Integrand size = 33, antiderivative size = 238 \[ \int \frac {\csc (e+f x) \sqrt {a+b \sin (e+f x)}}{c+c \sin (e+f x)} \, dx=\frac {E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (e+f x)}}{c f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {(a-b) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{c f \sqrt {a+b \sin (e+f x)}}+\frac {2 a \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{c f \sqrt {a+b \sin (e+f x)}}+\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (c+c \sin (e+f x))} \]
cos(f*x+e)*(a+b*sin(f*x+e))^(1/2)/f/(c+c*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)*(b/(a+b))^(1/2))*(a+b*sin(f*x+e))^(1/2)/c/f/((a+b*sin(f*x+e))/( a+b))^(1/2)+(a-b)*(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)*(b/(a+b))^(1/2))*((a+b*s in(f*x+e))/(a+b))^(1/2)/c/f/(a+b*sin(f*x+e))^(1/2)-2*a*(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),2,2^(1/2)*(b/(a+b))^(1/2))*((a+b*sin(f*x+e))/(a+b))^(1/2)/c/f/(a+b*s in(f*x+e))^(1/2)
Result contains complex when optimal does not.
Time = 19.05 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.90 \[ \int \frac {\csc (e+f x) \sqrt {a+b \sin (e+f x)}}{c+c \sin (e+f x)} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (-8 \sin \left (\frac {1}{2} (e+f x)\right ) \sqrt {a+b \sin (e+f x)}+\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (-\frac {2 i \left (-2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (e+f x)}\right )|\frac {a+b}{a-b}\right )+b \left (-2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (e+f x)}\right ),\frac {a+b}{a-b}\right )+b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (e+f x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sec (e+f x) \sqrt {-\frac {b (-1+\sin (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sin (e+f x))}{a-b}}}{a b \sqrt {-\frac {1}{a+b}}}+4 \sqrt {a+b \sin (e+f x)}-\frac {4 b \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{\sqrt {a+b \sin (e+f x)}}-\frac {2 (4 a+b) \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{\sqrt {a+b \sin (e+f x)}}\right )\right )}{4 c f (1+\sin (e+f x))} \]
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-8*Sin[(e + f*x)/2]*Sqrt[a + b*Sin [e + f*x]] + (Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(((-2*I)*(-2*a*(a - b)* EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[e + f*x]]], (a + b) /(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin [e + f*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[e + f*x]]], (a + b)/(a - b)]))*Sec[e + f*x]*Sqr t[-((b*(-1 + Sin[e + f*x]))/(a + b))]*Sqrt[-((b*(1 + Sin[e + f*x]))/(a - b ))])/(a*b*Sqrt[-(a + b)^(-1)]) + 4*Sqrt[a + b*Sin[e + f*x]] - (4*b*Ellipti cF[(-2*e + Pi - 2*f*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b) ])/Sqrt[a + b*Sin[e + f*x]] - (2*(4*a + b)*EllipticPi[2, (-2*e + Pi - 2*f* x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/Sqrt[a + b*Sin[e + f*x]])))/(4*c*f*(1 + Sin[e + f*x]))
Time = 1.77 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.15, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.515, Rules used = {3042, 3414, 3042, 3247, 27, 3042, 3231, 3042, 3134, 3042, 3132, 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.
| \(\displaystyle \int \frac {\csc (e+f x) \sqrt {a+b \sin (e+f x)}}{c \sin (e+f x)+c} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {a+b \sin (e+f x)}}{\sin (e+f x) (c \sin (e+f x)+c)}dx\) |
| \(\Big \downarrow \) 3414 |
| \(\displaystyle \frac {a \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)}}dx}{c}-(a-b) \int \frac {1}{\sqrt {a+b \sin (e+f x)} (\sin (e+f x) c+c)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \int \frac {1}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{c}-(a-b) \int \frac {1}{\sqrt {a+b \sin (e+f x)} (\sin (e+f x) c+c)}dx\) |
| \(\Big \downarrow \) 3247 |
| \(\displaystyle \frac {a \int \frac {1}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{c}-(a-b) \left (\frac {b \int -\frac {\sin (e+f x) c+c}{2 \sqrt {a+b \sin (e+f x)}}dx}{c^2 (a-b)}-\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \int \frac {1}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{c}-(a-b) \left (-\frac {b \int \frac {\sin (e+f x) c+c}{\sqrt {a+b \sin (e+f x)}}dx}{2 c^2 (a-b)}-\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \int \frac {1}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{c}-(a-b) \left (-\frac {b \int \frac {\sin (e+f x) c+c}{\sqrt {a+b \sin (e+f x)}}dx}{2 c^2 (a-b)}-\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\right )\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {a \int \frac {1}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{c}-(a-b) \left (-\frac {b \left (\frac {c \int \sqrt {a+b \sin (e+f x)}dx}{b}-\frac {c (a-b) \int \frac {1}{\sqrt {a+b \sin (e+f x)}}dx}{b}\right )}{2 c^2 (a-b)}-\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \int \frac {1}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{c}-(a-b) \left (-\frac {b \left (\frac {c \int \sqrt {a+b \sin (e+f x)}dx}{b}-\frac {c (a-b) \int \frac {1}{\sqrt {a+b \sin (e+f x)}}dx}{b}\right )}{2 c^2 (a-b)}-\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\right )\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {a \int \frac {1}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{c}-(a-b) \left (-\frac {b \left (\frac {c \sqrt {a+b \sin (e+f x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {c (a-b) \int \frac {1}{\sqrt {a+b \sin (e+f x)}}dx}{b}\right )}{2 c^2 (a-b)}-\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \int \frac {1}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{c}-(a-b) \left (-\frac {b \left (\frac {c \sqrt {a+b \sin (e+f x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {c (a-b) \int \frac {1}{\sqrt {a+b \sin (e+f x)}}dx}{b}\right )}{2 c^2 (a-b)}-\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\right )\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {a \int \frac {1}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{c}-(a-b) \left (-\frac {b \left (\frac {2 c \sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {c (a-b) \int \frac {1}{\sqrt {a+b \sin (e+f x)}}dx}{b}\right )}{2 c^2 (a-b)}-\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\right )\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {a \int \frac {1}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{c}-(a-b) \left (-\frac {b \left (\frac {2 c \sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {c (a-b) \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}}dx}{b \sqrt {a+b \sin (e+f x)}}\right )}{2 c^2 (a-b)}-\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \int \frac {1}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{c}-(a-b) \left (-\frac {b \left (\frac {2 c \sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {c (a-b) \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}}dx}{b \sqrt {a+b \sin (e+f x)}}\right )}{2 c^2 (a-b)}-\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\right )\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {a \int \frac {1}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{c}-(a-b) \left (-\frac {b \left (\frac {2 c \sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {2 c (a-b) \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b f \sqrt {a+b \sin (e+f x)}}\right )}{2 c^2 (a-b)}-\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\right )\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {a \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \int \frac {\csc (e+f x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}}dx}{c \sqrt {a+b \sin (e+f x)}}-(a-b) \left (-\frac {b \left (\frac {2 c \sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {2 c (a-b) \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b f \sqrt {a+b \sin (e+f x)}}\right )}{2 c^2 (a-b)}-\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \int \frac {1}{\sin (e+f x) \sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}}dx}{c \sqrt {a+b \sin (e+f x)}}-(a-b) \left (-\frac {b \left (\frac {2 c \sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {2 c (a-b) \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b f \sqrt {a+b \sin (e+f x)}}\right )}{2 c^2 (a-b)}-\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\right )\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {2 a \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{c f \sqrt {a+b \sin (e+f x)}}-(a-b) \left (-\frac {b \left (\frac {2 c \sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {2 c (a-b) \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b f \sqrt {a+b \sin (e+f x)}}\right )}{2 c^2 (a-b)}-\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\right )\) |
(2*a*EllipticPi[2, (e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(c*f*Sqrt[a + b*Sin[e + f*x]]) - (a - b)*(-((Cos[e + f*x]* Sqrt[a + b*Sin[e + f*x]])/((a - b)*f*(c + c*Sin[e + f*x]))) - (b*((2*c*Ell ipticE[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[e + f*x]])/(b*f*S qrt[(a + b*Sin[e + f*x])/(a + b)]) - (2*(a - b)*c*EllipticF[(e - Pi/2 + f* x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(b*f*Sqrt[a + b*S in[e + f*x]])))/(2*(a - b)*c^2))
3.1.29.3.1 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[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*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]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(b*c - a*d)*(a + b*Sin[e + f*x]))), x] + Simp[d/(a*(b*c - a*d)) Int[(c + d*Sin[e + f*x])^n*(a*n - b*(n + 1)*Sin[e + f*x]), x], x] /; Fre eQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[ c^2 - d^2, 0] && LtQ[n, 0] && (IntegerQ[2*n] || EqQ[c, 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[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/(sin[(e_.) + (f_.)*(x_)]*((c _) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[a/c Int[1/(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]), x], x] + Simp[(b*c - a*d)/c Int[1/(Sqrt [a + b*Sin[e + f*x]]*(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]
Time = 1.32 (sec) , antiderivative size = 593, normalized size of antiderivative = 2.49
| method | result | size |
| default | \(\frac {\sqrt {-\left (-b \sin \left (f x +e \right )-a \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (-\frac {2 \left (\frac {a}{b}-1\right ) \sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}\, \sqrt {\frac {\left (1-\sin \left (f x +e \right )\right ) b}{a +b}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) b}{a -b}}\, b \Pi \left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, -\frac {\left (-\frac {a}{b}+1\right ) b}{a}, \sqrt {\frac {a -b}{a +b}}\right )}{\sqrt {-\left (-b \sin \left (f x +e \right )-a \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}+\left (-a +b \right ) \left (-\frac {-b \left (\sin ^{2}\left (f x +e \right )\right )-a \sin \left (f x +e \right )+b \sin \left (f x +e \right )+a}{\left (a -b \right ) \sqrt {\left (1+\sin \left (f x +e \right )\right ) \left (\sin \left (f x +e \right )-1\right ) \left (-b \sin \left (f x +e \right )-a \right )}}-\frac {2 b \left (\frac {a}{b}-1\right ) \sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}\, \sqrt {\frac {\left (1-\sin \left (f x +e \right )\right ) b}{a +b}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) b}{a -b}}\, F\left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right )}{\left (2 a -2 b \right ) \sqrt {-\left (-b \sin \left (f x +e \right )-a \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}-\frac {b \left (\frac {a}{b}-1\right ) \sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}\, \sqrt {\frac {\left (1-\sin \left (f x +e \right )\right ) b}{a +b}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) b}{a -b}}\, \left (\left (-\frac {a}{b}-1\right ) E\left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right )+F\left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right )\right )}{\left (a -b \right ) \sqrt {-\left (-b \sin \left (f x +e \right )-a \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}\right )\right )}{c \cos \left (f x +e \right ) \sqrt {a +b \sin \left (f x +e \right )}\, f}\) | \(593\) |
(-(-b*sin(f*x+e)-a)*cos(f*x+e)^2)^(1/2)/c*(-2*(1/b*a-1)*((a+b*sin(f*x+e))/ (a-b))^(1/2)*(1/(a+b)*(1-sin(f*x+e))*b)^(1/2)*(1/(a-b)*(-sin(f*x+e)-1)*b)^ (1/2)/(-(-b*sin(f*x+e)-a)*cos(f*x+e)^2)^(1/2)*b*EllipticPi(((a+b*sin(f*x+e ))/(a-b))^(1/2),-(-1/b*a+1)*b/a,((a-b)/(a+b))^(1/2))+(-a+b)*(-(-b*sin(f*x+ e)^2-a*sin(f*x+e)+b*sin(f*x+e)+a)/(a-b)/((1+sin(f*x+e))*(sin(f*x+e)-1)*(-b *sin(f*x+e)-a))^(1/2)-2*b/(2*a-2*b)*(1/b*a-1)*((a+b*sin(f*x+e))/(a-b))^(1/ 2)*(1/(a+b)*(1-sin(f*x+e))*b)^(1/2)*(1/(a-b)*(-sin(f*x+e)-1)*b)^(1/2)/(-(- b*sin(f*x+e)-a)*cos(f*x+e)^2)^(1/2)*EllipticF(((a+b*sin(f*x+e))/(a-b))^(1/ 2),((a-b)/(a+b))^(1/2))-b/(a-b)*(1/b*a-1)*((a+b*sin(f*x+e))/(a-b))^(1/2)*( 1/(a+b)*(1-sin(f*x+e))*b)^(1/2)*(1/(a-b)*(-sin(f*x+e)-1)*b)^(1/2)/(-(-b*si n(f*x+e)-a)*cos(f*x+e)^2)^(1/2)*((-1/b*a-1)*EllipticE(((a+b*sin(f*x+e))/(a -b))^(1/2),((a-b)/(a+b))^(1/2))+EllipticF(((a+b*sin(f*x+e))/(a-b))^(1/2),( (a-b)/(a+b))^(1/2)))))/cos(f*x+e)/(a+b*sin(f*x+e))^(1/2)/f
Timed out. \[ \int \frac {\csc (e+f x) \sqrt {a+b \sin (e+f x)}}{c+c \sin (e+f x)} \, dx=\text {Timed out} \]
\[ \int \frac {\csc (e+f x) \sqrt {a+b \sin (e+f x)}}{c+c \sin (e+f x)} \, dx=\frac {\int \frac {\sqrt {a + b \sin {\left (e + f x \right )}}}{\sin ^{2}{\left (e + f x \right )} + \sin {\left (e + f x \right )}}\, dx}{c} \]
\[ \int \frac {\csc (e+f x) \sqrt {a+b \sin (e+f x)}}{c+c \sin (e+f x)} \, dx=\int { \frac {\sqrt {b \sin \left (f x + e\right ) + a}}{{\left (c \sin \left (f x + e\right ) + c\right )} \sin \left (f x + e\right )} \,d x } \]
\[ \int \frac {\csc (e+f x) \sqrt {a+b \sin (e+f x)}}{c+c \sin (e+f x)} \, dx=\int { \frac {\sqrt {b \sin \left (f x + e\right ) + a}}{{\left (c \sin \left (f x + e\right ) + c\right )} \sin \left (f x + e\right )} \,d x } \]
Timed out. \[ \int \frac {\csc (e+f x) \sqrt {a+b \sin (e+f x)}}{c+c \sin (e+f x)} \, dx=\int \frac {\sqrt {a+b\,\sin \left (e+f\,x\right )}}{\sin \left (e+f\,x\right )\,\left (c+c\,\sin \left (e+f\,x\right )\right )} \,d x \]