\(\int \csc (e+f x) \sqrt {a+b \sin (e+f x)} \, dx\) [205]

Optimal result

Integrand size = 21, antiderivative size = 128 \[ \int \csc (e+f x) \sqrt {a+b \sin (e+f x)} \, dx=\frac {2 b \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{f \sqrt {a+b \sin (e+f x)}}+\frac {2 a \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{f \sqrt {a+b \sin (e+f x)}} \]

[Out]

-2*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*sin(f*x+e))/(a+b))^(1/2)/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)/f/(a+b*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2882, 2742, 2740, 2886, 2884} \[ \int \csc (e+f x) \sqrt {a+b \sin (e+f x)} \, dx=\frac {2 b \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{f \sqrt {a+b \sin (e+f x)}}+\frac {2 a \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{f \sqrt {a+b \sin (e+f x)}} \]

[In]

Int[Csc[e + f*x]*Sqrt[a + b*Sin[e + f*x]],x]

[Out]

(2*b*EllipticF[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*
x]]) + (2*a*EllipticPi[2, (e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b
*Sin[e + f*x]])

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 2882

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

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]

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

Mathematica [A] (verified)

Time = 14.24 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.70 \[ \int \csc (e+f x) \sqrt {a+b \sin (e+f x)} \, dx=-\frac {2 \left (b \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 b}{a+b}\right )+a \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 b}{a+b}\right )\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{f \sqrt {a+b \sin (e+f x)}} \]

[In]

Integrate[Csc[e + f*x]*Sqrt[a + b*Sin[e + f*x]],x]

[Out]

(-2*(b*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*b)/(a + b)] + a*EllipticPi[2, (-2*e + Pi - 2*f*x)/4, (2*b)/(a + b)]
)*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]])

Maple [A] (verified)

Time = 1.15 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.32

method	result	size
default	\(\frac {2 \left (a -b \right ) \sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )+1\right ) b}{a -b}}\, \left (F\left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right )-\Pi \left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \frac {a -b}{a}, \sqrt {\frac {a -b}{a +b}}\right )\right )}{\cos \left (f x +e \right ) \sqrt {a +b \sin \left (f x +e \right )}\, f}\)	\(169\)

[In]

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

[Out]

2*(a-b)*((a+b*sin(f*x+e))/(a-b))^(1/2)*(-(sin(f*x+e)-1)*b/(a+b))^(1/2)*(-(sin(f*x+e)+1)*b/(a-b))^(1/2)*(Ellipt
icF(((a+b*sin(f*x+e))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))-EllipticPi(((a+b*sin(f*x+e))/(a-b))^(1/2),(a-b)/a,((a-
b)/(a+b))^(1/2)))/cos(f*x+e)/(a+b*sin(f*x+e))^(1/2)/f

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(a + b*sin(e + f*x))*csc(e + f*x), x)

Maxima [F]

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

[In]

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

[Out]

integrate(sqrt(b*sin(f*x + e) + a)*csc(f*x + e), x)

Giac [F]

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

[In]

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

[Out]

integrate(sqrt(b*sin(f*x + e) + a)*csc(f*x + e), x)

Mupad [F(-1)]

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

[In]

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

[Out]

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