Integrand size = 27, antiderivative size = 392 \[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=-\frac {2 d^2 \cos (e+f x)}{3 (b c-3 d) \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}-\frac {2 d^2 \left (7 b c^2-12 c d-3 b d^2\right ) \cos (e+f x)}{3 (b c-3 d)^2 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}-\frac {2 d \left (7 b c^2-12 c d-3 b d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 (b c-3 d)^2 \left (c^2-d^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 d \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 (b c-3 d) \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {2 b^2 \operatorname {EllipticPi}\left (\frac {2 b}{3+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(3+b) (b c-3 d)^2 f \sqrt {c+d \sin (e+f x)}} \]
-2/3*d^2*cos(f*x+e)/(-a*d+b*c)/(c^2-d^2)/f/(c+d*sin(f*x+e))^(3/2)-2/3*d^2* (-4*a*c*d+7*b*c^2-3*b*d^2)*cos(f*x+e)/(-a*d+b*c)^2/(c^2-d^2)^2/f/(c+d*sin( f*x+e))^(1/2)+2/3*d*(-4*a*c*d+7*b*c^2-3*b*d^2)*(sin(1/2*e+1/4*Pi+1/2*f*x)^ 2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticE(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*d+b*c)^2/(c^2-d^2)^2/f/(( c+d*sin(f*x+e))/(c+d))^(1/2)-2/3*d*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin (1/2*e+1/4*Pi+1/2*f*x)*EllipticF(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))/(c+d))^(1/2)/(-a*d+b*c)/(c^2-d^2)/f/(c+d*sin(f *x+e))^(1/2)-2*b^2*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/ 2*f*x)*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)/(-a*d+b*c)^2/f/(c+d*sin(f*x+e))^ (1/2)
Result contains complex when optimal does not.
Time = 6.61 (sec) , antiderivative size = 1049, normalized size of antiderivative = 2.68 \[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\frac {\sqrt {c+d \sin (e+f x)} \left (-\frac {2 d^2 \cos (e+f x)}{3 (b c-3 d) \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}-\frac {2 \left (7 b c^2 d^2 \cos (e+f x)-12 c d^3 \cos (e+f x)-3 b d^4 \cos (e+f x)\right )}{3 (b c-3 d)^2 \left (c^2-d^2\right )^2 (c+d \sin (e+f x))}\right )}{f}-\frac {-\frac {2 \left (-6 b^2 c^4+36 b c^3 d-54 c^2 d^2+19 b^2 c^2 d^2-24 b c d^3-18 d^4-9 b^2 d^4\right ) \operatorname {EllipticPi}\left (\frac {2 b}{3+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}}}{(3+b) \sqrt {c+d \sin (e+f x)}}-\frac {2 i \left (12 b^2 c^3 d+24 b c^2 d^2-72 c d^3-4 b^2 c d^3-24 b d^4\right ) \cos (e+f x) \left ((b c-3 d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )+3 d \operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-3 d},i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )\right ) \sqrt {\frac {d-d \sin (e+f x)}{c+d}} \sqrt {-\frac {d+d \sin (e+f x)}{c-d}} (-b c+3 d+b (c+d \sin (e+f x)))}{b (b c-3 d) d^2 \sqrt {-\frac {1}{c+d}} (3+b \sin (e+f x)) \sqrt {1-\sin ^2(e+f x)} \sqrt {-\frac {c^2-d^2-2 c (c+d \sin (e+f x))+(c+d \sin (e+f x))^2}{d^2}}}-\frac {2 i \left (-7 b^2 c^2 d^2+12 b c d^3+3 b^2 d^4\right ) \cos (e+f x) \cos (2 (e+f x)) \left (2 b (b c-3 d) (c-d) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )+d \left (2 (3+b) (b c-3 d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )-\left (-18+b^2\right ) d \operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-3 d},i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )\right )\right ) \sqrt {\frac {d-d \sin (e+f x)}{c+d}} \sqrt {-\frac {d+d \sin (e+f x)}{c-d}} (-b c+3 d+b (c+d \sin (e+f x)))}{b^2 (b c-3 d) d \sqrt {-\frac {1}{c+d}} (3+b \sin (e+f x)) \sqrt {1-\sin ^2(e+f x)} \left (-2 c^2+d^2+4 c (c+d \sin (e+f x))-2 (c+d \sin (e+f x))^2\right ) \sqrt {-\frac {c^2-d^2-2 c (c+d \sin (e+f x))+(c+d \sin (e+f x))^2}{d^2}}}}{6 (b c-3 d)^2 (c-d)^2 (c+d)^2 f} \]
(Sqrt[c + d*Sin[e + f*x]]*((-2*d^2*Cos[e + f*x])/(3*(b*c - 3*d)*(c^2 - d^2 )*(c + d*Sin[e + f*x])^2) - (2*(7*b*c^2*d^2*Cos[e + f*x] - 12*c*d^3*Cos[e + f*x] - 3*b*d^4*Cos[e + f*x]))/(3*(b*c - 3*d)^2*(c^2 - d^2)^2*(c + d*Sin[ e + f*x]))))/f - ((-2*(-6*b^2*c^4 + 36*b*c^3*d - 54*c^2*d^2 + 19*b^2*c^2*d ^2 - 24*b*c*d^3 - 18*d^4 - 9*b^2*d^4)*EllipticPi[(2*b)/(3 + b), (-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((3 + b)*Sqr t[c + d*Sin[e + f*x]]) - ((2*I)*(12*b^2*c^3*d + 24*b*c^2*d^2 - 72*c*d^3 - 4*b^2*c*d^3 - 24*b*d^4)*Cos[e + f*x]*((b*c - 3*d)*EllipticF[I*ArcSinh[Sqrt [-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + 3*d*Elliptic Pi[(b*(c + d))/(b*c - 3*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) + 3*d + b*(c + d*Sin[e + f*x])))/(b*(b *c - 3*d)*d^2*Sqrt[-(c + d)^(-1)]*(3 + 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*Sin[e + f*x])^ 2)/d^2)]) - ((2*I)*(-7*b^2*c^2*d^2 + 12*b*c*d^3 + 3*b^2*d^4)*Cos[e + f*x]* Cos[2*(e + f*x)]*(2*b*(b*c - 3*d)*(c - d)*EllipticE[I*ArcSinh[Sqrt[-(c + d )^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + d*(2*(3 + b)*(b*c - 3*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] - (-18 + b^2)*d*EllipticPi[(b*(c + d))/(b*c - 3*d), I*ArcSi nh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)]))*Sq...
Time = 3.65 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.10, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {3042, 3281, 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.
| \(\displaystyle \int \frac {1}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3281 |
| \(\displaystyle \frac {2 \int -\frac {-b d^2 \sin ^2(e+f x)+d (3 b c-a d) \sin (e+f x)+3 \left (a c d-b \left (c^2-d^2\right )\right )}{2 (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {-b d^2 \sin ^2(e+f x)+d (3 b c-a d) \sin (e+f x)+3 \left (a c d-b \left (c^2-d^2\right )\right )}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {-b d^2 \sin (e+f x)^2+d (3 b c-a d) \sin (e+f x)+3 \left (a c d-b \left (c^2-d^2\right )\right )}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle -\frac {\frac {2 \int -\frac {-b \left (7 b c^2-4 a d c-3 b d^2\right ) \sin ^2(e+f x) d^2+a^2 \left (3 c^2+d^2\right ) d^2+2 \left (-\left (\left (3 c^3-c d^2\right ) b^2\right )-2 a d \left (c^2-d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x) d+3 b^2 \left (c^2-d^2\right )^2-a b \left (6 c^3 d-2 c d^3\right )}{2 (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{\left (c^2-d^2\right ) (b c-a d)}+\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\int \frac {-b \left (7 b c^2-4 a d c-3 b d^2\right ) \sin ^2(e+f x) d^2+a^2 \left (3 c^2+d^2\right ) d^2-2 a b c \left (3 c^2-d^2\right ) d+2 \left (-\left (\left (3 c^3-c d^2\right ) b^2\right )-2 a d \left (c^2-d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x) d+3 b^2 \left (c^2-d^2\right )^2}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\int \frac {-b \left (7 b c^2-4 a d c-3 b d^2\right ) \sin (e+f x)^2 d^2+a^2 \left (3 c^2+d^2\right ) d^2-2 a b c \left (3 c^2-d^2\right ) d+2 \left (-\left (\left (3 c^3-c d^2\right ) b^2\right )-2 a d \left (c^2-d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x) d+3 b^2 \left (c^2-d^2\right )^2}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {-\frac {\int -\frac {b^2 (b c-a d) \left (c^2-d^2\right ) \sin (e+f x) d^2+b \left (c^2-d^2\right ) \left (3 \left (c^2-d^2\right ) b^2+a c d b-a^2 d^2\right ) d}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-d \left (-4 a c d+7 b c^2-3 b d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\int \frac {b^2 (b c-a d) \left (c^2-d^2\right ) \sin (e+f x) d^2+b \left (c^2-d^2\right ) \left (3 \left (c^2-d^2\right ) b^2+a c d b-a^2 d^2\right ) d}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-d \left (-4 a c d+7 b c^2-3 b d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\int \frac {b^2 (b c-a d) \left (c^2-d^2\right ) \sin (e+f x) d^2+b \left (c^2-d^2\right ) \left (3 \left (c^2-d^2\right ) b^2+a c d b-a^2 d^2\right ) d}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-d \left (-4 a c d+7 b c^2-3 b d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\int \frac {b^2 (b c-a d) \left (c^2-d^2\right ) \sin (e+f x) d^2+b \left (c^2-d^2\right ) \left (3 \left (c^2-d^2\right ) b^2+a c d b-a^2 d^2\right ) d}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {d \left (-4 a c d+7 b c^2-3 b d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\int \frac {b^2 (b c-a d) \left (c^2-d^2\right ) \sin (e+f x) d^2+b \left (c^2-d^2\right ) \left (3 \left (c^2-d^2\right ) b^2+a c d b-a^2 d^2\right ) d}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {d \left (-4 a c d+7 b c^2-3 b d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\int \frac {b^2 (b c-a d) \left (c^2-d^2\right ) \sin (e+f x) d^2+b \left (c^2-d^2\right ) \left (3 \left (c^2-d^2\right ) b^2+a c d b-a^2 d^2\right ) d}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {2 d \left (-4 a c d+7 b c^2-3 b d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {3 b^3 d \left (c^2-d^2\right )^2 \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx+b d^2 \left (c^2-d^2\right ) (b c-a d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {2 d \left (-4 a c d+7 b c^2-3 b d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {3 b^3 d \left (c^2-d^2\right )^2 \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx+b d^2 \left (c^2-d^2\right ) (b c-a d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {2 d \left (-4 a c d+7 b c^2-3 b d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {3 b^3 d \left (c^2-d^2\right )^2 \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx+\frac {b d^2 \left (c^2-d^2\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}}{b d}-\frac {2 d \left (-4 a c d+7 b c^2-3 b d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {3 b^3 d \left (c^2-d^2\right )^2 \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx+\frac {b d^2 \left (c^2-d^2\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}}{b d}-\frac {2 d \left (-4 a c d+7 b c^2-3 b d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {3 b^3 d \left (c^2-d^2\right )^2 \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx+\frac {2 b d^2 \left (c^2-d^2\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}}{b d}-\frac {2 d \left (-4 a c d+7 b c^2-3 b d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\frac {3 b^3 d \left (c^2-d^2\right )^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}+\frac {2 b d^2 \left (c^2-d^2\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}}{b d}-\frac {2 d \left (-4 a c d+7 b c^2-3 b d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\frac {3 b^3 d \left (c^2-d^2\right )^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}+\frac {2 b d^2 \left (c^2-d^2\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}}{b d}-\frac {2 d \left (-4 a c d+7 b c^2-3 b d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\frac {6 b^3 d \left (c^2-d^2\right )^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f (a+b) \sqrt {c+d \sin (e+f x)}}+\frac {2 b d^2 \left (c^2-d^2\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}}{b d}-\frac {2 d \left (-4 a c d+7 b c^2-3 b d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\) |
(-2*d^2*Cos[e + f*x])/(3*(b*c - a*d)*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3 /2)) - ((2*d^2*(7*b*c^2 - 4*a*c*d - 3*b*d^2)*Cos[e + f*x])/((b*c - a*d)*(c ^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) - ((-2*d*(7*b*c^2 - 4*a*c*d - 3*b*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*d^2*(b*c - a*d)*(c^2 - d^2) *EllipticF[(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]]) + (6*b^3*d*(c^2 - d^2)^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)))/(3*(b*c - a*d)*(c^2 - d^2))
3.8.50.3.1 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[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. \(1071\) vs. \(2(474)=948\).
Time = 7.98 (sec) , antiderivative size = 1072, normalized size of antiderivative = 2.73
(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*(2*b/(a*d-b*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)*(1/(c-d)*(-sin(f*x+e) -1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(-c/d+a/b)*EllipticPi (((c+d*sin(f*x+e))/(c-d))^(1/2),(-c/d+1)/(-c/d+a/b),((c-d)/(c+d))^(1/2))+d /(a*d-b*c)*(2/3/(c^2-d^2)/d*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(sin(f *x+e)+c/d)^2+8/3*d*cos(f*x+e)^2/(c^2-d^2)^2*c/(-(-d*sin(f*x+e)-c)*cos(f*x+ e)^2)^(1/2)+2*(3*c^2+d^2)/(3*c^4-6*c^2*d^2+3*d^4)*(c/d-1)*((c+d*sin(f*x+e) )/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^ (1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/ (c-d))^(1/2),((c-d)/(c+d))^(1/2))+8/3*c*d/(c^2-d^2)^2*(c/d-1)*((c+d*sin(f* x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1) *d)^(1/2)/(-(-d*sin(f*x+e)-c)*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*d/(a*d-b*c)^2*(2*d*cos(f*x+e)^2/(c ^2-d^2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2*c/(c^2-d^2)*(c/d-1)*((c+ d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f *x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*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)*(1/(c-d)*(-sin(f*x +e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*Ellip...
Timed out. \[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\int \frac {1}{\left (a+b\,\sin \left (e+f\,x\right )\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]