (8 a^3 c d^4-8 a b^2 c d^4-4 a^2 b \left (5 c^2-3 d^2\right ) d^3-b^3 \left (15 d^5-26 c^2 d^3+3 c^4 d\right )\right ) \sin ^2(e+f x)-2 d \left (4 c d^3 a^4-7 b d^2 \left (c^2-d^2\right ) a^3-b^2 c d \left (9 c^2-d^2\right ) a^2-b^3 \left (3 c^4-13 d^2 c^2+10 d^4\right ) a+b^4 c d \left (9 c^2-5 d^2\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{\left (c^2-d^2\right ) (b c-a d)}-\frac {2 \left (8 a^3 c d^4-4 a^2 b d^3 \left (5 c^2-3 d^2\right )-8 a b^2 c d^4-b^3 \left (3 c^4 d-26 c^2 d^3+15 d^5\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {-2 \left (d^5+3 c^2 d^3\right ) a^4+2 b c d^2 \left (9 c^2-5 d^2\right ) a^3-2 b^2 d \left (9 c^4-21 d^2 c^2+8 d^4\right ) a^2+2 b^3 c \left (3 c^4-15 d^2 c^2+8 d^4\right ) a+15 b^4 d \left (c^2-d^2\right )^2-b \left (8 a^3 c d^4-8 a b^2 c d^4-4 a^2 b \left (5 c^2-3 d^2\right ) d^3-b^3 \left (15 d^5-26 c^2 d^3+3 c^4 d\right )\right ) \sin (e+f x)^2-2 d \left (4 c d^3 a^4-7 b d^2 \left (c^2-d^2\right ) a^3-b^2 c d \left (9 c^2-d^2\right ) a^2-b^3 \left (3 c^4-13 d^2 c^2+10 d^4\right ) a+b^4 c d \left (9 c^2-5 d^2\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{\left (c^2-d^2\right ) (b c-a d)}-\frac {2 \left (8 a^3 c d^4-4 a^2 b d^3 \left (5 c^2-3 d^2\right )-8 a b^2 c d^4-b^3 \left (3 c^4 d-26 c^2 d^3+15 d^5\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {\frac {\frac {-\left (\left (8 a^3 c d^3-4 a^2 b d^2 \left (5 c^2-3 d^2\right )-8 a b^2 c d^3-\left (b^3 \left (3 c^4-26 c^2 d^2+15 d^4\right )\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx\right )-\frac {\int \frac {d (b c-a d) \left (c^2-d^2\right ) \left (\left (3 c^2-5 d^2\right ) b^2+2 a^2 d^2\right ) \sin (e+f x) b^2+d \left (c^2-d^2\right ) \left (-2 d^3 a^4+2 b c d^2 a^3+2 b^2 d \left (9 c^2-8 d^2\right ) a^2-b^3 c \left (3 c^2-d^2\right ) a-15 b^4 d \left (c^2-d^2\right )\right ) b}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}}{\left (c^2-d^2\right ) (b c-a d)}-\frac {2 \left (8 a^3 c d^4-4 a^2 b d^3 \left (5 c^2-3 d^2\right )-8 a b^2 c d^4-b^3 \left (3 c^4 d-26 c^2 d^3+15 d^5\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {-\left (\left (8 a^3 c d^3-4 a^2 b d^2 \left (5 c^2-3 d^2\right )-8 a b^2 c d^3-\left (b^3 \left (3 c^4-26 c^2 d^2+15 d^4\right )\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx\right )-\frac {\int \frac {d (b c-a d) \left (c^2-d^2\right ) \left (\left (3 c^2-5 d^2\right ) b^2+2 a^2 d^2\right ) \sin (e+f x) b^2+d \left (c^2-d^2\right ) \left (-2 d^3 a^4+2 b c d^2 a^3+2 b^2 d \left (9 c^2-8 d^2\right ) a^2-b^3 c \left (3 c^2-d^2\right ) a-15 b^4 d \left (c^2-d^2\right )\right ) b}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}}{\left (c^2-d^2\right ) (b c-a d)}-\frac {2 \left (8 a^3 c d^4-4 a^2 b d^3 \left (5 c^2-3 d^2\right )-8 a b^2 c d^4-b^3 \left (3 c^4 d-26 c^2 d^3+15 d^5\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\left (8 a^3 c d^3-4 a^2 b d^2 \left (5 c^2-3 d^2\right )-8 a b^2 c d^3-\left (b^3 \left (3 c^4-26 c^2 d^2+15 d^4\right )\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\int \frac {d (b c-a d) \left (c^2-d^2\right ) \left (\left (3 c^2-5 d^2\right ) b^2+2 a^2 d^2\right ) \sin (e+f x) b^2+d \left (c^2-d^2\right ) \left (-2 d^3 a^4+2 b c d^2 a^3+2 b^2 d \left (9 c^2-8 d^2\right ) a^2-b^3 c \left (3 c^2-d^2\right ) a-15 b^4 d \left (c^2-d^2\right )\right ) b}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}}{\left (c^2-d^2\right ) (b c-a d)}-\frac {2 \left (8 a^3 c d^4-4 a^2 b d^3 \left (5 c^2-3 d^2\right )-8 a b^2 c d^4-b^3 \left (3 c^4 d-26 c^2 d^3+15 d^5\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\left (8 a^3 c d^3-4 a^2 b d^2 \left (5 c^2-3 d^2\right )-8 a b^2 c d^3-\left (b^3 \left (3 c^4-26 c^2 d^2+15 d^4\right )\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\int \frac {d (b c-a d) \left (c^2-d^2\right ) \left (\left (3 c^2-5 d^2\right ) b^2+2 a^2 d^2\right ) \sin (e+f x) b^2+d \left (c^2-d^2\right ) \left (-2 d^3 a^4+2 b c d^2 a^3+2 b^2 d \left (9 c^2-8 d^2\right ) a^2-b^3 c \left (3 c^2-d^2\right ) a-15 b^4 d \left (c^2-d^2\right )\right ) b}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}}{\left (c^2-d^2\right ) (b c-a d)}-\frac {2 \left (8 a^3 c d^4-4 a^2 b d^3 \left (5 c^2-3 d^2\right )-8 a b^2 c d^4-b^3 \left (3 c^4 d-26 c^2 d^3+15 d^5\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\int \frac {d (b c-a d) \left (c^2-d^2\right ) \left (\left (3 c^2-5 d^2\right ) b^2+2 a^2 d^2\right ) \sin (e+f x) b^2+d \left (c^2-d^2\right ) \left (-2 d^3 a^4+2 b c d^2 a^3+2 b^2 d \left (9 c^2-8 d^2\right ) a^2-b^3 c \left (3 c^2-d^2\right ) a-15 b^4 d \left (c^2-d^2\right )\right ) b}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {2 \left (8 a^3 c d^3-4 a^2 b d^2 \left (5 c^2-3 d^2\right )-8 a b^2 c d^3-\left (b^3 \left (3 c^4-26 c^2 d^2+15 d^4\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}-\frac {2 \left (8 a^3 c d^4-4 a^2 b d^3 \left (5 c^2-3 d^2\right )-8 a b^2 c d^4-b^3 \left (3 c^4 d-26 c^2 d^3+15 d^5\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {\frac {\frac {-\frac {b d \left (c^2-d^2\right ) (b c-a d) \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx-3 b^3 d \left (c^2-d^2\right )^2 \left (-7 a^2 d+2 a b c+5 b^2 d\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {2 \left (8 a^3 c d^3-4 a^2 b d^2 \left (5 c^2-3 d^2\right )-8 a b^2 c d^3-\left (b^3 \left (3 c^4-26 c^2 d^2+15 d^4\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}-\frac {2 \left (8 a^3 c d^4-4 a^2 b d^3 \left (5 c^2-3 d^2\right )-8 a b^2 c d^4-b^3 \left (3 c^4 d-26 c^2 d^3+15 d^5\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {-\frac {b d \left (c^2-d^2\right ) (b c-a d) \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx-3 b^3 d \left (c^2-d^2\right )^2 \left (-7 a^2 d+2 a b c+5 b^2 d\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {2 \left (8 a^3 c d^3-4 a^2 b d^2 \left (5 c^2-3 d^2\right )-8 a b^2 c d^3-\left (b^3 \left (3 c^4-26 c^2 d^2+15 d^4\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}-\frac {2 \left (8 a^3 c d^4-4 a^2 b d^3 \left (5 c^2-3 d^2\right )-8 a b^2 c d^4-b^3 \left (3 c^4 d-26 c^2 d^3+15 d^5\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\frac {b d \left (c^2-d^2\right ) (b c-a d) \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}-3 b^3 d \left (c^2-d^2\right )^2 \left (-7 a^2 d+2 a b c+5 b^2 d\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {2 \left (8 a^3 c d^3-4 a^2 b d^2 \left (5 c^2-3 d^2\right )-8 a b^2 c d^3-\left (b^3 \left (3 c^4-26 c^2 d^2+15 d^4\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}-\frac {2 \left (8 a^3 c d^4-4 a^2 b d^3 \left (5 c^2-3 d^2\right )-8 a b^2 c d^4-b^3 \left (3 c^4 d-26 c^2 d^3+15 d^5\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\frac {b d \left (c^2-d^2\right ) (b c-a d) \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}-3 b^3 d \left (c^2-d^2\right )^2 \left (-7 a^2 d+2 a b c+5 b^2 d\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {2 \left (8 a^3 c d^3-4 a^2 b d^2 \left (5 c^2-3 d^2\right )-8 a b^2 c d^3-\left (b^3 \left (3 c^4-26 c^2 d^2+15 d^4\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}-\frac {2 \left (8 a^3 c d^4-4 a^2 b d^3 \left (5 c^2-3 d^2\right )-8 a b^2 c d^4-b^3 \left (3 c^4 d-26 c^2 d^3+15 d^5\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\frac {2 b d \left (c^2-d^2\right ) (b c-a d) \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}-3 b^3 d \left (c^2-d^2\right )^2 \left (-7 a^2 d+2 a b c+5 b^2 d\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {2 \left (8 a^3 c d^3-4 a^2 b d^2 \left (5 c^2-3 d^2\right )-8 a b^2 c d^3-\left (b^3 \left (3 c^4-26 c^2 d^2+15 d^4\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}-\frac {2 \left (8 a^3 c d^4-4 a^2 b d^3 \left (5 c^2-3 d^2\right )-8 a b^2 c d^4-b^3 \left (3 c^4 d-26 c^2 d^3+15 d^5\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\frac {2 b d \left (c^2-d^2\right ) (b c-a d) \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}-\frac {3 b^3 d \left (c^2-d^2\right )^2 \left (-7 a^2 d+2 a b c+5 b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}}{b d}-\frac {2 \left (8 a^3 c d^3-4 a^2 b d^2 \left (5 c^2-3 d^2\right )-8 a b^2 c d^3-\left (b^3 \left (3 c^4-26 c^2 d^2+15 d^4\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}-\frac {2 \left (8 a^3 c d^4-4 a^2 b d^3 \left (5 c^2-3 d^2\right )-8 a b^2 c d^4-b^3 \left (3 c^4 d-26 c^2 d^3+15 d^5\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\frac {2 b d \left (c^2-d^2\right ) (b c-a d) \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}-\frac {3 b^3 d \left (c^2-d^2\right )^2 \left (-7 a^2 d+2 a b c+5 b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}}{b d}-\frac {2 \left (8 a^3 c d^3-4 a^2 b d^2 \left (5 c^2-3 d^2\right )-8 a b^2 c d^3-\left (b^3 \left (3 c^4-26 c^2 d^2+15 d^4\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}-\frac {2 \left (8 a^3 c d^4-4 a^2 b d^3 \left (5 c^2-3 d^2\right )-8 a b^2 c d^4-b^3 \left (3 c^4 d-26 c^2 d^3+15 d^5\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}+\frac {\frac {2 d \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}+\frac {\frac {-\frac {\frac {2 b d \left (c^2-d^2\right ) (b c-a d) \left (2 a^2 d^2+b^2 \left (3 c^2-5 d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}-\frac {6 b^3 d \left (c^2-d^2\right )^2 \left (-7 a^2 d+2 a b c+5 b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f (a+b) \sqrt {c+d \sin (e+f x)}}}{b d}-\frac {2 \left (8 a^3 c d^3-4 a^2 b d^2 \left (5 c^2-3 d^2\right )-8 a b^2 c d^3-\left (b^3 \left (3 c^4-26 c^2 d^2+15 d^4\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}-\frac {2 \left (8 a^3 c d^4-4 a^2 b d^3 \left (5 c^2-3 d^2\right )-8 a b^2 c d^4-b^3 \left (3 c^4 d-26 c^2 d^3+15 d^5\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right ) (b c-a d)}}{2 \left (a^2-b^2\right ) (b c-a d)}\) |
Input:
Int[1/((a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(5/2)),x]
Output:
(b^2*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])*(c + d* Sin[e + f*x])^(3/2)) + ((2*d*(2*a^2*d^2 + b^2*(3*c^2 - 5*d^2))*Cos[e + f*x ])/(3*(b*c - a*d)*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + ((-2*(8*a^3* c*d^4 - 8*a*b^2*c*d^4 - 4*a^2*b*d^3*(5*c^2 - 3*d^2) - b^3*(3*c^4*d - 26*c^ 2*d^3 + 15*d^5))*Cos[e + f*x])/((b*c - a*d)*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + ((-2*(8*a^3*c*d^3 - 8*a*b^2*c*d^3 - 4*a^2*b*d^2*(5*c^2 - 3*d^2 ) - b^3*(3*c^4 - 26*c^2*d^2 + 15*d^4))*EllipticE[(e - Pi/2 + f*x)/2, (2*d) /(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - ((2*b*d*(b*c - a*d)*(c^2 - d^2)*(2*a^2*d^2 + b^2*(3*c^2 - 5*d^2))*Ellip ticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] )/(f*Sqrt[c + d*Sin[e + f*x]]) - (6*b^3*d*(2*a*b*c - 7*a^2*d + 5*b^2*d)*(c ^2 - d^2)^2*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*S qrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*f*Sqrt[c + d*Sin[e + f*x]]))/( b*d))/((b*c - a*d)*(c^2 - d^2)))/(3*(b*c - a*d)*(c^2 - d^2)))/(2*(a^2 - b^ 2)*(b*c - a*d))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f* x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2 ))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))*Sin[e + f*x] - b^2*d*(m + n + 3)*Si n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a* d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(IntegerQ[2* n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) || EqQ[a, 0])))
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], 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[c + d, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt [c + d*Sin[e + f*x]] Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + d))*Sin[e + f*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[c + d, 0]
Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[ B/d Int[(a + b*Sin[e + f*x])^m, x], x] - Simp[(B*c - A*d)/d Int[(a + b* Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
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[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d) Int[Sqrt[a + b*Sin[e + f*x]], x] , x] - Simp[1/(b*d) Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre eQ[{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]
Leaf count of result is larger than twice the leaf count of optimal. \(1722\) vs. \(2(644)=1288\).
Time = 4.64 (sec) , antiderivative size = 1723, normalized size of antiderivative = 2.61
Input:
int(1/(a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(5/2),x,method=_RETURNVERBOSE)
Output:
(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*(b^2/(a*d-b*c)^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) )-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)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*si n(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))-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)*(-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*si n(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)*(-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+a/b),((c-d)/(c+d))^(1/2)))+d^2/(a*d-b*c)^2*(2/3/(c^2-d^2)/d*(-(-c-d*sin (f*x+e))*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+c/d)^2+8/3*d*cos(f*x+e)^2/(c^2-d^ 2)^2*c/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)+2*(3*c^2+d^2)/(3*c^4-6*c^2* d^2+3*d^4)*(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))+8/3*c*d /(c^2-d^2)^2*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(...
Timed out. \[ \int \frac {1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(5/2),x, algorithm="fricas ")
Output:
Timed out
Timed out. \[ \int \frac {1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(a+b*sin(f*x+e))**2/(c+d*sin(f*x+e))**(5/2),x)
Output:
Timed out
Timed out. \[ \int \frac {1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(5/2),x, algorithm="maxima ")
Output:
Timed out
\[ \int \frac {1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(5/2),x, algorithm="giac")
Output:
integrate(1/((b*sin(f*x + e) + a)^2*(d*sin(f*x + e) + c)^(5/2)), x)
Timed out. \[ \int \frac {1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx=\int \frac {1}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \] Input:
int(1/((a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^(5/2)),x)
Output:
int(1/((a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^(5/2)), x)
\[ \int \frac {1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx=\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}}{\sin \left (f x +e \right )^{5} b^{2} d^{3}+2 \sin \left (f x +e \right )^{4} a b \,d^{3}+3 \sin \left (f x +e \right )^{4} b^{2} c \,d^{2}+\sin \left (f x +e \right )^{3} a^{2} d^{3}+6 \sin \left (f x +e \right )^{3} a b c \,d^{2}+3 \sin \left (f x +e \right )^{3} b^{2} c^{2} d +3 \sin \left (f x +e \right )^{2} a^{2} c \,d^{2}+6 \sin \left (f x +e \right )^{2} a b \,c^{2} d +\sin \left (f x +e \right )^{2} b^{2} c^{3}+3 \sin \left (f x +e \right ) a^{2} c^{2} d +2 \sin \left (f x +e \right ) a b \,c^{3}+a^{2} c^{3}}d x \] Input:
int(1/(a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(5/2),x)
Output:
int(sqrt(sin(e + f*x)*d + c)/(sin(e + f*x)**5*b**2*d**3 + 2*sin(e + f*x)** 4*a*b*d**3 + 3*sin(e + f*x)**4*b**2*c*d**2 + sin(e + f*x)**3*a**2*d**3 + 6 *sin(e + f*x)**3*a*b*c*d**2 + 3*sin(e + f*x)**3*b**2*c**2*d + 3*sin(e + f* x)**2*a**2*c*d**2 + 6*sin(e + f*x)**2*a*b*c**2*d + sin(e + f*x)**2*b**2*c* *3 + 3*sin(e + f*x)*a**2*c**2*d + 2*sin(e + f*x)*a*b*c**3 + a**2*c**3),x)