\(\int \csc ^2(e+f x) \sqrt {a+b \sec (e+f x)} \, dx\) [250]

Optimal result

Integrand size = 23, antiderivative size = 121 \[ \int \csc ^2(e+f x) \sqrt {a+b \sec (e+f x)} \, dx=\frac {\sqrt {a+b} \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{f}-\frac {\cot (e+f x) \sqrt {a+b \sec (e+f x)}}{f} \]

[Out]

cot(f*x+e)*EllipticF((a+b*sec(f*x+e))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(f*x+e))/(a+
b))^(1/2)*(-b*(1+sec(f*x+e))/(a-b))^(1/2)/f-cot(f*x+e)*(a+b*sec(f*x+e))^(1/2)/f

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3960, 3917} \[ \int \csc ^2(e+f x) \sqrt {a+b \sec (e+f x)} \, dx=\frac {\sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{f}-\frac {\cot (e+f x) \sqrt {a+b \sec (e+f x)}}{f} \]

[In]

Int[Csc[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]],x]

[Out]

(Sqrt[a + b]*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1
- Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/f - (Cot[e + f*x]*Sqrt[a + b*Sec[e + f*x]])
/f

Rule 3917

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(Rt[a + b, 2]/(b*
f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin
[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 3960

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)/cos[(e_.) + (f_.)*(x_)]^2, x_Symbol] :> Simp[Tan[e + f*x]*((a
+ b*Csc[e + f*x])^m/f), x] + Dist[b*m, Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e
, f, m}, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {\cot (e+f x) \sqrt {a+b \sec (e+f x)}}{f}+\frac {1}{2} b \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx \\ & = \frac {\sqrt {a+b} \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{f}-\frac {\cot (e+f x) \sqrt {a+b \sec (e+f x)}}{f} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.74 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.99 \[ \int \csc ^2(e+f x) \sqrt {a+b \sec (e+f x)} \, dx=\frac {-\left ((b+a \cos (e+f x)) \csc (e+f x) \sqrt {\frac {1}{1+\sec (e+f x)}}\right )+b \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {\frac {a+b \sec (e+f x)}{(a+b) (1+\sec (e+f x))}}}{f \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {a+b \sec (e+f x)}} \]

[In]

Integrate[Csc[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]],x]

[Out]

(-((b + a*Cos[e + f*x])*Csc[e + f*x]*Sqrt[(1 + Sec[e + f*x])^(-1)]) + b*EllipticF[ArcSin[Tan[(e + f*x)/2]], (a
- b)/(a + b)]*Sqrt[(a + b*Sec[e + f*x])/((a + b)*(1 + Sec[e + f*x]))])/(f*Sqrt[(1 + Sec[e + f*x])^(-1)]*Sqrt[
a + b*Sec[e + f*x]])

Maple [A] (verified)

Time = 6.64 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.78

method	result	size
default	\(-\frac {\sqrt {a +b \sec \left (f x +e \right )}\, \left (\operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, b \cos \left (f x +e \right )+\operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, b +a \cos \left (f x +e \right ) \cot \left (f x +e \right )+b \cot \left (f x +e \right )\right )}{f \left (b +a \cos \left (f x +e \right )\right )}\)	\(215\)

[In]

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

[Out]

-1/f*(a+b*sec(f*x+e))^(1/2)/(b+a*cos(f*x+e))*(EllipticF(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*(cos(f*x+e)
/(cos(f*x+e)+1))^(1/2)*(1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*b*cos(f*x+e)+EllipticF(cot(f*x+e)-csc(f
*x+e),((a-b)/(a+b))^(1/2))*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*(1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*b
+a*cos(f*x+e)*cot(f*x+e)+b*cot(f*x+e))

Fricas [F]

\[ \int \csc ^2(e+f x) \sqrt {a+b \sec (e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right ) + a} \csc \left (f x + e\right )^{2} \,d x } \]

[In]

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

[Out]

integral(sqrt(b*sec(f*x + e) + a)*csc(f*x + e)^2, x)

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(a + b*sec(e + f*x))*csc(e + f*x)**2, x)

Maxima [F]

\[ \int \csc ^2(e+f x) \sqrt {a+b \sec (e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right ) + a} \csc \left (f x + e\right )^{2} \,d x } \]

[In]

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

[Out]

integrate(sqrt(b*sec(f*x + e) + a)*csc(f*x + e)^2, x)

Giac [F]

\[ \int \csc ^2(e+f x) \sqrt {a+b \sec (e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right ) + a} \csc \left (f x + e\right )^{2} \,d x } \]

[In]

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

[Out]

integrate(sqrt(b*sec(f*x + e) + a)*csc(f*x + e)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \csc ^2(e+f x) \sqrt {a+b \sec (e+f x)} \, dx=\int \frac {\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}}{{\sin \left (e+f\,x\right )}^2} \,d x \]

[In]

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

[Out]

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