Integrand size = 29, antiderivative size = 894 \[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2}} \, dx=\frac {2 b^2 \cos (e+f x)}{3 \left (9-b^2\right ) (b c-3 d) f (3+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}}+\frac {4 b^2 \left (6 b c-45 d+3 b^2 d\right ) \cos (e+f x)}{3 \left (9-b^2\right )^2 (b c-3 d)^2 f \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {2 d \left (81 d^3+9 b^2 d \left (11 c^2-13 d^2\right )-b^4 d \left (7 c^2-8 d^2\right )-12 b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x) \sqrt {3+b \sin (e+f x)}}{3 \left (9-b^2\right )^2 (b c-3 d)^3 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}-\frac {8 \left (243 c d^4-54 b^2 c d^4+3 b^4 c \left (c^4-2 c^2 d^2+2 d^4\right )+b^5 d \left (2 c^4-7 c^2 d^2+4 d^4\right )-9 b^3 d \left (3 c^4-12 c^2 d^2+7 d^4\right )-81 b \left (3 c^2 d^3-2 d^5\right )\right ) E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{3 \sqrt {3+b} \left (9-b^2\right ) (b c-3 d)^5 (c-d)^2 (c+d)^{3/2} f}-\frac {2 \left (81 d^3 (3 c+d)-243 b d^2 \left (c^2-d^2\right )+9 b^2 d \left (9 c^3-18 c^2 d-15 c d^2+16 d^3\right )+b^4 \left (c^4-9 c^3 d+16 c^2 d^2+12 c d^3-16 d^4\right )-9 b^3 \left (c^4-5 c^2 d^2+4 d^4\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{3 \sqrt {3+b} \left (9-b^2\right ) (b c-3 d)^4 (c-d)^2 (c+d)^{3/2} f} \]
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))^(3/2)+4/3*b^2*(-5*a^2*d+2*a*b*c+3*b^2*d)*cos(f*x+e)/(a^2-b^2)^2/(- a*d+b*c)^2/f/(c+d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e))^(1/2)-2/3*d*(a^4*d^3+ a^2*b^2*d*(11*c^2-13*d^2)-b^4*d*(7*c^2-8*d^2)-4*a*b^3*c*(c^2-d^2))*cos(f*x +e)*(a+b*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))^(3/2)-8/3*(a^5*c*d^4-2*a^3*b^2*c*d^4+a*b^4*c*(c^4-2*c^2*d^2+2*d^4)+ b^5*d*(2*c^4-7*c^2*d^2+4*d^4)-a^2*b^3*d*(3*c^4-12*c^2*d^2+7*d^4)-a^4*b*(3* c^2*d^3-2*d^5))*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)^2/(c+d)^ (3/2)/(-a*d+b*c)^5/f/(a+b)^(1/2)-2/3*(a^4*d^3*(3*c+d)-9*a^3*b*d^2*(c^2-d^2 )+a^2*b^2*d*(9*c^3-18*c^2*d-15*c*d^2+16*d^3)+b^4*(c^4-9*c^3*d+16*c^2*d^2+1 2*c*d^3-16*d^4)-3*a*b^3*(c^4-5*c^2*d^2+4*d^4))*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)^2/(c+d)^(3/2)/(-a*d+b*c)^4/f/(a+b)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(2530\) vs. \(2(894)=1788\).
Time = 7.74 (sec) , antiderivative size = 2530, normalized size of antiderivative = 2.83 \[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2}} \, dx=\text {Result too large to show} \]
(Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*((-2*b^4*Cos[e + f*x])/ (3*(-9 + b^2)*(b*c - 3*d)^3*(3 + b*Sin[e + f*x])^2) + (8*(3*b^5*c*Cos[e + f*x] - 27*b^4*d*Cos[e + f*x] + 2*b^6*d*Cos[e + f*x]))/(3*(-9 + b^2)^2*(b*c - 3*d)^4*(3 + b*Sin[e + f*x])) - (2*d^4*Cos[e + f*x])/(3*(b*c - 3*d)^3*(c ^2 - d^2)*(c + d*Sin[e + f*x])^2) - (8*(3*b*c^2*d^4*Cos[e + f*x] - 3*c*d^5 *Cos[e + f*x] - 2*b*d^6*Cos[e + f*x]))/(3*(b*c - 3*d)^4*(c^2 - d^2)^2*(c + d*Sin[e + f*x]))))/f + ((-4*(-(b*c) + 3*d)*(27*b^4*c^6 + b^6*c^6 - 324*b^ 3*c^5*d + 24*b^5*c^5*d + 1458*b^2*c^4*d^2 - 369*b^4*c^4*d^2 + 15*b^6*c^4*d ^2 - 2916*b*c^3*d^3 + 1296*b^3*c^3*d^3 - 84*b^5*c^3*d^3 + 2187*c^2*d^4 - 3 321*b^2*c^2*d^4 + 666*b^4*c^2*d^4 - 32*b^6*c^2*d^4 + 1944*b*c*d^5 - 756*b^ 3*c*d^5 + 48*b^5*c*d^5 + 729*d^6 + 1215*b^2*d^6 - 288*b^4*d^6 + 16*b^6*d^6 )*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqr t[((-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*Si n[e + f*x]))/(-(b*c) + 3*d)])/((3 + b)*(c + d)*Sqrt[3 + b*Sin[e + f*x]]*Sq rt[c + d*Sin[e + f*x]]) - 4*(-(b*c) + 3*d)*(12*b^5*c^6 - 72*b^4*c^5*d + 8* b^6*c^5*d - 324*b^3*c^4*d^2 - 972*b^2*c^3*d^3 + 360*b^4*c^3*d^3 - 28*b^6*c ^3*d^3 - 1944*b*c^2*d^4 + 1080*b^3*c^2*d^4 - 60*b^5*c^2*d^4 + 2916*c*d^...
Time = 4.01 (sec) , antiderivative size = 983, normalized size of antiderivative = 1.10, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {3042, 3281, 27, 3042, 3534, 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.
| \(\displaystyle \int \frac {1}{(a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3281 |
| \(\displaystyle \frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}}-\frac {2 \int -\frac {-4 b^2 d \sin ^2(e+f x)-b (b c-3 a d) \sin (e+f x)+3 \left (-d a^2+b c a+2 b^2 d\right )}{2 (a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2}}dx}{3 \left (a^2-b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {-4 b^2 d \sin ^2(e+f x)-b (b c-3 a d) \sin (e+f x)+3 \left (-d a^2+b c a+2 b^2 d\right )}{(a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2}}dx}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {-4 b^2 d \sin (e+f x)^2-b (b c-3 a d) \sin (e+f x)+3 \left (-d a^2+b c a+2 b^2 d\right )}{(a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2}}dx}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {4 b^2 \left (-5 a^2 d+2 a b c+3 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {2 \int \frac {-3 d^2 a^4+6 b c d a^3-3 b^2 \left (c^2-13 d^2\right ) a^2-14 b^3 c d a+4 b^2 d \left (-5 d a^2+2 b c a+3 b^2 d\right ) \sin ^2(e+f x)-b^4 \left (c^2+24 d^2\right )+2 b \left (3 d^2 a^3+5 b c d a^2-b^2 \left (2 c^2+d^2\right ) a-5 b^3 c d\right ) \sin (e+f x)}{2 \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}dx}{\left (a^2-b^2\right ) (b c-a d)}}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {4 b^2 \left (-5 a^2 d+2 a b c+3 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {\int \frac {-3 d^2 a^4+6 b c d a^3-3 b^2 \left (c^2-13 d^2\right ) a^2-14 b^3 c d a+4 b^2 d \left (-5 d a^2+2 b c a+3 b^2 d\right ) \sin ^2(e+f x)-b^4 \left (c^2+24 d^2\right )+2 b \left (3 d^2 a^3+5 b c d a^2-b^2 \left (2 c^2+d^2\right ) a-5 b^3 c d\right ) \sin (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}dx}{\left (a^2-b^2\right ) (b c-a d)}}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {4 b^2 \left (-5 a^2 d+2 a b c+3 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {\int \frac {-3 d^2 a^4+6 b c d a^3-3 b^2 \left (c^2-13 d^2\right ) a^2-14 b^3 c d a+4 b^2 d \left (-5 d a^2+2 b c a+3 b^2 d\right ) \sin (e+f x)^2-b^4 \left (c^2+24 d^2\right )+2 b \left (3 d^2 a^3+5 b c d a^2-b^2 \left (2 c^2+d^2\right ) a-5 b^3 c d\right ) \sin (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}dx}{\left (a^2-b^2\right ) (b c-a d)}}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {4 b^2 \left (-5 a^2 d+2 a b c+3 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {\frac {2 \int \frac {3 \left (3 c d^3 a^5-b \left (9 c^2 d^2-8 d^4\right ) a^4+3 b^2 c d \left (3 c^2-5 d^2\right ) a^3-b^3 \left (3 c^4-33 d^2 c^2+28 d^4\right ) a^2-3 b^4 c d \left (3 c^2-4 d^2\right ) a-b^5 \left (c^4+16 d^2 c^2-16 d^4\right )+\left (-d^4 a^5+3 b c d^3 a^4+b^2 d^2 \left (9 c^2-7 d^2\right ) a^3+3 b^3 c d \left (3 c^2-5 d^2\right ) a^2-b^4 \left (4 c^4+d^2 c^2-4 d^4\right ) a-3 b^5 c d \left (3 c^2-4 d^2\right )\right ) \sin (e+f x)\right )}{2 \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (a^4 d^3+a^2 b^2 d \left (11 c^2-13 d^2\right )-4 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-8 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}}{\left (a^2-b^2\right ) (b c-a d)}}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {4 b^2 \left (-5 a^2 d+2 a b c+3 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {\frac {\int \frac {3 c d^3 a^5-b \left (9 c^2 d^2-8 d^4\right ) a^4+3 b^2 c d \left (3 c^2-5 d^2\right ) a^3-b^3 \left (3 c^4-33 d^2 c^2+28 d^4\right ) a^2-3 b^4 c d \left (3 c^2-4 d^2\right ) a-b^5 \left (c^4+16 d^2 c^2-16 d^4\right )+\left (-d^4 a^5+3 b c d^3 a^4+b^2 d^2 \left (9 c^2-7 d^2\right ) a^3+3 b^3 c d \left (3 c^2-5 d^2\right ) a^2-b^4 \left (4 c^4+d^2 c^2-4 d^4\right ) a-3 b^5 c d \left (3 c^2-4 d^2\right )\right ) \sin (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{\left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (a^4 d^3+a^2 b^2 d \left (11 c^2-13 d^2\right )-4 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-8 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}}{\left (a^2-b^2\right ) (b c-a d)}}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {4 b^2 \left (-5 a^2 d+2 a b c+3 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {\frac {\int \frac {3 c d^3 a^5-b \left (9 c^2 d^2-8 d^4\right ) a^4+3 b^2 c d \left (3 c^2-5 d^2\right ) a^3-b^3 \left (3 c^4-33 d^2 c^2+28 d^4\right ) a^2-3 b^4 c d \left (3 c^2-4 d^2\right ) a-b^5 \left (c^4+16 d^2 c^2-16 d^4\right )+\left (-d^4 a^5+3 b c d^3 a^4+b^2 d^2 \left (9 c^2-7 d^2\right ) a^3+3 b^3 c d \left (3 c^2-5 d^2\right ) a^2-b^4 \left (4 c^4+d^2 c^2-4 d^4\right ) a-3 b^5 c d \left (3 c^2-4 d^2\right )\right ) \sin (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{\left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (a^4 d^3+a^2 b^2 d \left (11 c^2-13 d^2\right )-4 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-8 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}}{\left (a^2-b^2\right ) (b c-a d)}}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \frac {\frac {4 b^2 \left (-5 a^2 d+2 a b c+3 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {\frac {\frac {(a-b) \left (a^4 d^3 (3 c+d)-9 a^3 b d^2 \left (c^2-d^2\right )+a^2 b^2 d \left (9 c^3-18 c^2 d-15 c d^2+16 d^3\right )-3 a b^3 \left (c^4-5 c^2 d^2+4 d^4\right )+b^4 \left (c^4-9 c^3 d+16 c^2 d^2+12 c d^3-16 d^4\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{c-d}-\frac {4 \left (a^5 c d^4-a^4 b \left (3 c^2 d^3-2 d^5\right )-2 a^3 b^2 c d^4-a^2 b^3 d \left (3 c^4-12 c^2 d^2+7 d^4\right )+a b^4 c \left (c^4-2 c^2 d^2+2 d^4\right )+b^5 d \left (2 c^4-7 c^2 d^2+4 d^4\right )\right ) \int \frac {\sin (e+f x)+1}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{c-d}}{\left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (a^4 d^3+a^2 b^2 d \left (11 c^2-13 d^2\right )-4 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-8 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}}{\left (a^2-b^2\right ) (b c-a d)}}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {4 b^2 \left (-5 a^2 d+2 a b c+3 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {\frac {\frac {(a-b) \left (a^4 d^3 (3 c+d)-9 a^3 b d^2 \left (c^2-d^2\right )+a^2 b^2 d \left (9 c^3-18 c^2 d-15 c d^2+16 d^3\right )-3 a b^3 \left (c^4-5 c^2 d^2+4 d^4\right )+b^4 \left (c^4-9 c^3 d+16 c^2 d^2+12 c d^3-16 d^4\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{c-d}-\frac {4 \left (a^5 c d^4-a^4 b \left (3 c^2 d^3-2 d^5\right )-2 a^3 b^2 c d^4-a^2 b^3 d \left (3 c^4-12 c^2 d^2+7 d^4\right )+a b^4 c \left (c^4-2 c^2 d^2+2 d^4\right )+b^5 d \left (2 c^4-7 c^2 d^2+4 d^4\right )\right ) \int \frac {\sin (e+f x)+1}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{c-d}}{\left (c^2-d^2\right ) (b c-a d)}+\frac {2 d \left (a^4 d^3+a^2 b^2 d \left (11 c^2-13 d^2\right )-4 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-8 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}}{\left (a^2-b^2\right ) (b c-a d)}}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x)}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3297 |
| \(\displaystyle \frac {2 \cos (e+f x) b^2}{3 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}}+\frac {\frac {4 b^2 \left (-5 d a^2+2 b c a+3 b^2 d\right ) \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {\frac {2 d \left (d^3 a^4+b^2 d \left (11 c^2-13 d^2\right ) a^2-4 b^3 c \left (c^2-d^2\right ) a-b^4 d \left (7 c^2-8 d^2\right )\right ) \sqrt {a+b \sin (e+f x)} \cos (e+f x)}{(b c-a d) \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {\frac {2 (a-b) \sqrt {a+b} \left (d^3 (3 c+d) a^4-9 b d^2 \left (c^2-d^2\right ) a^3+b^2 d \left (9 c^3-18 d c^2-15 d^2 c+16 d^3\right ) a^2-3 b^3 \left (c^4-5 d^2 c^2+4 d^4\right ) a+b^4 \left (c^4-9 d c^3+16 d^2 c^2+12 d^3 c-16 d^4\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(c-d) \sqrt {c+d} (b c-a d) f}-\frac {4 \left (c d^4 a^5-b \left (3 c^2 d^3-2 d^5\right ) a^4-2 b^2 c d^4 a^3-b^3 d \left (3 c^4-12 d^2 c^2+7 d^4\right ) a^2+b^4 c \left (c^4-2 d^2 c^2+2 d^4\right ) a+b^5 d \left (2 c^4-7 d^2 c^2+4 d^4\right )\right ) \int \frac {\sin (e+f x)+1}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{c-d}}{(b c-a d) \left (c^2-d^2\right )}}{\left (a^2-b^2\right ) (b c-a d)}}{3 \left (a^2-b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 3475 |
| \(\displaystyle \frac {2 \cos (e+f x) b^2}{3 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}}+\frac {\frac {4 b^2 \left (-5 d a^2+2 b c a+3 b^2 d\right ) \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {\frac {2 d \left (d^3 a^4+b^2 d \left (11 c^2-13 d^2\right ) a^2-4 b^3 c \left (c^2-d^2\right ) a-b^4 d \left (7 c^2-8 d^2\right )\right ) \sqrt {a+b \sin (e+f x)} \cos (e+f x)}{(b c-a d) \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {\frac {8 (a-b) \sqrt {a+b} \left (c d^4 a^5-b \left (3 c^2 d^3-2 d^5\right ) a^4-2 b^2 c d^4 a^3-b^3 d \left (3 c^4-12 d^2 c^2+7 d^4\right ) a^2+b^4 c \left (c^4-2 d^2 c^2+2 d^4\right ) a+b^5 d \left (2 c^4-7 d^2 c^2+4 d^4\right )\right ) E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(c-d) \sqrt {c+d} (b c-a d)^2 f}+\frac {2 (a-b) \sqrt {a+b} \left (d^3 (3 c+d) a^4-9 b d^2 \left (c^2-d^2\right ) a^3+b^2 d \left (9 c^3-18 d c^2-15 d^2 c+16 d^3\right ) a^2-3 b^3 \left (c^4-5 d^2 c^2+4 d^4\right ) a+b^4 \left (c^4-9 d c^3+16 d^2 c^2+12 d^3 c-16 d^4\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(c-d) \sqrt {c+d} (b c-a d) f}}{(b c-a d) \left (c^2-d^2\right )}}{\left (a^2-b^2\right ) (b c-a d)}}{3 \left (a^2-b^2\right ) (b c-a d)}\) |
(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)*(c + d*Sin[e + f*x])^(3/2)) + ((4*b^2*(2*a*b*c - 5*a^2*d + 3*b^2*d)*Cos [e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[ e + f*x])^(3/2)) - ((2*d*(a^4*d^3 + a^2*b^2*d*(11*c^2 - 13*d^2) - b^4*d*(7 *c^2 - 8*d^2) - 4*a*b^3*c*(c^2 - d^2))*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x ]])/((b*c - a*d)*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + ((8*(a - b)*S qrt[a + b]*(a^5*c*d^4 - 2*a^3*b^2*c*d^4 + a*b^4*c*(c^4 - 2*c^2*d^2 + 2*d^4 ) + b^5*d*(2*c^4 - 7*c^2*d^2 + 4*d^4) - a^2*b^3*d*(3*c^4 - 12*c^2*d^2 + 7* d^4) - a^4*b*(3*c^2*d^3 - 2*d^5))*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]*(b* c - a*d)^2*f) + (2*(a - b)*Sqrt[a + b]*(a^4*d^3*(3*c + d) - 9*a^3*b*d^2*(c ^2 - d^2) + a^2*b^2*d*(9*c^3 - 18*c^2*d - 15*c*d^2 + 16*d^3) + b^4*(c^4 - 9*c^3*d + 16*c^2*d^2 + 12*c*d^3 - 16*d^4) - 3*a*b^3*(c^4 - 5*c^2*d^2 + 4*d ^4))*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))]*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])))...
3.8.100.3.1 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])))
result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 24.23 (sec) , antiderivative size = 902545, normalized size of antiderivative = 1009.56
\[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
integral(-sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(b^3*d^3*cos(f *x + e)^6 - 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + (a^2*b + b^3)*d^3)*cos(f*x + e) ^4 - (a^3 + 3*a*b^2)*c^3 - 3*(3*a^2*b + b^3)*c^2*d - 3*(a^3 + 3*a*b^2)*c*d ^2 - (3*a^2*b + b^3)*d^3 + 3*(a*b^2*c^3 + (3*a^2*b + 2*b^3)*c^2*d + (a^3 + 6*a*b^2)*c*d^2 + (2*a^2*b + b^3)*d^3)*cos(f*x + e)^2 - (3*(b^3*c*d^2 + a* b^2*d^3)*cos(f*x + e)^4 + (3*a^2*b + b^3)*c^3 + 3*(a^3 + 3*a*b^2)*c^2*d + 3*(3*a^2*b + b^3)*c*d^2 + (a^3 + 3*a*b^2)*d^3 - (b^3*c^3 + 9*a*b^2*c^2*d + 3*(3*a^2*b + 2*b^3)*c*d^2 + (a^3 + 6*a*b^2)*d^3)*cos(f*x + e)^2)*sin(f*x + e)), x)
Timed out. \[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2}} \, dx=\int \frac {1}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]