Integrand size = 29, antiderivative size = 655 \[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=\frac {2 b^2 \cos (e+f x)}{3 \left (9-b^2\right ) (b c-3 d) f (3+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}+\frac {8 b^2 \left (3 b c-18 d+b^2 d\right ) \cos (e+f x)}{3 \left (9-b^2\right )^2 (b c-3 d)^2 f \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (243 d^3-b^4 d \left (5 c^2-8 d^2\right )+27 b^2 d \left (3 c^2-5 d^2\right )-12 b^3 c \left (c^2-d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{3 \sqrt {3+b} \left (9-b^2\right ) (b c-3 d)^4 (c-d) \sqrt {c+d} f}-\frac {2 \left (27 b (2 c-3 d) d-81 d^2-9 b^2 \left (c^2-2 d^2\right )+b^3 \left (c^2-6 c d+8 d^2\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{3 \sqrt {3+b} \left (9-b^2\right ) (b c-3 d)^3 (c-d) \sqrt {c+d} f} \]
2/3*(3*a^4*d^3-b^4*d*(5*c^2-8*d^2)+3*a^2*b^2*d*(3*c^2-5*d^2)-4*a*b^3*c*(c^ 2-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*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)/(a^2-b^2)/(c-d)/(-a*d+b*c)^4/f/( a+b)^(1/2)/(c+d)^(1/2)-2/3*(3*a^2*b*(2*c-3*d)*d-3*a^3*d^2-3*a*b^2*(c^2-2*d ^2)+b^3*(c^2-6*c*d+8*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*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)/(a^2-b^2)/(c- d)/(-a*d+b*c)^3/f/(a+b)^(1/2)/(c+d)^(1/2)+2/3*b^2*cos(f*x+e)/(a^2-b^2)/(-a *d+b*c)/f/(a+b*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(1/2)+8/3*b^2*(-2*a^2*d+ a*b*c+b^2*d)*cos(f*x+e)/(a^2-b^2)^2/(-a*d+b*c)^2/f/(a+b*sin(f*x+e))^(1/2)/ (c+d*sin(f*x+e))^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(2255\) vs. \(2(655)=1310\).
Time = 6.96 (sec) , antiderivative size = 2255, normalized size of antiderivative = 3.44 \[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=\text {Result too large to show} \]
(Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*((-2*b^3*Cos[e + f*x])/ (3*(-9 + b^2)*(b*c - 3*d)^2*(3 + b*Sin[e + f*x])^2) + (2*(12*b^4*c*Cos[e + f*x] - 81*b^3*d*Cos[e + f*x] + 5*b^5*d*Cos[e + f*x]))/(3*(-9 + b^2)^2*(b* c - 3*d)^3*(3 + b*Sin[e + f*x])) - (2*d^4*Cos[e + f*x])/((b*c - 3*d)^3*(c^ 2 - d^2)*(c + d*Sin[e + f*x]))))/f + ((-4*(-(b*c) + 3*d)*(27*b^3*c^4 + b^5 *c^4 - 243*b^2*c^3*d + 15*b^4*c^3*d + 729*b*c^2*d^2 - 180*b^3*c^2*d^2 + 7* b^5*c^2*d^2 - 729*c*d^3 + 405*b^2*c*d^3 - 24*b^4*c*d^3 - 729*b*d^4 + 153*b ^3*d^4 - 8*b^5*d^4)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*El lipticF[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^4*c^4 - 45*b^3*c^3*d + 5*b^5*c^3*d - 243*b^2*c^2*d^2 + 3*b^4*c^2*d^2 - 243*b*c*d^ 3 + 99*b^3*c*d^3 - 8*b^5*c*d^3 - 729*d^4 + 405*b^2*d^4 - 24*b^4*d^4)*((Sqr t[((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)]/Sq rt[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*...
Time = 2.42 (sec) , antiderivative size = 720, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {3042, 3281, 27, 3042, 3534, 27, 3042, 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.
\(\displaystyle \int \frac {1}{(a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}}dx\) |
\(\Big \downarrow \) 3281 |
\(\displaystyle \frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}-\frac {2 \int -\frac {-3 d a^2+3 b c a-2 b^2 d \sin ^2(e+f x)+4 b^2 d-b (b c-3 a d) \sin (e+f x)}{2 (a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}}dx}{3 \left (a^2-b^2\right ) (b c-a d)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {-3 d a^2+3 b c a-2 b^2 d \sin ^2(e+f x)+4 b^2 d-b (b c-3 a d) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}}dx}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {-3 d a^2+3 b c a-2 b^2 d \sin (e+f x)^2+4 b^2 d-b (b c-3 a d) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}}dx}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}\) |
\(\Big \downarrow \) 3534 |
\(\displaystyle \frac {\frac {8 b^2 \left (-2 a^2 d+a b c+b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {2 \int \frac {-3 d^2 a^4+6 b c d a^3-3 b^2 \left (c^2-5 d^2\right ) a^2-6 b^3 c d a-b^4 \left (c^2+8 d^2\right )+2 b \left (3 d^2 a^3+3 b c d a^2-b^2 \left (2 c^2+d^2\right ) a-3 b^3 c d\right ) \sin (e+f x)}{2 \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{\left (a^2-b^2\right ) (b c-a d)}}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {8 b^2 \left (-2 a^2 d+a b c+b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {\int \frac {-3 d^2 a^4+6 b c d a^3-3 b^2 \left (c^2-5 d^2\right ) a^2-6 b^3 c d a-b^4 \left (c^2+8 d^2\right )+2 b \left (3 d^2 a^3+3 b c d a^2-b^2 \left (2 c^2+d^2\right ) a-3 b^3 c d\right ) \sin (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{\left (a^2-b^2\right ) (b c-a d)}}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {8 b^2 \left (-2 a^2 d+a b c+b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {\int \frac {-3 d^2 a^4+6 b c d a^3-3 b^2 \left (c^2-5 d^2\right ) a^2-6 b^3 c d a-b^4 \left (c^2+8 d^2\right )+2 b \left (3 d^2 a^3+3 b c d a^2-b^2 \left (2 c^2+d^2\right ) a-3 b^3 c d\right ) \sin (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{\left (a^2-b^2\right ) (b c-a d)}}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}\) |
\(\Big \downarrow \) 3477 |
\(\displaystyle \frac {\frac {8 b^2 \left (-2 a^2 d+a b c+b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\left (3 a^4 d^3+3 a^2 b^2 d \left (3 c^2-5 d^2\right )-4 a b^3 c \left (c^2-d^2\right )-b^4 d \left (5 c^2-8 d^2\right )\right ) \int \frac {\sin (e+f x)+1}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{c-d}+\frac {(a-b) \left (-3 a^3 d^2+3 a^2 b d (2 c-3 d)-3 a b^2 \left (c^2-2 d^2\right )+b^3 \left (c^2-6 c d+8 d^2\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{c-d}}{\left (a^2-b^2\right ) (b c-a d)}}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {8 b^2 \left (-2 a^2 d+a b c+b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\left (3 a^4 d^3+3 a^2 b^2 d \left (3 c^2-5 d^2\right )-4 a b^3 c \left (c^2-d^2\right )-b^4 d \left (5 c^2-8 d^2\right )\right ) \int \frac {\sin (e+f x)+1}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{c-d}+\frac {(a-b) \left (-3 a^3 d^2+3 a^2 b d (2 c-3 d)-3 a b^2 \left (c^2-2 d^2\right )+b^3 \left (c^2-6 c d+8 d^2\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{c-d}}{\left (a^2-b^2\right ) (b c-a d)}}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}\) |
\(\Big \downarrow \) 3297 |
\(\displaystyle \frac {\frac {8 b^2 \left (-2 a^2 d+a b c+b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\left (3 a^4 d^3+3 a^2 b^2 d \left (3 c^2-5 d^2\right )-4 a b^3 c \left (c^2-d^2\right )-b^4 d \left (5 c^2-8 d^2\right )\right ) \int \frac {\sin (e+f x)+1}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{c-d}+\frac {2 (a-b) \sqrt {a+b} \left (-3 a^3 d^2+3 a^2 b d (2 c-3 d)-3 a b^2 \left (c^2-2 d^2\right )+b^3 \left (c^2-6 c d+8 d^2\right )\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (c-d) \sqrt {c+d} (b c-a d)}}{\left (a^2-b^2\right ) (b c-a d)}}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}\) |
\(\Big \downarrow \) 3475 |
\(\displaystyle \frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}+\frac {\frac {8 b^2 \left (-2 a^2 d+a b c+b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {\frac {2 (a-b) \sqrt {a+b} \left (-3 a^3 d^2+3 a^2 b d (2 c-3 d)-3 a b^2 \left (c^2-2 d^2\right )+b^3 \left (c^2-6 c d+8 d^2\right )\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (c-d) \sqrt {c+d} (b c-a d)}-\frac {2 (a-b) \sqrt {a+b} \left (3 a^4 d^3+3 a^2 b^2 d \left (3 c^2-5 d^2\right )-4 a b^3 c \left (c^2-d^2\right )-b^4 d \left (5 c^2-8 d^2\right )\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (c-d) \sqrt {c+d} (b c-a d)^2}}{\left (a^2-b^2\right ) (b c-a d)}}{3 \left (a^2-b^2\right ) (b c-a d)}\) |
(2*b^2*Cos[e + f*x])/(3*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^(3/ 2)*Sqrt[c + d*Sin[e + f*x]]) + ((8*b^2*(a*b*c - 2*a^2*d + b^2*d)*Cos[e + f *x])/((a^2 - b^2)*(b*c - a*d)*f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - ((-2*(a - b)*Sqrt[a + b]*(3*a^4*d^3 - b^4*d*(5*c^2 - 8*d^2) + 3 *a^2*b^2*d*(3*c^2 - 5*d^2) - 4*a*b^3*c*(c^2 - 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 + Si n[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)^2*f) + (2*(a - b)*Sqrt[a + b]*(3*a^2*b*(2*c - 3 *d)*d - 3*a^3*d^2 - 3*a*b^2*(c^2 - 2*d^2) + b^3*(c^2 - 6*c*d + 8*d^2))*Ell ipticF[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*Si n[e + f*x]))/((c - d)*Sqrt[c + d]*(b*c - a*d)*f))/((a^2 - b^2)*(b*c - a*d) ))/(3*(a^2 - b^2)*(b*c - a*d))
3.8.99.3.1 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)*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/(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_)])^(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])))
Leaf count of result is larger than twice the leaf count of optimal. \(373778\) vs. \(2(642)=1284\).
Time = 16.30 (sec) , antiderivative size = 373779, normalized size of antiderivative = 570.65
\[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
integral(sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/((2*b^3*c*d + 3 *a*b^2*d^2)*cos(f*x + e)^4 + (a^3 + 3*a*b^2)*c^2 + 2*(3*a^2*b + b^3)*c*d + (a^3 + 3*a*b^2)*d^2 - (3*a*b^2*c^2 + 2*(3*a^2*b + 2*b^3)*c*d + (a^3 + 6*a *b^2)*d^2)*cos(f*x + e)^2 + (b^3*d^2*cos(f*x + e)^4 + (3*a^2*b + b^3)*c^2 + 2*(a^3 + 3*a*b^2)*c*d + (3*a^2*b + b^3)*d^2 - (b^3*c^2 + 6*a*b^2*c*d + ( 3*a^2*b + 2*b^3)*d^2)*cos(f*x + e)^2)*sin(f*x + e)), x)
Timed out. \[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]