Integrand size = 27, antiderivative size = 682 \[ \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx=-\frac {d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-b^4 d \left (7 c^2-15 d^2\right )-6 a b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{4 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}+\frac {b^2 \left (6 a b c-11 a^2 d+5 b^2 d\right ) \cos (e+f x)}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-b^4 d \left (7 c^2-15 d^2\right )-6 a b^3 c \left (c^2-d^2\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {b \left (6 a b c-11 a^2 d+5 b^2 d\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f \sqrt {c+d \sin (e+f x)}}-\frac {b \left (28 a^3 b c d-4 a b^3 c d-35 a^4 d^2-2 a^2 b^2 \left (4 c^2-19 d^2\right )-b^4 \left (4 c^2+15 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{4 (a-b)^2 (a+b)^3 (b c-a d)^3 f \sqrt {c+d \sin (e+f x)}} \] Output:
-1/4*d*(8*a^4*d^3+a^2*b^2*d*(13*c^2-29*d^2)-b^4*d*(7*c^2-15*d^2)-6*a*b^3*c *(c^2-d^2))*cos(f*x+e)/(a^2-b^2)^2/(-a*d+b*c)^3/(c^2-d^2)/f/(c+d*sin(f*x+e ))^(1/2)+1/2*b^2*cos(f*x+e)/(a^2-b^2)/(-a*d+b*c)/f/(a+b*sin(f*x+e))^2/(c+d *sin(f*x+e))^(1/2)+1/4*b^2*(-11*a^2*d+6*a*b*c+5*b^2*d)*cos(f*x+e)/(a^2-b^2 )^2/(-a*d+b*c)^2/f/(a+b*sin(f*x+e))/(c+d*sin(f*x+e))^(1/2)+1/4*(8*a^4*d^3+ a^2*b^2*d*(13*c^2-29*d^2)-b^4*d*(7*c^2-15*d^2)-6*a*b^3*c*(c^2-d^2))*Ellipt icE(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)/(a^2-b^2)^2/(-a*d+b*c)^3/(c^2-d^2)/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-1/ 4*b*(-11*a^2*d+6*a*b*c+5*b^2*d)*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)^2/(-a*d+b*c)^ 2/f/(c+d*sin(f*x+e))^(1/2)+1/4*b*(28*a^3*b*c*d-4*a*b^3*c*d-35*a^4*d^2-2*a^ 2*b^2*(4*c^2-19*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/(a+b)^3/(-a*d+b*c)^3/f/(c+d*sin(f*x+e))^(1/2)
Result contains complex when optimal does not.
Time = 7.94 (sec) , antiderivative size = 1318, normalized size of antiderivative = 1.93 \[ \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx =\text {Too large to display} \] Input:
Integrate[1/((a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(3/2)),x]
Output:
(Sqrt[c + d*Sin[e + f*x]]*((b^3*Cos[e + f*x])/(2*(a^2 - b^2)*(-(b*c) + a*d )^2*(a + b*Sin[e + f*x])^2) - (6*a*b^4*c*Cos[e + f*x] - 13*a^2*b^3*d*Cos[e + f*x] + 7*b^5*d*Cos[e + f*x])/(4*(a^2 - b^2)^2*(-(b*c) + a*d)^3*(a + b*S in[e + f*x])) - (2*d^4*Cos[e + f*x])/((b*c - a*d)^3*(c^2 - d^2)*(c + d*Sin [e + f*x]))))/f + ((-2*(-16*a^2*b^3*c^4 - 8*b^5*c^4 + 48*a^3*b^2*c^3*d - 1 8*a*b^4*c^3*d - 48*a^4*b*c^2*d^2 + 95*a^2*b^3*c^2*d^2 - 29*b^5*c^2*d^2 + 1 6*a^5*c*d^3 - 80*a^3*b^2*c*d^3 + 34*a*b^4*c*d^3 + 56*a^4*b*d^4 - 95*a^2*b^ 3*d^4 + 45*b^5*d^4)*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^3*c^3*d - 4*b^5*c^3*d + 48*a^3*b^2*c^2*d^2 - 24*a *b^4*c^2*d^2 + 16*a^4*b*c*d^3 - 12*a^2*b^3*c*d^3 + 20*b^5*c*d^3 + 16*a^5*d ^4 - 80*a^3*b^2*d^4 + 40*a*b^4*d^4)*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*Si n[e + f*x])^2)/d^2)]) - ((2*I)*(6*a*b^4*c^3*d - 13*a^2*b^3*c^2*d^2 + 7*b^5 *c^2*d^2 - 6*a*b^4*c*d^3 - 8*a^4*b*d^4 + 29*a^2*b^3*d^4 - 15*b^5*d^4)*C...
Time = 6.52 (sec) , antiderivative size = 705, normalized size of antiderivative = 1.03, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {3042, 3281, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3538, 25, 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.
| \(\displaystyle \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3281 |
| \(\displaystyle \frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}-\frac {\int -\frac {-4 d a^2+4 b c a-3 b^2 d \sin ^2(e+f x)+5 b^2 d-2 b (b c-2 a d) \sin (e+f x)}{2 (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}}dx}{2 \left (a^2-b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {-4 d a^2+4 b c a-3 b^2 d \sin ^2(e+f x)+5 b^2 d-2 b (b c-2 a d) \sin (e+f x)}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}}dx}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {-4 d a^2+4 b c a-3 b^2 d \sin (e+f x)^2+5 b^2 d-2 b (b c-2 a d) \sin (e+f x)}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}}dx}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {b^2 \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\int \frac {-8 d^2 a^4+16 b c d a^3-b^2 \left (8 c^2-29 d^2\right ) a^2-10 b^3 c d a+b^2 d \left (-11 d a^2+6 b c a+5 b^2 d\right ) \sin ^2(e+f x)-b^4 \left (4 c^2+15 d^2\right )-2 b d \left (-8 d a^3+3 b c a^2+2 b^2 d a+3 b^3 c\right ) \sin (e+f x)}{2 (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}dx}{\left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {b^2 \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\int \frac {-8 d^2 a^4+16 b c d a^3-b^2 \left (8 c^2-29 d^2\right ) a^2-10 b^3 c d a+b^2 d \left (-11 d a^2+6 b c a+5 b^2 d\right ) \sin ^2(e+f x)-b^4 \left (4 c^2+15 d^2\right )-2 b d \left (-8 d a^3+3 b c a^2+2 b^2 d a+3 b^3 c\right ) \sin (e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}dx}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {b^2 \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\int \frac {-8 d^2 a^4+16 b c d a^3-b^2 \left (8 c^2-29 d^2\right ) a^2-10 b^3 c d a+b^2 d \left (-11 d a^2+6 b c a+5 b^2 d\right ) \sin (e+f x)^2-b^4 \left (4 c^2+15 d^2\right )-2 b d \left (-8 d a^3+3 b c a^2+2 b^2 d a+3 b^3 c\right ) \sin (e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}dx}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {b^2 \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {2 \int \frac {8 c d^3 a^5-24 b d^2 \left (c^2-d^2\right ) a^4+8 b^2 c d \left (3 c^2-5 d^2\right ) a^3-b^3 \left (8 c^4-41 d^2 c^2+33 d^4\right ) a^2-2 b^4 c d \left (3 c^2-7 d^2\right ) a+b d \left (8 d^3 a^4+b^2 d \left (13 c^2-29 d^2\right ) a^2-6 b^3 c \left (c^2-d^2\right ) a-b^4 d \left (7 c^2-15 d^2\right )\right ) \sin ^2(e+f x)-b^5 \left (4 c^4+11 d^2 c^2-15 d^4\right )+2 d \left (4 d^3 a^5+4 b c d^2 a^4+4 b^2 d \left (3 c^2-5 d^2\right ) a^3-b^3 c \left (5 c^2+3 d^2\right ) a^2-2 b^4 d \left (3 c^2-5 d^2\right ) a-b^5 c \left (c^2-5 d^2\right )\right ) \sin (e+f x)}{2 (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{\left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {b^2 \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\int \frac {8 c d^3 a^5-24 b d^2 \left (c^2-d^2\right ) a^4+8 b^2 c d \left (3 c^2-5 d^2\right ) a^3-b^3 \left (8 c^4-41 d^2 c^2+33 d^4\right ) a^2-2 b^4 c d \left (3 c^2-7 d^2\right ) a+b d \left (8 d^3 a^4+b^2 d \left (13 c^2-29 d^2\right ) a^2-6 b^3 c \left (c^2-d^2\right ) a-b^4 d \left (7 c^2-15 d^2\right )\right ) \sin ^2(e+f x)-b^5 \left (4 c^4+11 d^2 c^2-15 d^4\right )+2 d \left (4 d^3 a^5+4 b c d^2 a^4+4 b^2 d \left (3 c^2-5 d^2\right ) a^3-b^3 c \left (5 c^2+3 d^2\right ) a^2-2 b^4 d \left (3 c^2-5 d^2\right ) a-b^5 c \left (c^2-5 d^2\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{\left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {b^2 \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\int \frac {8 c d^3 a^5-24 b d^2 \left (c^2-d^2\right ) a^4+8 b^2 c d \left (3 c^2-5 d^2\right ) a^3-b^3 \left (8 c^4-41 d^2 c^2+33 d^4\right ) a^2-2 b^4 c d \left (3 c^2-7 d^2\right ) a+b d \left (8 d^3 a^4+b^2 d \left (13 c^2-29 d^2\right ) a^2-6 b^3 c \left (c^2-d^2\right ) a-b^4 d \left (7 c^2-15 d^2\right )\right ) \sin (e+f x)^2-b^5 \left (4 c^4+11 d^2 c^2-15 d^4\right )+2 d \left (4 d^3 a^5+4 b c d^2 a^4+4 b^2 d \left (3 c^2-5 d^2\right ) a^3-b^3 c \left (5 c^2+3 d^2\right ) a^2-2 b^4 d \left (3 c^2-5 d^2\right ) a-b^5 c \left (c^2-5 d^2\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{\left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {\frac {b^2 \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx-\frac {\int -\frac {d (b c-a d) \left (-11 d a^2+6 b c a+5 b^2 d\right ) \left (c^2-d^2\right ) \sin (e+f x) b^3+d \left (c^2-d^2\right ) \left (-24 d^2 a^4+11 b c d a^3-b^2 \left (2 c^2-33 d^2\right ) a^2+b^3 c d a-b^4 \left (4 c^2+15 d^2\right )\right ) b^2}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}}{\left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {b^2 \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx+\frac {\int \frac {d (b c-a d) \left (-11 d a^2+6 b c a+5 b^2 d\right ) \left (c^2-d^2\right ) \sin (e+f x) b^3+d \left (c^2-d^2\right ) \left (-24 d^2 a^4+11 b c d a^3-b^2 \left (2 c^2-33 d^2\right ) a^2+b^3 c d a-b^4 \left (4 c^2+15 d^2\right )\right ) b^2}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}}{\left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {b^2 \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx+\frac {\int \frac {d (b c-a d) \left (-11 d a^2+6 b c a+5 b^2 d\right ) \left (c^2-d^2\right ) \sin (e+f x) b^3+d \left (c^2-d^2\right ) \left (-24 d^2 a^4+11 b c d a^3-b^2 \left (2 c^2-33 d^2\right ) a^2+b^3 c d a-b^4 \left (4 c^2+15 d^2\right )\right ) b^2}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}}{\left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {b^2 \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\frac {\left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\int \frac {d (b c-a d) \left (-11 d a^2+6 b c a+5 b^2 d\right ) \left (c^2-d^2\right ) \sin (e+f x) b^3+d \left (c^2-d^2\right ) \left (-24 d^2 a^4+11 b c d a^3-b^2 \left (2 c^2-33 d^2\right ) a^2+b^3 c d a-b^4 \left (4 c^2+15 d^2\right )\right ) b^2}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}}{\left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {b^2 \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\frac {\left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\int \frac {d (b c-a d) \left (-11 d a^2+6 b c a+5 b^2 d\right ) \left (c^2-d^2\right ) \sin (e+f x) b^3+d \left (c^2-d^2\right ) \left (-24 d^2 a^4+11 b c d a^3-b^2 \left (2 c^2-33 d^2\right ) a^2+b^3 c d a-b^4 \left (4 c^2+15 d^2\right )\right ) b^2}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}}{\left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {b^2 \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\frac {\int \frac {d (b c-a d) \left (-11 d a^2+6 b c a+5 b^2 d\right ) \left (c^2-d^2\right ) \sin (e+f x) b^3+d \left (c^2-d^2\right ) \left (-24 d^2 a^4+11 b c d a^3-b^2 \left (2 c^2-33 d^2\right ) a^2+b^3 c d a-b^4 \left (4 c^2+15 d^2\right )\right ) b^2}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}+\frac {2 \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {\frac {b^2 \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\frac {b^2 d \left (c^2-d^2\right ) (b c-a d) \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx+b^2 d \left (c^2-d^2\right ) \left (-35 a^4 d^2+28 a^3 b c d-2 a^2 b^2 \left (4 c^2-19 d^2\right )-4 a b^3 c d-b^4 \left (4 c^2+15 d^2\right )\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}+\frac {2 \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {b^2 \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\frac {b^2 d \left (c^2-d^2\right ) (b c-a d) \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx+b^2 d \left (c^2-d^2\right ) \left (-35 a^4 d^2+28 a^3 b c d-2 a^2 b^2 \left (4 c^2-19 d^2\right )-4 a b^3 c d-b^4 \left (4 c^2+15 d^2\right )\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}+\frac {2 \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {b^2 \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\frac {\frac {b^2 d \left (c^2-d^2\right ) (b c-a d) \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}+b^2 d \left (c^2-d^2\right ) \left (-35 a^4 d^2+28 a^3 b c d-2 a^2 b^2 \left (4 c^2-19 d^2\right )-4 a b^3 c d-b^4 \left (4 c^2+15 d^2\right )\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}+\frac {2 \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {b^2 \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\frac {\frac {b^2 d \left (c^2-d^2\right ) (b c-a d) \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}+b^2 d \left (c^2-d^2\right ) \left (-35 a^4 d^2+28 a^3 b c d-2 a^2 b^2 \left (4 c^2-19 d^2\right )-4 a b^3 c d-b^4 \left (4 c^2+15 d^2\right )\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}+\frac {2 \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {b^2 \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\frac {b^2 d \left (c^2-d^2\right ) \left (-35 a^4 d^2+28 a^3 b c d-2 a^2 b^2 \left (4 c^2-19 d^2\right )-4 a b^3 c d-b^4 \left (4 c^2+15 d^2\right )\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx+\frac {2 b^2 d \left (c^2-d^2\right ) (b c-a d) \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}}{b d}+\frac {2 \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {\frac {b^2 \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\frac {\frac {b^2 d \left (c^2-d^2\right ) \left (-35 a^4 d^2+28 a^3 b c d-2 a^2 b^2 \left (4 c^2-19 d^2\right )-4 a b^3 c d-b^4 \left (4 c^2+15 d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}+\frac {2 b^2 d \left (c^2-d^2\right ) (b c-a d) \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}}{b d}+\frac {2 \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {b^2 \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\frac {\frac {b^2 d \left (c^2-d^2\right ) \left (-35 a^4 d^2+28 a^3 b c d-2 a^2 b^2 \left (4 c^2-19 d^2\right )-4 a b^3 c d-b^4 \left (4 c^2+15 d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}+\frac {2 b^2 d \left (c^2-d^2\right ) (b c-a d) \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}}{b d}+\frac {2 \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}+\frac {\frac {b^2 \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {2 d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}+\frac {\frac {2 \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\frac {2 b^2 d \left (c^2-d^2\right ) (b c-a d) \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}+\frac {2 b^2 d \left (c^2-d^2\right ) \left (-35 a^4 d^2+28 a^3 b c d-2 a^2 b^2 \left (4 c^2-19 d^2\right )-4 a b^3 c d-b^4 \left (4 c^2+15 d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f (a+b) \sqrt {c+d \sin (e+f x)}}}{b d}}{\left (c^2-d^2\right ) (b c-a d)}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}\) |
Input:
Int[1/((a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(3/2)),x]
Output:
(b^2*Cos[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^2*Sqr t[c + d*Sin[e + f*x]]) + ((b^2*(6*a*b*c - 11*a^2*d + 5*b^2*d)*Cos[e + f*x] )/((a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]] ) - ((2*d*(8*a^4*d^3 + a^2*b^2*d*(13*c^2 - 29*d^2) - b^4*d*(7*c^2 - 15*d^2 ) - 6*a*b^3*c*(c^2 - d^2))*Cos[e + f*x])/((b*c - a*d)*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + ((2*(8*a^4*d^3 + a^2*b^2*d*(13*c^2 - 29*d^2) - b^4*d *(7*c^2 - 15*d^2) - 6*a*b^3*c*(c^2 - d^2))*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^2*d*(b*c - a*d)*(6*a*b*c - 11*a^2*d + 5*b^2*d)*(c^2 - d^2)*El lipticF[(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]]) + (2*b^2*d*(c^2 - d^2)*(28*a^3*b*c*d - 4 *a*b^3*c*d - 35*a^4*d^2 - 2*a^2*b^2*(4*c^2 - 19*d^2) - b^4*(4*c^2 + 15*d^2 ))*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)*f*Sqrt[c + d*Sin[e + f*x]]))/(b*d))/((b *c - a*d)*(c^2 - d^2)))/(2*(a^2 - b^2)*(b*c - a*d)))/(4*(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. \(2089\) vs. \(2(660)=1320\).
Time = 4.93 (sec) , antiderivative size = 2090, normalized size of antiderivative = 3.06
Input:
int(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(3/2),x,method=_RETURNVERBOSE)
Output:
(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*(d^3/(a*d-b*c)^3*(2*d*cos(f*x+e)^2 /(c^2-d^2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)+2/(c^2-d^2)*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)*EllipticF(((c+d *sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+2/(c^2-d^2)*d*(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*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))))-b/(a*d-b*c)*(-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))*c os(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))...
Timed out. \[ \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(3/2),x, algorithm="fricas ")
Output:
Timed out
Timed out. \[ \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(a+b*sin(f*x+e))**3/(c+d*sin(f*x+e))**(3/2),x)
Output:
Timed out
Timed out. \[ \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(3/2),x, algorithm="maxima ")
Output:
Timed out
\[ \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(3/2),x, algorithm="giac")
Output:
integrate(1/((b*sin(f*x + e) + a)^3*(d*sin(f*x + e) + c)^(3/2)), x)
Timed out. \[ \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^3\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \] Input:
int(1/((a + b*sin(e + f*x))^3*(c + d*sin(e + f*x))^(3/2)),x)
Output:
int(1/((a + b*sin(e + f*x))^3*(c + d*sin(e + f*x))^(3/2)), x)
\[ \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx=\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}}{\sin \left (f x +e \right )^{5} b^{3} d^{2}+3 \sin \left (f x +e \right )^{4} a \,b^{2} d^{2}+2 \sin \left (f x +e \right )^{4} b^{3} c d +3 \sin \left (f x +e \right )^{3} a^{2} b \,d^{2}+6 \sin \left (f x +e \right )^{3} a \,b^{2} c d +\sin \left (f x +e \right )^{3} b^{3} c^{2}+\sin \left (f x +e \right )^{2} a^{3} d^{2}+6 \sin \left (f x +e \right )^{2} a^{2} b c d +3 \sin \left (f x +e \right )^{2} a \,b^{2} c^{2}+2 \sin \left (f x +e \right ) a^{3} c d +3 \sin \left (f x +e \right ) a^{2} b \,c^{2}+a^{3} c^{2}}d x \] Input:
int(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(3/2),x)
Output:
int(sqrt(sin(e + f*x)*d + c)/(sin(e + f*x)**5*b**3*d**2 + 3*sin(e + f*x)** 4*a*b**2*d**2 + 2*sin(e + f*x)**4*b**3*c*d + 3*sin(e + f*x)**3*a**2*b*d**2 + 6*sin(e + f*x)**3*a*b**2*c*d + sin(e + f*x)**3*b**3*c**2 + sin(e + f*x) **2*a**3*d**2 + 6*sin(e + f*x)**2*a**2*b*c*d + 3*sin(e + f*x)**2*a*b**2*c* *2 + 2*sin(e + f*x)*a**3*c*d + 3*sin(e + f*x)*a**2*b*c**2 + a**3*c**2),x)