Integrand size = 33, antiderivative size = 246 \[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\frac {E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (e+f x)}}{(a-b) c f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {\operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{c f \sqrt {a+b \sin (e+f x)}}+\frac {2 \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{c f \sqrt {a+b \sin (e+f x)}}+\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{(a-b) f (c+c \sin (e+f x))} \] Output:
-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)/(a-b)/c/f/((a+b*sin(f*x+e))/(a+b))^(1/2)-InverseJacobiAM(1/2*e- 1/4*Pi+1/2*f*x,2^(1/2)*(b/(a+b))^(1/2))*((a+b*sin(f*x+e))/(a+b))^(1/2)/c/f /(a+b*sin(f*x+e))^(1/2)-2*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*sin(f*x+e))^(1/2)+ cos(f*x+e)*(a+b*sin(f*x+e))^(1/2)/(a-b)/f/(c+c*sin(f*x+e))
Result contains complex when optimal does not.
Time = 9.41 (sec) , antiderivative size = 625, normalized size of antiderivative = 2.54 \[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=-\frac {2 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a+b \sin (e+f x)}}{(a-b) f (c+c \sin (e+f x))}-\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \left (\frac {4 b \operatorname {EllipticF}\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{\sqrt {a+b \sin (e+f x)}}-\frac {2 (-4 a+3 b) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{\sqrt {a+b \sin (e+f x)}}-\frac {2 i b \cos (e+f x) \cos (2 (e+f x)) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (e+f x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (e+f x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (e+f x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sqrt {\frac {b-b \sin (e+f x)}{a+b}} \sqrt {-\frac {b+b \sin (e+f x)}{a-b}}}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\sin ^2(e+f x)} \left (-2 a^2+b^2+4 a (a+b \sin (e+f x))-2 (a+b \sin (e+f x))^2\right ) \sqrt {-\frac {a^2-b^2-2 a (a+b \sin (e+f x))+(a+b \sin (e+f x))^2}{b^2}}}-\frac {2 \cot (e+f x) \sqrt {a+b \sin (e+f x)} \sin (2 (e+f x))}{1-\sin ^2(e+f x)}\right )}{4 (a-b) f (c+c \sin (e+f x))} \] Input:
Integrate[Csc[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*(c + c*Sin[e + f*x])),x]
Output:
(-2*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[a + b*Sin[ e + f*x]])/((a - b)*f*(c + c*Sin[e + f*x])) - ((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*((4*b*EllipticF[(-e + Pi/2 - f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/Sqrt[a + b*Sin[e + f*x]] - (2*(-4*a + 3*b)*Elli pticPi[2, (-e + Pi/2 - f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/Sqrt[a + b*Sin[e + f*x]] - ((2*I)*b*Cos[e + f*x]*Cos[2*(e + f*x)]* (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)]*Sq rt[a + b*Sin[e + f*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcS inh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[e + f*x]]], (a + b)/(a - b)]))*Sqrt [(b - b*Sin[e + f*x])/(a + b)]*Sqrt[-((b + b*Sin[e + f*x])/(a - b))])/(a*S qrt[-(a + b)^(-1)]*Sqrt[1 - Sin[e + f*x]^2]*(-2*a^2 + b^2 + 4*a*(a + b*Sin [e + f*x]) - 2*(a + b*Sin[e + f*x])^2)*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Sin[ e + f*x]) + (a + b*Sin[e + f*x])^2)/b^2)]) - (2*Cot[e + f*x]*Sqrt[a + b*Si n[e + f*x]]*Sin[2*(e + f*x)])/(1 - Sin[e + f*x]^2)))/(4*(a - b)*f*(c + c*S in[e + f*x]))
Time = 1.76 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.07, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.515, Rules used = {3042, 3420, 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.
| \(\displaystyle \int \frac {\csc (e+f x)}{(c \sin (e+f x)+c) \sqrt {a+b \sin (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (e+f x) (c \sin (e+f x)+c) \sqrt {a+b \sin (e+f x)}}dx\) |
| \(\Big \downarrow \) 3420 |
| \(\displaystyle \frac {\int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)}}dx}{c}-\int \frac {1}{\sqrt {a+b \sin (e+f x)} (\sin (e+f x) c+c)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {1}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{c}-\int \frac {1}{\sqrt {a+b \sin (e+f x)} (\sin (e+f x) c+c)}dx\) |
| \(\Big \downarrow \) 3247 |
| \(\displaystyle -\frac {b \int -\frac {\sin (e+f x) c+c}{2 \sqrt {a+b \sin (e+f x)}}dx}{c^2 (a-b)}+\frac {\int \frac {1}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{c}+\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \int \frac {\sin (e+f x) c+c}{\sqrt {a+b \sin (e+f x)}}dx}{2 c^2 (a-b)}+\frac {\int \frac {1}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{c}+\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b \int \frac {\sin (e+f x) c+c}{\sqrt {a+b \sin (e+f x)}}dx}{2 c^2 (a-b)}+\frac {\int \frac {1}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{c}+\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {b \left (\frac {c \int \sqrt {a+b \sin (e+f x)}dx}{b}-\frac {c (a-b) \int \frac {1}{\sqrt {a+b \sin (e+f x)}}dx}{b}\right )}{2 c^2 (a-b)}+\frac {\int \frac {1}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{c}+\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b \left (\frac {c \int \sqrt {a+b \sin (e+f x)}dx}{b}-\frac {c (a-b) \int \frac {1}{\sqrt {a+b \sin (e+f x)}}dx}{b}\right )}{2 c^2 (a-b)}+\frac {\int \frac {1}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{c}+\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {b \left (\frac {c \sqrt {a+b \sin (e+f x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {c (a-b) \int \frac {1}{\sqrt {a+b \sin (e+f x)}}dx}{b}\right )}{2 c^2 (a-b)}+\frac {\int \frac {1}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{c}+\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b \left (\frac {c \sqrt {a+b \sin (e+f x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {c (a-b) \int \frac {1}{\sqrt {a+b \sin (e+f x)}}dx}{b}\right )}{2 c^2 (a-b)}+\frac {\int \frac {1}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{c}+\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {b \left (\frac {2 c \sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {c (a-b) \int \frac {1}{\sqrt {a+b \sin (e+f x)}}dx}{b}\right )}{2 c^2 (a-b)}+\frac {\int \frac {1}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{c}+\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {b \left (\frac {2 c \sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {c (a-b) \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}}dx}{b \sqrt {a+b \sin (e+f x)}}\right )}{2 c^2 (a-b)}+\frac {\int \frac {1}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{c}+\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b \left (\frac {2 c \sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {c (a-b) \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}}dx}{b \sqrt {a+b \sin (e+f x)}}\right )}{2 c^2 (a-b)}+\frac {\int \frac {1}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{c}+\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\int \frac {1}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{c}+\frac {b \left (\frac {2 c \sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {2 c (a-b) \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b f \sqrt {a+b \sin (e+f x)}}\right )}{2 c^2 (a-b)}+\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {\sqrt {\frac {a+b \sin (e+f x)}{a+b}} \int \frac {\csc (e+f x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}}dx}{c \sqrt {a+b \sin (e+f x)}}+\frac {b \left (\frac {2 c \sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {2 c (a-b) \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b f \sqrt {a+b \sin (e+f x)}}\right )}{2 c^2 (a-b)}+\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {\frac {a+b \sin (e+f x)}{a+b}} \int \frac {1}{\sin (e+f x) \sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}}dx}{c \sqrt {a+b \sin (e+f x)}}+\frac {b \left (\frac {2 c \sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {2 c (a-b) \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b f \sqrt {a+b \sin (e+f x)}}\right )}{2 c^2 (a-b)}+\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {b \left (\frac {2 c \sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {2 c (a-b) \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b f \sqrt {a+b \sin (e+f x)}}\right )}{2 c^2 (a-b)}+\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}+\frac {2 \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{c f \sqrt {a+b \sin (e+f x)}}\) |
Input:
Int[Csc[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*(c + c*Sin[e + f*x])),x]
Output:
(2*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]]) + (Cos[e + f*x]*Sqrt[a + b*Si n[e + f*x]])/((a - b)*f*(c + c*Sin[e + f*x])) + (b*((2*c*EllipticE[(e - Pi /2 + f*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[e + f*x]])/(b*f*Sqrt[(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*Sin[e + f*x]])) )/(2*(a - b)*c^2)
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[1/(sin[(e_.) + (f_.)*(x_)]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*( (c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[1/c Int[1/(Sin[ e + f*x]*Sqrt[a + b*Sin[e + f*x]]), x], x] - Simp[d/c Int[1/(Sqrt[a + b*S in[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]
Leaf count of result is larger than twice the leaf count of optimal. \(583\) vs. \(2(241)=482\).
Time = 1.15 (sec) , antiderivative size = 584, normalized size of antiderivative = 2.37
| method | result | size |
| default | \(\frac {\sqrt {-\left (-a -b \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}\, \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 {b \left (1+\sin \left (f x +e \right )\right )}{a -b}}\, b \operatorname {EllipticPi}\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 (-a -b \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}\, a}+\frac {-\sin \left (f x +e \right )^{2} b -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 (-a -b \sin \left (f x +e \right )\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 {b \left (1+\sin \left (f x +e \right )\right )}{a -b}}\, \operatorname {EllipticF}\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 (-a -b \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}+\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 {b \left (1+\sin \left (f x +e \right )\right )}{a -b}}\, \left (\left (-\frac {a}{b}-1\right ) \operatorname {EllipticE}\left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right )+\operatorname {EllipticF}\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 (-a -b \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}\right )}{c \cos \left (f x +e \right ) \sqrt {a +b \sin \left (f x +e \right )}\, f}\) | \(584\) |
Input:
int(csc(f*x+e)/(a+b*sin(f*x+e))^(1/2)/(c+c*sin(f*x+e)),x,method=_RETURNVER BOSE)
Output:
(-(-a-b*sin(f*x+e))*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)*(-b*(1+sin(f*x+e))/(a-b))^(1 /2)/(-(-a-b*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*b/a*EllipticPi(((a+b*sin(f*x+e ))/(a-b))^(1/2),-(-1/b*a+1)*b/a,((a-b)/(a+b))^(1/2))+(-sin(f*x+e)^2*b-a*si n(f*x+e)+b*sin(f*x+e)+a)/(a-b)/((1+sin(f*x+e))*(sin(f*x+e)-1)*(-a-b*sin(f* x+e)))^(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)*(-b*(1+sin(f*x+e))/(a-b))^(1/2)/(-(-a-b*sin(f*x +e))*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)*(-b*(1+sin(f*x+e))/(a-b))^(1/2)/(-(-a-b*sin(f*x+e))*c os(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. \[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\text {Timed out} \] Input:
integrate(csc(f*x+e)/(a+b*sin(f*x+e))^(1/2)/(c+c*sin(f*x+e)),x, algorithm= "fricas")
Output:
Timed out
\[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\frac {\int \frac {\csc {\left (e + f x \right )}}{\sqrt {a + b \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )} + \sqrt {a + b \sin {\left (e + f x \right )}}}\, dx}{c} \] Input:
integrate(csc(f*x+e)/(a+b*sin(f*x+e))**(1/2)/(c+c*sin(f*x+e)),x)
Output:
Integral(csc(e + f*x)/(sqrt(a + b*sin(e + f*x))*sin(e + f*x) + sqrt(a + b* sin(e + f*x))), x)/c
\[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\int { \frac {\csc \left (f x + e\right )}{\sqrt {b \sin \left (f x + e\right ) + a} {\left (c \sin \left (f x + e\right ) + c\right )}} \,d x } \] Input:
integrate(csc(f*x+e)/(a+b*sin(f*x+e))^(1/2)/(c+c*sin(f*x+e)),x, algorithm= "maxima")
Output:
integrate(csc(f*x + e)/(sqrt(b*sin(f*x + e) + a)*(c*sin(f*x + e) + c)), x)
\[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\int { \frac {\csc \left (f x + e\right )}{\sqrt {b \sin \left (f x + e\right ) + a} {\left (c \sin \left (f x + e\right ) + c\right )}} \,d x } \] Input:
integrate(csc(f*x+e)/(a+b*sin(f*x+e))^(1/2)/(c+c*sin(f*x+e)),x, algorithm= "giac")
Output:
integrate(csc(f*x + e)/(sqrt(b*sin(f*x + e) + a)*(c*sin(f*x + e) + c)), x)
Timed out. \[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\int \frac {1}{\sin \left (e+f\,x\right )\,\sqrt {a+b\,\sin \left (e+f\,x\right )}\,\left (c+c\,\sin \left (e+f\,x\right )\right )} \,d x \] Input:
int(1/(sin(e + f*x)*(a + b*sin(e + f*x))^(1/2)*(c + c*sin(e + f*x))),x)
Output:
int(1/(sin(e + f*x)*(a + b*sin(e + f*x))^(1/2)*(c + c*sin(e + f*x))), x)
\[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\frac {\int \frac {\sqrt {\sin \left (f x +e \right ) b +a}\, \csc \left (f x +e \right )}{\sin \left (f x +e \right )^{2} b +a \sin \left (f x +e \right )+\sin \left (f x +e \right ) b +a}d x}{c} \] Input:
int(csc(f*x+e)/(a+b*sin(f*x+e))^(1/2)/(c+c*sin(f*x+e)),x)
Output:
int((sqrt(sin(e + f*x)*b + a)*csc(e + f*x))/(sin(e + f*x)**2*b + sin(e + f *x)*a + sin(e + f*x)*b + a),x)/c