Integrand size = 27, antiderivative size = 296 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+b \sin (e+f x))^2} \, dx=\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{\left (9-b^2\right ) f (3+b \sin (e+f x))}+\frac {E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{\left (9-b^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {(b c-3 d) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{b \left (9-b^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {\left (6 b c-9 d-b^2 d\right ) \operatorname {EllipticPi}\left (\frac {2 b}{3+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(3-b) b (3+b)^2 f \sqrt {c+d \sin (e+f x)}} \]
b*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/(a^2-b^2)/f/(a+b*sin(f*x+e))-(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^2-b^2)/f /((c+d*sin(f*x+e))/(c+d))^(1/2)+(-a*d+b*c)*(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)/b/(a^2-b^2)/f/(c+d*sin(f* x+e))^(1/2)-(-a^2*d+2*a*b*c-b^2*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)*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)/b/(a+b)^2/f/(c+d *sin(f*x+e))^(1/2)
Result contains complex when optimal does not.
Time = 16.61 (sec) , antiderivative size = 575, normalized size of antiderivative = 1.94 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+b \sin (e+f x))^2} \, dx=\frac {\frac {24 i \left ((b c-3 d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )+3 d \operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-3 d},i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )\right ) \sec (e+f x) \sqrt {-\frac {d (-1+\sin (e+f x))}{c+d}} \sqrt {-\frac {d (1+\sin (e+f x))}{c-d}}}{b (b c-3 d) \sqrt {-\frac {1}{c+d}}}+\frac {2 i \left (-2 b (b c-3 d) (c-d) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )+d \left (-2 (3+b) (b c-3 d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )+\left (-18+b^2\right ) d \operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-3 d},i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )\right )\right ) \sec (e+f x) \sqrt {-\frac {d (-1+\sin (e+f x))}{c+d}} \sqrt {-\frac {d (1+\sin (e+f x))}{c-d}}}{b (b c-3 d) d \sqrt {-\frac {1}{c+d}}}-\frac {4 b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3+b \sin (e+f x)}+\frac {2 (12 c-b d) \operatorname {EllipticPi}\left (\frac {2 b}{3+b},\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(3+b) \sqrt {c+d \sin (e+f x)}}}{4 \left (-9+b^2\right ) f} \]
(((24*I)*((b*c - 3*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*S in[e + f*x]]], (c + d)/(c - d)] + 3*d*EllipticPi[(b*(c + d))/(b*c - 3*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)]) *Sec[e + f*x]*Sqrt[-((d*(-1 + Sin[e + f*x]))/(c + d))]*Sqrt[-((d*(1 + Sin[ e + f*x]))/(c - d))])/(b*(b*c - 3*d)*Sqrt[-(c + d)^(-1)]) + ((2*I)*(-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*ArcSi nh[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*ArcSinh[Sqrt[-(c + d)^(-1)] *Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)]))*Sec[e + f*x]*Sqrt[-((d*(-1 + Sin[e + f*x]))/(c + d))]*Sqrt[-((d*(1 + Sin[e + f*x]))/(c - d))])/(b*(b* c - 3*d)*d*Sqrt[-(c + d)^(-1)]) - (4*b*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x ]])/(3 + b*Sin[e + f*x]) + (2*(12*c - b*d)*EllipticPi[(2*b)/(3 + b), (-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((3 + b)*Sqrt[c + d*Sin[e + f*x]]))/(4*(-9 + b^2)*f)
Time = 2.12 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.01, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3275, 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.
| \(\displaystyle \int \frac {\sqrt {c+d \sin (e+f x)}}{(a+b \sin (e+f x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {c+d \sin (e+f x)}}{(a+b \sin (e+f x))^2}dx\) |
| \(\Big \downarrow \) 3275 |
| \(\displaystyle \frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\int -\frac {b d \sin ^2(e+f x)+2 a d \sin (e+f x)+2 a c-b d}{2 (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{a^2-b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {b d \sin ^2(e+f x)+2 a d \sin (e+f x)+2 a c-b d}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{2 \left (a^2-b^2\right )}+\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {b d \sin (e+f x)^2+2 a d \sin (e+f x)+2 a c-b d}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{2 \left (a^2-b^2\right )}+\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {\int \sqrt {c+d \sin (e+f x)}dx-\frac {\int -\frac {b d (a c-b d)-b d (b c-a d) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}}{2 \left (a^2-b^2\right )}+\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {b d (a c-b d)-b d (b c-a d) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}+\int \sqrt {c+d \sin (e+f x)}dx}{2 \left (a^2-b^2\right )}+\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {b d (a c-b d)-b d (b c-a d) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}+\int \sqrt {c+d \sin (e+f x)}dx}{2 \left (a^2-b^2\right )}+\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {\int \frac {b d (a c-b d)-b d (b c-a d) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}+\frac {\sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right )}+\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {b d (a c-b d)-b d (b c-a d) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}+\frac {\sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right )}+\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {\int \frac {b d (a c-b d)-b d (b c-a d) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}+\frac {2 \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right )}+\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {\frac {d \left (a^2 (-d)+2 a b c-b^2 d\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx-d (b c-a d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{b d}+\frac {2 \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right )}+\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {d \left (a^2 (-d)+2 a b c-b^2 d\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx-d (b c-a d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{b d}+\frac {2 \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right )}+\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {d \left (a^2 (-d)+2 a b c-b^2 d\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx-\frac {d (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}}{b d}+\frac {2 \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right )}+\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {d \left (a^2 (-d)+2 a b c-b^2 d\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx-\frac {d (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}}{b d}+\frac {2 \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right )}+\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {d \left (a^2 (-d)+2 a b c-b^2 d\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx-\frac {2 d (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}}{b d}+\frac {2 \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right )}+\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {\frac {\frac {d \left (a^2 (-d)+2 a b c-b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}-\frac {2 d (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}}{b d}+\frac {2 \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right )}+\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {d \left (a^2 (-d)+2 a b c-b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}-\frac {2 d (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}}{b d}+\frac {2 \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right )}+\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}+\frac {\frac {\frac {2 d \left (a^2 (-d)+2 a b c-b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f (a+b) \sqrt {c+d \sin (e+f x)}}-\frac {2 d (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}}{b d}+\frac {2 \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right )}\) |
(b*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/((a^2 - b^2)*f*(a + b*Sin[e + f* x])) + ((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*d*(b*c - a*d)*Ellipt icF[(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]]) + (2*d*(2*a*b*c - a^2*d - b^2*d)*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))/(2*(a^2 - b^2))
3.8.54.3.1 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)*Cos[e + f*x]*(a + b*Sin[e + f*x] )^(m + 1)*((c + d*Sin[e + f*x])^n/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^ (n - 1)*Simp[a*c*(m + 1) + b*d*n + (a*d*(m + 1) - b*c*(m + 2))*Sin[e + f*x] - b*d*(m + n + 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] && LtQ[m, -1] && LtQ[0, n, 1] && IntegersQ[2*m, 2*n]
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_)] + (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. \(871\) vs. \(2(393)=786\).
Time = 6.21 (sec) , antiderivative size = 872, normalized size of antiderivative = 2.95
| method | result | size |
| default | \(\frac {\sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (\frac {2 d \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, \Pi \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \frac {-\frac {c}{d}+1}{-\frac {c}{d}+\frac {a}{b}}, \sqrt {\frac {c -d}{c +d}}\right )}{b^{2} \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (-\frac {c}{d}+\frac {a}{b}\right )}+\frac {\left (-d a +c b \right ) \left (-\frac {b^{2} \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}{\left (a^{3} d -a^{2} b c -a \,b^{2} d +b^{3} c \right ) \left (a +b \sin \left (f x +e \right )\right )}-\frac {a d \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{\left (a^{3} d -a^{2} b c -a \,b^{2} d +b^{3} c \right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}-\frac {b d \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, \left (\left (-\frac {c}{d}-1\right ) E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )+F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )\right )}{\left (a^{3} d -a^{2} b c -a \,b^{2} d +b^{3} c \right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}+\frac {\left (3 a^{2} d -2 a b c -b^{2} d \right ) \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, \Pi \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \frac {-\frac {c}{d}+1}{-\frac {c}{d}+\frac {a}{b}}, \sqrt {\frac {c -d}{c +d}}\right )}{\left (a^{3} d -a^{2} b c -a \,b^{2} d +b^{3} c \right ) b \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (-\frac {c}{d}+\frac {a}{b}\right )}\right )}{b}\right )}{\cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\) | \(872\) |
(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*(2*d/b^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*s in(f*x+e))/(c-d))^(1/2),(-c/d+1)/(-c/d+a/b),((c-d)/(c+d))^(1/2))+(-a*d+b*c )/b*(-b^2/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^ (1/2)/(a+b*sin(f*x+e))-a*d/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)*(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))-b*d/(a^3*d-a^2*b*c-a*b^2*d+b^3*c) *(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))+Ellip ticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)))+(3*a^2*d-2*a*b*c -b^2*d)/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)/b*(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))))/cos(f*x+e)/(c+d* sin(f*x+e))^(1/2)/f
Timed out. \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+b \sin (e+f x))^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+b \sin (e+f x))^2} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+b \sin (e+f x))^2} \, dx=\int { \frac {\sqrt {d \sin \left (f x + e\right ) + c}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+b \sin (e+f x))^2} \, dx=\int { \frac {\sqrt {d \sin \left (f x + e\right ) + c}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+b \sin (e+f x))^2} \, dx=\int \frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]