Integrand size = 29, antiderivative size = 797 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+b \sin (e+f x))^{3/2}} \, dx=\frac {(c-d) \sqrt {c+d} \left (2 b^2 c^2-12 b c d+27 d^2-b^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right )|\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{(3-b) b^2 \sqrt {3+b} (b c-3 d) f}+\frac {(5 b c-9 d) d \sqrt {c+d} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(3+b) d},\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right ),\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{b^3 \sqrt {3+b} f}+\frac {2 (b c-3 d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b \left (9-b^2\right ) f \sqrt {3+b \sin (e+f x)}}+\frac {\left (12 b c d-27 d^2-b^2 \left (2 c^2-d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b \left (9-b^2\right ) f \sqrt {3+b \sin (e+f x)}}-\frac {\sqrt {3+b} \left (27 d^2-6 b d (c+3 d)-b^2 \left (2 c^2-6 c d-d^2\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(3-b) b^3 \sqrt {c+d} f} \]
(c-d)*(3*a^2*d^2-4*a*b*c*d+2*b^2*c^2-b^2*d^2)*EllipticE((a+b)^(1/2)*(c+d*s in(f*x+e))^(1/2)/(c+d)^(1/2)/(a+b*sin(f*x+e))^(1/2),((a-b)*(c+d)/(a+b)/(c- d))^(1/2))*sec(f*x+e)*(a+b*sin(f*x+e))*(c+d)^(1/2)*(-(-a*d+b*c)*(1-sin(f*x +e))/(c+d)/(a+b*sin(f*x+e)))^(1/2)*((-a*d+b*c)*(1+sin(f*x+e))/(c-d)/(a+b*s in(f*x+e)))^(1/2)/(a-b)/b^2/(-a*d+b*c)/f/(a+b)^(1/2)+d*(-3*a*d+5*b*c)*Elli pticPi((a+b)^(1/2)*(c+d*sin(f*x+e))^(1/2)/(c+d)^(1/2)/(a+b*sin(f*x+e))^(1/ 2),b*(c+d)/(a+b)/d,((a-b)*(c+d)/(a+b)/(c-d))^(1/2))*sec(f*x+e)*(a+b*sin(f* x+e))*(c+d)^(1/2)*(-(-a*d+b*c)*(1-sin(f*x+e))/(c+d)/(a+b*sin(f*x+e)))^(1/2 )*((-a*d+b*c)*(1+sin(f*x+e))/(c-d)/(a+b*sin(f*x+e)))^(1/2)/b^3/f/(a+b)^(1/ 2)-(3*a^2*d^2-2*a*b*d*(c+3*d)-b^2*(2*c^2-6*c*d-d^2))*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-s in(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)/(a-b)/b^3/f/(c+d)^(1/2)+2*(-a*d+b*c)^2*cos(f*x+e) *(c+d*sin(f*x+e))^(1/2)/b/(a^2-b^2)/f/(a+b*sin(f*x+e))^(1/2)+(4*a*b*c*d-3* a^2*d^2-b^2*(2*c^2-d^2))*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/b/(a^2-b^2)/f/( a+b*sin(f*x+e))^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(1963\) vs. \(2(797)=1594\).
Time = 8.67 (sec) , antiderivative size = 1963, normalized size of antiderivative = 2.46 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+b \sin (e+f x))^{3/2}} \, dx =\text {Too large to display} \]
(-2*(b^2*c^2*Cos[e + f*x] - 6*b*c*d*Cos[e + f*x] + 9*d^2*Cos[e + f*x])*Sqr t[c + d*Sin[e + f*x]])/(b*(-9 + b^2)*f*Sqrt[3 + b*Sin[e + f*x]]) - ((-4*(- (b*c) + 3*d)*(6*b*c^3 - 4*b^2*c^2*d + 6*b*c*d^2 + 9*d^3 - 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)*(2*b^2*c^3 - 6*b*c^2*d + 36*c*d^2 - 6*b^ 2*c*d^2 + 6*b*d^3)*((Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*E llipticF[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]]) - (Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticPi[(-(b*c) + 3*d)/((3 + b)*d), 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...
Time = 3.62 (sec) , antiderivative size = 835, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {3042, 3271, 27, 3042, 3540, 25, 3042, 3532, 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.
| \(\displaystyle \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+b \sin (e+f x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+b \sin (e+f x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3271 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) \sqrt {a+b \sin (e+f x)}}-\frac {2 \int \frac {a^2 d^3+3 b^2 c^2 d+\left (-\left (\left (2 c^2-d^2\right ) b^2\right )+4 a c d b-3 a^2 d^2\right ) \sin ^2(e+f x) d-a b c \left (c^2+3 d^2\right )-\left (\left (c^3-3 c d^2\right ) b^2-a d \left (c^2-d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{2 \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) \sqrt {a+b \sin (e+f x)}}-\frac {\int \frac {a^2 d^3+3 b^2 c^2 d+\left (-\left (\left (2 c^2-d^2\right ) b^2\right )+4 a c d b-3 a^2 d^2\right ) \sin ^2(e+f x) d-a b c \left (c^2+3 d^2\right )-\left (\left (c^3-3 c d^2\right ) b^2-a d \left (c^2-d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) \sqrt {a+b \sin (e+f x)}}-\frac {\int \frac {a^2 d^3+3 b^2 c^2 d+\left (-\left (\left (2 c^2-d^2\right ) b^2\right )+4 a c d b-3 a^2 d^2\right ) \sin (e+f x)^2 d-a b c \left (c^2+3 d^2\right )-\left (\left (c^3-3 c d^2\right ) b^2-a d \left (c^2-d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3540 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) \sqrt {a+b \sin (e+f x)}}-\frac {\frac {\int -\frac {\left (a^2-b^2\right ) (5 b c-3 a d) \sin ^2(e+f x) d^3+2 \left (a^2-b^2\right ) c (3 b c-a d) \sin (e+f x) d^2+\left (a^2-b^2\right ) \left (2 b c^3-b d^2 c+a d^3\right ) d}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {\left (-3 a^2 d^2+4 a b c d-\left (b^2 \left (2 c^2-d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) \sqrt {a+b \sin (e+f x)}}-\frac {-\frac {\int \frac {\left (a^2-b^2\right ) (5 b c-3 a d) \sin ^2(e+f x) d^3+2 \left (a^2-b^2\right ) c (3 b c-a d) \sin (e+f x) d^2+\left (a^2-b^2\right ) \left (2 b c^3-b d^2 c+a d^3\right ) d}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {\left (-3 a^2 d^2+4 a b c d-\left (b^2 \left (2 c^2-d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) \sqrt {a+b \sin (e+f x)}}-\frac {-\frac {\int \frac {\left (a^2-b^2\right ) (5 b c-3 a d) \sin (e+f x)^2 d^3+2 \left (a^2-b^2\right ) c (3 b c-a d) \sin (e+f x) d^2+\left (a^2-b^2\right ) \left (2 b c^3-b d^2 c+a d^3\right ) d}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {\left (-3 a^2 d^2+4 a b c d-\left (b^2 \left (2 c^2-d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3532 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) \sqrt {a+b \sin (e+f x)}}-\frac {-\frac {\frac {\int \frac {6 b \left (a^2-b^2\right ) d^2 \sin (e+f x) (b c-a d)^2+\left (a^2-b^2\right ) d \left (2 b^2 c^2+2 a b d c-3 a^2 d^2-b^2 d^2\right ) (b c-a d)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+\frac {d^3 \left (a^2-b^2\right ) (5 b c-3 a d) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}-\frac {\left (-3 a^2 d^2+4 a b c d-\left (b^2 \left (2 c^2-d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) \sqrt {a+b \sin (e+f x)}}-\frac {-\frac {\frac {\int \frac {6 b \left (a^2-b^2\right ) d^2 \sin (e+f x) (b c-a d)^2+\left (a^2-b^2\right ) d \left (2 b^2 c^2+2 a b d c-3 a^2 d^2-b^2 d^2\right ) (b c-a d)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+\frac {d^3 \left (a^2-b^2\right ) (5 b c-3 a d) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}-\frac {\left (-3 a^2 d^2+4 a b c d-\left (b^2 \left (2 c^2-d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3290 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) \sqrt {a+b \sin (e+f x)}}-\frac {-\frac {\frac {\int \frac {6 b \left (a^2-b^2\right ) d^2 \sin (e+f x) (b c-a d)^2+\left (a^2-b^2\right ) d \left (2 b^2 c^2+2 a b d c-3 a^2 d^2-b^2 d^2\right ) (b c-a d)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+\frac {2 d^2 \left (a^2-b^2\right ) \sqrt {c+d} (5 b c-3 a d) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b^2 f \sqrt {a+b}}}{2 d}-\frac {\left (-3 a^2 d^2+4 a b c d-\left (b^2 \left (2 c^2-d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) \sqrt {a+b \sin (e+f x)}}-\frac {-\frac {\frac {b d (a+b) (b c-a d) \left (-3 a^2 d^2+4 a b c d-\left (b^2 \left (2 c^2-d^2\right )\right )\right ) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx-d (a+b) (b c-a d) \left (3 a^2 d^2-2 a b d (c+3 d)-\left (b^2 \left (2 c^2-6 c d-d^2\right )\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+\frac {2 d^2 \left (a^2-b^2\right ) \sqrt {c+d} (5 b c-3 a d) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b^2 f \sqrt {a+b}}}{2 d}-\frac {\left (-3 a^2 d^2+4 a b c d-\left (b^2 \left (2 c^2-d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) \sqrt {a+b \sin (e+f x)}}-\frac {-\frac {\frac {b d (a+b) (b c-a d) \left (-3 a^2 d^2+4 a b c d-\left (b^2 \left (2 c^2-d^2\right )\right )\right ) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx-d (a+b) (b c-a d) \left (3 a^2 d^2-2 a b d (c+3 d)-\left (b^2 \left (2 c^2-6 c d-d^2\right )\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+\frac {2 d^2 \left (a^2-b^2\right ) \sqrt {c+d} (5 b c-3 a d) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b^2 f \sqrt {a+b}}}{2 d}-\frac {\left (-3 a^2 d^2+4 a b c d-\left (b^2 \left (2 c^2-d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3297 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) \sqrt {a+b \sin (e+f x)}}-\frac {-\frac {\frac {b d (a+b) (b c-a d) \left (-3 a^2 d^2+4 a b c d-\left (b^2 \left (2 c^2-d^2\right )\right )\right ) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx-\frac {2 d (a+b)^{3/2} \left (3 a^2 d^2-2 a b d (c+3 d)-\left (b^2 \left (2 c^2-6 c d-d^2\right )\right )\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f \sqrt {c+d}}}{b^2}+\frac {2 d^2 \left (a^2-b^2\right ) \sqrt {c+d} (5 b c-3 a d) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b^2 f \sqrt {a+b}}}{2 d}-\frac {\left (-3 a^2 d^2+4 a b c d-\left (b^2 \left (2 c^2-d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3475 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b \left (a^2-b^2\right ) f \sqrt {a+b \sin (e+f x)}}-\frac {-\frac {\left (-\left (\left (2 c^2-d^2\right ) b^2\right )+4 a c d b-3 a^2 d^2\right ) \sqrt {c+d \sin (e+f x)} \cos (e+f x)}{f \sqrt {a+b \sin (e+f x)}}-\frac {\frac {2 \left (a^2-b^2\right ) \sqrt {c+d} (5 b c-3 a d) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x)) d^2}{b^2 \sqrt {a+b} f}+\frac {-\frac {2 d \left (-\left (\left (2 c^2-6 d c-d^2\right ) b^2\right )-2 a d (c+3 d) b+3 a^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x)) (a+b)^{3/2}}{\sqrt {c+d} f}-\frac {2 b (c-d) d \sqrt {c+d} \left (-\left (\left (2 c^2-d^2\right ) b^2\right )+4 a c d b-3 a^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x)) \sqrt {a+b}}{(b c-a d) f}}{b^2}}{2 d}}{b \left (a^2-b^2\right )}\) |
(2*(b*c - a*d)^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(b*(a^2 - b^2)*f*S qrt[a + b*Sin[e + f*x]]) - (-(((4*a*b*c*d - 3*a^2*d^2 - b^2*(2*c^2 - d^2)) *Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*Sqrt[a + b*Sin[e + f*x]])) - (( 2*(a^2 - b^2)*d^2*Sqrt[c + d]*(5*b*c - 3*a*d)*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqr t[-(((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]))]*(a + b*S in[e + f*x]))/(b^2*Sqrt[a + b]*f) + ((-2*b*Sqrt[a + b]*(c - d)*d*Sqrt[c + d]*(4*a*b*c*d - 3*a^2*d^2 - b^2*(2*c^2 - d^2))*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*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]))/((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]))]*(a + b*Sin[e + f*x]))/((b*c - a*d )*f) - (2*(a + b)^(3/2)*d*(3*a^2*d^2 - 2*a*b*d*(c + 3*d) - b^2*(2*c^2 - 6* c*d - d^2))*EllipticF[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))]*Se c[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]))/(Sqrt[c + d]*f))/b^2)/(2*d))/(b*(a^2 - b^2)...
3.8.89.3.1 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]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(Sqrt[c + d*Sin[e + f *x]]/(d*f*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[1/(2*d) Int[(1/((a + b*Si n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]))*Simp[2*a*A*d - C*(b*c - a*d) - 2*(a*c*C - d*(A*b + a*B))*Sin[e + f*x] + (2*b*B*d - C*(b*c + a*d))*Sin[e + f*x]^2, 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 = 39.75 (sec) , antiderivative size = 849240, normalized size of antiderivative = 1065.55
\[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+b \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
integral((d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)*sqrt(b*sin( f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2), x)
Timed out. \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+b \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+b \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+b \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+b \sin (e+f x))^{3/2}} \, dx=\int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]