\(\int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx\) [30]

Optimal result

Integrand size = 33, antiderivative size = 246 \[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\frac {E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (e+f x)}}{(a-b) c f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {\operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{c f \sqrt {a+b \sin (e+f x)}}+\frac {2 \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{c f \sqrt {a+b \sin (e+f x)}}+\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{(a-b) f (c+c \sin (e+f x))} \]

[Out]

cos(f*x+e)*(a+b*sin(f*x+e))^(1/2)/(a-b)/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*P
i+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)/(a-b)/c/f/((a+b
*sin(f*x+e))/(a+b))^(1/2)+(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)*EllipticF(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))/(a+b))^(1/2)/c/f/(a+b*sin(f*x+e))^(1/2)-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),2,2^(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)

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3020, 2886, 2884, 2847, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (a-b) (c \sin (e+f x)+c)}-\frac {\sqrt {\frac {a+b \sin (e+f x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{c f \sqrt {a+b \sin (e+f x)}}+\frac {\sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{c f (a-b) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}+\frac {2 \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{c f \sqrt {a+b \sin (e+f x)}} \]

[In]

Int[Csc[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*(c + c*Sin[e + f*x])),x]

[Out]

(EllipticE[(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)]) - (EllipticF[(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]]) + (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*Sin[e + f*x]])/((a - b)*f*(c + c*Sin[e + f*x]))

Rule 2732

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]

Rule 2734

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[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]

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2742

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[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]

Rule 2831

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[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]

Rule 2847

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] + Dist[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] /; FreeQ[{a, b, c, d, e, f}, x] && Ne
Q[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])

Rule 2884

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]

Rule 2886

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist
[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] && N
eQ[c^2 - d^2, 0] &&  !GtQ[c + d, 0]

Rule 3020

Int[1/(sin[(e_.) + (f_.)*(x_)]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)
])), x_Symbol] :> Dist[1/c, Int[1/(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]), x], x] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = \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)} (c+c \sin (e+f x))} \, dx \\ & = \frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{(a-b) f (c+c \sin (e+f x))}-\frac {b \int \frac {-\frac {c}{2}-\frac {1}{2} c \sin (e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx}{(a-b) c^2}+\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 {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))}-\frac {\int \frac {1}{\sqrt {a+b \sin (e+f x)}} \, dx}{2 c}+\frac {\int \sqrt {a+b \sin (e+f x)} \, dx}{2 (a-b) c} \\ & = \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))}+\frac {\sqrt {a+b \sin (e+f x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}} \, dx}{2 (a-b) c \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {\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}{2 c \sqrt {a+b \sin (e+f x)}} \\ & = \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))} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.99 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.88 \[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (-8 \sin \left (\frac {1}{2} (e+f x)\right ) \sqrt {a+b \sin (e+f x)}-\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\frac {2 i \left (-2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (e+f x)}\right )|\frac {a+b}{a-b}\right )+b \left (-2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (e+f x)}\right ),\frac {a+b}{a-b}\right )+b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (e+f x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sec (e+f x) \sqrt {-\frac {b (-1+\sin (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sin (e+f x))}{a-b}}}{a b \sqrt {-\frac {1}{a+b}}}-4 \sqrt {a+b \sin (e+f x)}+\frac {4 b \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{\sqrt {a+b \sin (e+f x)}}+\frac {2 (4 a-3 b) \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{\sqrt {a+b \sin (e+f x)}}\right )\right )}{4 (a-b) c f (1+\sin (e+f x))} \]

[In]

Integrate[Csc[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*(c + c*Sin[e + f*x])),x]

[Out]

((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*E
llipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[e + f*x]]], (a + b)/(a - b)]))*Sec[e + f*x]*
Sqrt[-((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*EllipticF[(-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 - 3*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*(a - b)*c*f*(1 + Sin[e + f*x]))

Maple [A] (verified)

Time = 1.25 (sec) , antiderivative size = 587, normalized size of antiderivative = 2.39

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 )}\, a}+\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 )}{c \cos \left (f x +e \right ) \sqrt {a +b \sin \left (f x +e \right )}\, f}\)	\(587\)

[In]

int(1/sin(f*x+e)/(c+c*sin(f*x+e))/(a+b*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

(-(-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/a*EllipticPi(((a+b*sin(f
*x+e))/(a-b))^(1/2),-(-1/b*a+1)*b/a,((a-b)/(a+b))^(1/2))+(-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)*Ell
ipticF(((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*sin(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

Fricas [F(-1)]

Timed out. \[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\text {Timed out} \]

[In]

integrate(1/sin(f*x+e)/(c+c*sin(f*x+e))/(a+b*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\frac {\int \frac {1}{\sqrt {a + b \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )} + \sqrt {a + b \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}}\, dx}{c} \]

[In]

integrate(1/sin(f*x+e)/(c+c*sin(f*x+e))/(a+b*sin(f*x+e))**(1/2),x)

[Out]

Integral(1/(sqrt(a + b*sin(e + f*x))*sin(e + f*x)**2 + sqrt(a + b*sin(e + f*x))*sin(e + f*x)), x)/c

Maxima [F]

\[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\int { \frac {1}{\sqrt {b \sin \left (f x + e\right ) + a} {\left (c \sin \left (f x + e\right ) + c\right )} \sin \left (f x + e\right )} \,d x } \]

[In]

integrate(1/sin(f*x+e)/(c+c*sin(f*x+e))/(a+b*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*sin(f*x + e) + a)*(c*sin(f*x + e) + c)*sin(f*x + e)), x)

Giac [F]

\[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\int { \frac {1}{\sqrt {b \sin \left (f x + e\right ) + a} {\left (c \sin \left (f x + e\right ) + c\right )} \sin \left (f x + e\right )} \,d x } \]

[In]

integrate(1/sin(f*x+e)/(c+c*sin(f*x+e))/(a+b*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*sin(f*x + e) + a)*(c*sin(f*x + e) + c)*sin(f*x + e)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\int \frac {1}{\sin \left (e+f\,x\right )\,\sqrt {a+b\,\sin \left (e+f\,x\right )}\,\left (c+c\,\sin \left (e+f\,x\right )\right )} \,d x \]

[In]

int(1/(sin(e + f*x)*(a + b*sin(e + f*x))^(1/2)*(c + c*sin(e + f*x))),x)

[Out]

int(1/(sin(e + f*x)*(a + b*sin(e + f*x))^(1/2)*(c + c*sin(e + f*x))), x)