\(\int \frac {\sec (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx\) [255]

Optimal result

Integrand size = 19, antiderivative size = 47 \[ \int \frac {\sec (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\frac {\operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sec (e+f x) \sqrt {\sin (2 e+2 f x)}}{f \sqrt {d \tan (e+f x)}} \]

[Out]

-(sin(e+1/4*Pi+f*x)^2)^(1/2)/sin(e+1/4*Pi+f*x)*EllipticF(cos(e+1/4*Pi+f*x),2^(1/2))*sec(f*x+e)*sin(2*f*x+2*e)^
(1/2)/f/(d*tan(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2694, 2653, 2720} \[ \int \frac {\sec (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\frac {\sqrt {\sin (2 e+2 f x)} \sec (e+f x) \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{f \sqrt {d \tan (e+f x)}} \]

[In]

Int[Sec[e + f*x]/Sqrt[d*Tan[e + f*x]],x]

[Out]

(EllipticF[e - Pi/4 + f*x, 2]*Sec[e + f*x]*Sqrt[Sin[2*e + 2*f*x]])/(f*Sqrt[d*Tan[e + f*x]])

Rule 2653

Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[Sqrt[Sin[2*
e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b*Cos[e + f*x]]), Int[1/Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b,
e, f}, x]

Rule 2694

Int[sec[(e_.) + (f_.)*(x_)]/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[e + f*x]]/(Sqrt[Co
s[e + f*x]]*Sqrt[b*Tan[e + f*x]]), Int[1/(Sqrt[Cos[e + f*x]]*Sqrt[Sin[e + f*x]]), x], x] /; FreeQ[{b, e, f}, x
]

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {\sin (e+f x)} \int \frac {1}{\sqrt {\cos (e+f x)} \sqrt {\sin (e+f x)}} \, dx}{\sqrt {\cos (e+f x)} \sqrt {d \tan (e+f x)}} \\ & = \frac {\left (\sec (e+f x) \sqrt {\sin (2 e+2 f x)}\right ) \int \frac {1}{\sqrt {\sin (2 e+2 f x)}} \, dx}{\sqrt {d \tan (e+f x)}} \\ & = \frac {\operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sec (e+f x) \sqrt {\sin (2 e+2 f x)}}{f \sqrt {d \tan (e+f x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.18 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.64 \[ \int \frac {\sec (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=-\frac {2 \sqrt [4]{-1} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (e+f x)}\right ),-1\right ) \sec ^3(e+f x) \sqrt {\tan (e+f x)}}{f \sqrt {d \tan (e+f x)} \left (1+\tan ^2(e+f x)\right )^{3/2}} \]

[In]

Integrate[Sec[e + f*x]/Sqrt[d*Tan[e + f*x]],x]

[Out]

(-2*(-1)^(1/4)*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[e + f*x]]], -1]*Sec[e + f*x]^3*Sqrt[Tan[e + f*x]])/(f*S
qrt[d*Tan[e + f*x]]*(1 + Tan[e + f*x]^2)^(3/2))

Maple [A] (verified)

Time = 0.97 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.19

method	result	size
default	\(\frac {F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )+1}\, \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \left (1+\sec \left (f x +e \right )\right ) \sqrt {2}}{f \sqrt {d \tan \left (f x +e \right )}}\)	\(103\)

[In]

int(sec(f*x+e)/(d*tan(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/f*EllipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*(cot(f*x+e)-csc(f*x+e))^(1/2)*(cot(f*x+e)-csc(f*x+
e)+1)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)/(d*tan(f*x+e))^(1/2)*(1+sec(f*x+e))*2^(1/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.21 \[ \int \frac {\sec (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=-\frac {\sqrt {i \, d} F(\arcsin \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\,|\,-1) + \sqrt {-i \, d} F(\arcsin \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\,|\,-1)}{d f} \]

[In]

integrate(sec(f*x+e)/(d*tan(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

-(sqrt(I*d)*elliptic_f(arcsin(cos(f*x + e) + I*sin(f*x + e)), -1) + sqrt(-I*d)*elliptic_f(arcsin(cos(f*x + e)
- I*sin(f*x + e)), -1))/(d*f)

Sympy [F]

\[ \int \frac {\sec (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\int \frac {\sec {\left (e + f x \right )}}{\sqrt {d \tan {\left (e + f x \right )}}}\, dx \]

[In]

integrate(sec(f*x+e)/(d*tan(f*x+e))**(1/2),x)

[Out]

Integral(sec(e + f*x)/sqrt(d*tan(e + f*x)), x)

Maxima [F]

\[ \int \frac {\sec (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\int { \frac {\sec \left (f x + e\right )}{\sqrt {d \tan \left (f x + e\right )}} \,d x } \]

[In]

integrate(sec(f*x+e)/(d*tan(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(sec(f*x + e)/sqrt(d*tan(f*x + e)), x)

Giac [F]

\[ \int \frac {\sec (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\int { \frac {\sec \left (f x + e\right )}{\sqrt {d \tan \left (f x + e\right )}} \,d x } \]

[In]

integrate(sec(f*x+e)/(d*tan(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(sec(f*x + e)/sqrt(d*tan(f*x + e)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sec (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\int \frac {1}{\cos \left (e+f\,x\right )\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}} \,d x \]

[In]

int(1/(cos(e + f*x)*(d*tan(e + f*x))^(1/2)),x)

[Out]

int(1/(cos(e + f*x)*(d*tan(e + f*x))^(1/2)), x)