Integrand size = 29, antiderivative size = 727 \[ \int \frac {(3+b \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{5/2}} \, dx=\frac {2 (b c-3 d)^2 \cos (e+f x) \sqrt {3+b \sin (e+f x)}}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {2 (3-b) \sqrt {3+b} \left (3 b c^2+12 c d-7 b d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{3 (c-d)^2 d^2 (c+d)^{3/2} f}-\frac {2 \sqrt {3+b} \left (9 d^2 (3 c+d)+3 b d \left (3 c^2-4 c d-7 d^2\right )+b^2 \left (3 c^3-6 c^2 d-2 c d^2+9 d^3\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{3 (c-d)^2 d^3 (c+d)^{3/2} f}+\frac {2 b^2 \sqrt {3+b} \operatorname {EllipticPi}\left (\frac {(3+b) d}{b (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{d^3 \sqrt {c+d} f} \]
2/3*(-a*d+b*c)^2*cos(f*x+e)*(a+b*sin(f*x+e))^(1/2)/d/(c^2-d^2)/f/(c+d*sin( f*x+e))^(3/2)+2/3*(a-b)*(4*a*c*d+3*b*c^2-7*b*d^2)*EllipticE((c+d)^(1/2)*(a +b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(c+d*sin(f*x+e))^(1/2),((a+b)*(c-d)/(a-b) /(c+d))^(1/2))*sec(f*x+e)*(c+d*sin(f*x+e))*(a+b)^(1/2)*((-a*d+b*c)*(1-sin( f*x+e))/(a+b)/(c+d*sin(f*x+e)))^(1/2)*(-(-a*d+b*c)*(1+sin(f*x+e))/(a-b)/(c +d*sin(f*x+e)))^(1/2)/(c-d)^2/d^2/(c+d)^(3/2)/f-2/3*(a^2*d^2*(3*c+d)+a*b*d *(3*c^2-4*c*d-7*d^2)+b^2*(3*c^3-6*c^2*d-2*c*d^2+9*d^3))*EllipticF((c+d)^(1 /2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(c+d*sin(f*x+e))^(1/2),((a+b)*(c-d) /(a-b)/(c+d))^(1/2))*sec(f*x+e)*(c+d*sin(f*x+e))*(a+b)^(1/2)*((-a*d+b*c)*( 1-sin(f*x+e))/(a+b)/(c+d*sin(f*x+e)))^(1/2)*(-(-a*d+b*c)*(1+sin(f*x+e))/(a -b)/(c+d*sin(f*x+e)))^(1/2)/(c-d)^2/d^3/(c+d)^(3/2)/f+2*b^2*EllipticPi((c+ d)^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(c+d*sin(f*x+e))^(1/2),(a+b)*d /b/(c+d),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*sec(f*x+e)*(c+d*sin(f*x+e))*(a+b )^(1/2)*((-a*d+b*c)*(1-sin(f*x+e))/(a+b)/(c+d*sin(f*x+e)))^(1/2)*(-(-a*d+b *c)*(1+sin(f*x+e))/(a-b)/(c+d*sin(f*x+e)))^(1/2)/d^3/f/(c+d)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(2118\) vs. \(2(727)=1454\).
Time = 11.47 (sec) , antiderivative size = 2118, normalized size of antiderivative = 2.91 \[ \int \frac {(3+b \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{5/2}} \, dx=\text {Result too large to show} \]
(Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*((-2*(b^2*c^2*Cos[e + f *x] - 6*b*c*d*Cos[e + f*x] + 9*d^2*Cos[e + f*x]))/(3*d*(-c^2 + d^2)*(c + d *Sin[e + f*x])^2) - (2*(3*b^2*c^3*Cos[e + f*x] + 3*b*c^2*d*Cos[e + f*x] - 36*c*d^2*Cos[e + f*x] - 7*b^2*c*d^2*Cos[e + f*x] + 21*b*d^3*Cos[e + f*x])) /(3*d*(-c^2 + d^2)^2*(c + d*Sin[e + f*x]))))/f + ((-4*(-(b*c) + 3*d)*(-(b^ 3*c^3) + 81*c^2*d + 6*b^2*c^2*d - 72*b*c*d^2 + b^3*c*d^2 + 27*d^3 + 6*b^2* d^3)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[ Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + b)*(-c + d))]*Sec[e + f*x]*Sin[(- e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d *Sin[e + f*x]))/(-(b*c) + 3*d)])/((3 + b)*(c + d)*Sqrt[3 + b*Sin[e + f*x]] *Sqrt[c + d*Sin[e + f*x]]) - 4*(-(b*c) + 3*d)*(-12*b^2*c^3 + 27*b*c^2*d + b^3*c^2*d + 108*c*d^2 - 63*b*d^3 + 3*b^3*d^3)*((Sqrt[((c + d)*Cot[(-e + Pi /2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3* d))/((3 + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*Sqrt[ ((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)] )/((3 + b)*(c + d)*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) -...
Time = 3.06 (sec) , antiderivative size = 795, normalized size of antiderivative = 1.09, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {3042, 3271, 27, 3042, 3532, 25, 3042, 3290, 3477, 3042, 3297, 3475}
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.
| \(\displaystyle \int \frac {(a+b \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3271 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {2 \int \frac {-3 c d a^3+7 b d^2 a^2-5 b^2 c d a+b^3 c^2-3 b^3 \left (c^2-d^2\right ) \sin ^2(e+f x)-\left (-d^2 a^3+5 b c d a^2+b^2 \left (2 c^2-9 d^2\right ) a+3 b^3 c d\right ) \sin (e+f x)}{2 \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{3 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {\int \frac {-3 c d a^3+7 b d^2 a^2-5 b^2 c d a+b^3 c^2-3 b^3 \left (c^2-d^2\right ) \sin ^2(e+f x)-\left (-d^2 a^3+5 b c d a^2+b^2 \left (2 c^2-9 d^2\right ) a+3 b^3 c d\right ) \sin (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{3 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {\int \frac {-3 c d a^3+7 b d^2 a^2-5 b^2 c d a+b^3 c^2-3 b^3 \left (c^2-d^2\right ) \sin (e+f x)^2-\left (-d^2 a^3+5 b c d a^2+b^2 \left (2 c^2-9 d^2\right ) a+3 b^3 c d\right ) \sin (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{3 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3532 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {\frac {\int -\frac {-7 a^2 b d^4+3 a^3 c d^3+5 a b^2 c d^3-(b c-a d) \left (6 b^2 c^2+4 a b d c-a^2 d^2-9 b^2 d^2\right ) \sin (e+f x) d-b^3 \left (3 c^4-2 c^2 d^2\right )}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{d^2}-\frac {3 b^3 \left (c^2-d^2\right ) \int \frac {\sqrt {c+d \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)}}dx}{d^2}}{3 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {-\frac {\int \frac {-7 a^2 b d^4+3 a^3 c d^3+5 a b^2 c d^3-(b c-a d) \left (\left (6 c^2-9 d^2\right ) b^2+4 a c d b-a^2 d^2\right ) \sin (e+f x) d-b^3 \left (3 c^4-2 c^2 d^2\right )}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{d^2}-\frac {3 b^3 \left (c^2-d^2\right ) \int \frac {\sqrt {c+d \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)}}dx}{d^2}}{3 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {-\frac {\int \frac {-7 a^2 b d^4+3 a^3 c d^3+5 a b^2 c d^3-(b c-a d) \left (\left (6 c^2-9 d^2\right ) b^2+4 a c d b-a^2 d^2\right ) \sin (e+f x) d-b^3 \left (3 c^4-2 c^2 d^2\right )}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{d^2}-\frac {3 b^3 \left (c^2-d^2\right ) \int \frac {\sqrt {c+d \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)}}dx}{d^2}}{3 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3290 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {-\frac {\int \frac {-7 a^2 b d^4+3 a^3 c d^3+5 a b^2 c d^3-(b c-a d) \left (\left (6 c^2-9 d^2\right ) b^2+4 a c d b-a^2 d^2\right ) \sin (e+f x) d-b^3 \left (3 c^4-2 c^2 d^2\right )}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{d^2}-\frac {6 b^2 \sqrt {a+b} \left (c^2-d^2\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {(a+b) d}{b (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{d^2 f \sqrt {c+d}}}{3 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {-\frac {\frac {\left (a^3 d^3 (3 c+d)-a^2 b d^3 (5 c+7 d)-a b^2 d^2 \left (2 c^2-5 c d-9 d^2\right )+b^3 (-c) \left (3 c^3-6 c^2 d-2 c d^2+9 d^3\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{c-d}-\frac {d (b c-a d)^2 \left (4 a c d+3 b c^2-7 b d^2\right ) \int \frac {\sin (e+f x)+1}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{c-d}}{d^2}-\frac {6 b^2 \sqrt {a+b} \left (c^2-d^2\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {(a+b) d}{b (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{d^2 f \sqrt {c+d}}}{3 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {-\frac {\frac {\left (a^3 d^3 (3 c+d)-a^2 b d^3 (5 c+7 d)-a b^2 d^2 \left (2 c^2-5 c d-9 d^2\right )+b^3 (-c) \left (3 c^3-6 c^2 d-2 c d^2+9 d^3\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{c-d}-\frac {d (b c-a d)^2 \left (4 a c d+3 b c^2-7 b d^2\right ) \int \frac {\sin (e+f x)+1}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{c-d}}{d^2}-\frac {6 b^2 \sqrt {a+b} \left (c^2-d^2\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {(a+b) d}{b (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{d^2 f \sqrt {c+d}}}{3 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3297 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {-\frac {\frac {2 \sqrt {a+b} \left (a^3 d^3 (3 c+d)-a^2 b d^3 (5 c+7 d)-a b^2 d^2 \left (2 c^2-5 c d-9 d^2\right )+b^3 (-c) \left (3 c^3-6 c^2 d-2 c d^2+9 d^3\right )\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (c-d) \sqrt {c+d} (b c-a d)}-\frac {d (b c-a d)^2 \left (4 a c d+3 b c^2-7 b d^2\right ) \int \frac {\sin (e+f x)+1}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{c-d}}{d^2}-\frac {6 b^2 \sqrt {a+b} \left (c^2-d^2\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {(a+b) d}{b (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{d^2 f \sqrt {c+d}}}{3 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3475 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}-\frac {-\frac {6 \sqrt {a+b} \left (c^2-d^2\right ) \operatorname {EllipticPi}\left (\frac {(a+b) d}{b (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x)) b^2}{d^2 \sqrt {c+d} f}-\frac {\frac {2 (a-b) \sqrt {a+b} d \left (3 b c^2+4 a d c-7 b d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(c-d) \sqrt {c+d} f}+\frac {2 \sqrt {a+b} \left (-c \left (3 c^3-6 d c^2-2 d^2 c+9 d^3\right ) b^3-a d^2 \left (2 c^2-5 d c-9 d^2\right ) b^2-a^2 d^3 (5 c+7 d) b+a^3 d^3 (3 c+d)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(c-d) \sqrt {c+d} (b c-a d) f}}{d^2}}{3 d \left (c^2-d^2\right )}\) |
(2*(b*c - a*d)^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(3*d*(c^2 - d^2)*f *(c + d*Sin[e + f*x])^(3/2)) - ((-6*b^2*Sqrt[a + b]*(c^2 - d^2)*EllipticPi [((a + b)*d)/(b*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(S qrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d)) ]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(d^2*Sqrt[c + d]*f) - ((2*(a - b)*Sqrt[a + b]*d*(3*b*c^2 + 4*a*c*d - 7*b*d^2)*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d ))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/( (a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((c - d)*Sqrt[c + d]*f ) + (2*Sqrt[a + b]*(a^3*d^3*(3*c + d) - a^2*b*d^3*(5*c + 7*d) - a*b^2*d^2* (2*c^2 - 5*c*d - 9*d^2) - b^3*c*(3*c^3 - 6*c^2*d - 2*c*d^2 + 9*d^3))*Ellip ticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d *Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[( (b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b *c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[ e + f*x]))/((c - d)*Sqrt[c + d]*(b*c - a*d)*f))/d^2)/(3*d*(c^2 - d^2))
3.8.82.3.1 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])
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[2*((a + b*Sin[e + f*x])/(d*f*Rt[(a + b)/ (c + d), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x])))]*EllipticPi[b*((c + d)/(d*(a + b))), ArcSin[Rt[(a + b)/( c + d), 2]*(Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]])], (a - b)*(( c + d)/((a + b)*(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] && PosQ[(a + b)/(c + d)]
Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_ .) + (f_.)*(x_)]]), x_Symbol] :> Simp[2*((c + d*Sin[e + f*x])/(f*(b*c - a*d )*Rt[(c + d)/(a + b), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 - Sin[e + f*x] )/((a + b)*(c + d*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 + Sin[e + f*x])/ ((a - b)*(c + d*Sin[e + f*x])))]*EllipticF[ArcSin[Rt[(c + d)/(a + b), 2]*(S qrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]])], (a + b)*((c - d)/((a - b)*(c + d)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && N eQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/(a + b)]
Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_) + (b_.)*sin[(e_.) + (f_.) *(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Sim p[-2*A*(c - d)*((a + b*Sin[e + f*x])/(f*(b*c - a*d)^2*Rt[(a + b)/(c + d), 2 ]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x])))]*EllipticE[ArcSin[Rt[(a + b)/(c + d), 2]*(Sqrt[c + d*Sin[e + f*x]] /Sqrt[a + b*Sin[e + f*x]])], (a - b)*((c + d)/((a + b)*(c - d)))], x] /; Fr eeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(a + b)/(c + d)]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> S imp[(A - B)/(a - b) Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f* x]]), x], x] - Simp[(A*b - a*B)/(a - b) Int[(1 + Sin[e + f*x])/((a + b*Si n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e , f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[A, B]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_.) + (d_.)*sin[(e _.) + (f_.)*(x_)]]), x_Symbol] :> Simp[C/b^2 Int[Sqrt[a + b*Sin[e + f*x]] /Sqrt[c + d*Sin[e + f*x]], x], x] + Simp[1/b^2 Int[(A*b^2 - a^2*C + b*(b* B - 2*a*C)*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x ]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] & & NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 53.88 (sec) , antiderivative size = 1389526, normalized size of antiderivative = 1911.31
Timed out. \[ \int \frac {(3+b \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(3+b \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(3+b \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(3+b \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(3+b \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{5/2}} \, dx=\int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{5/2}}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]