\(\int \frac {\sec (e+f x) \sqrt {a+b \sec (e+f x)}}{c+c \sec (e+f x)} \, dx\) [273]

Optimal result

Integrand size = 33, antiderivative size = 95 \[ \int \frac {\sec (e+f x) \sqrt {a+b \sec (e+f x)}}{c+c \sec (e+f x)} \, dx=\frac {E\left (\arcsin \left (\frac {\tan (e+f x)}{1+\sec (e+f x)}\right )|\frac {a-b}{a+b}\right ) \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {a+b \sec (e+f x)}}{c f \sqrt {\frac {a+b \sec (e+f x)}{(a+b) (1+\sec (e+f x))}}} \]

[Out]

EllipticE(tan(f*x+e)/(1+sec(f*x+e)),((a-b)/(a+b))^(1/2))*(1/(1+sec(f*x+e)))^(1/2)*(a+b*sec(f*x+e))^(1/2)/c/f/(
(a+b*sec(f*x+e))/(a+b)/(1+sec(f*x+e)))^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {4053} \[ \int \frac {\sec (e+f x) \sqrt {a+b \sec (e+f x)}}{c+c \sec (e+f x)} \, dx=\frac {\sqrt {\frac {1}{\sec (e+f x)+1}} \sqrt {a+b \sec (e+f x)} E\left (\arcsin \left (\frac {\tan (e+f x)}{\sec (e+f x)+1}\right )|\frac {a-b}{a+b}\right )}{c f \sqrt {\frac {a+b \sec (e+f x)}{(a+b) (\sec (e+f x)+1)}}} \]

[In]

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

[Out]

(EllipticE[ArcSin[Tan[e + f*x]/(1 + Sec[e + f*x])], (a - b)/(a + b)]*Sqrt[(1 + Sec[e + f*x])^(-1)]*Sqrt[a + b*
Sec[e + f*x]])/(c*f*Sqrt[(a + b*Sec[e + f*x])/((a + b)*(1 + Sec[e + f*x]))])

Rule 4053

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

Rubi steps \begin{align*} \text {integral}& = \frac {E\left (\arcsin \left (\frac {\tan (e+f x)}{1+\sec (e+f x)}\right )|\frac {a-b}{a+b}\right ) \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {a+b \sec (e+f x)}}{c f \sqrt {\frac {a+b \sec (e+f x)}{(a+b) (1+\sec (e+f x))}}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(264\) vs. \(2(95)=190\).

Time = 6.44 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.78 \[ \int \frac {\sec (e+f x) \sqrt {a+b \sec (e+f x)}}{c+c \sec (e+f x)} \, dx=\frac {\cos ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {\sec (e+f x)} \sqrt {a+b \sec (e+f x)} \left (\frac {2 \sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}} E\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a-b}{a+b}\right ) \sec ^4\left (\frac {1}{2} (e+f x)\right ) \sqrt {1+\sec (e+f x)}}{\left (\frac {1}{1+\cos (e+f x)}\right )^{3/2} \sqrt {\frac {b+a \cos (e+f x)}{(a+b) (1+\cos (e+f x))}}}+\frac {\sec ^5\left (\frac {1}{2} (e+f x)\right ) \sqrt {1+\sec (e+f x)} \left (-\sin \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {3}{2} (e+f x)\right )\right )}{\left (\frac {1}{1+\cos (e+f x)}\right )^{3/2}}-8 \sqrt {\sec (e+f x)} \left (\sin (e+f x)-\tan \left (\frac {1}{2} (e+f x)\right )\right )\right )}{4 c f (1+\sec (e+f x))} \]

[In]

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

[Out]

(Cos[(e + f*x)/2]^2*Sqrt[Sec[e + f*x]]*Sqrt[a + b*Sec[e + f*x]]*((2*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*Elli
pticE[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]*Sec[(e + f*x)/2]^4*Sqrt[1 + Sec[e + f*x]])/(((1 + Cos[e + f*x
])^(-1))^(3/2)*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]) + (Sec[(e + f*x)/2]^5*Sqrt[1 + Sec[e +
f*x]]*(-Sin[(e + f*x)/2] + Sin[(3*(e + f*x))/2]))/((1 + Cos[e + f*x])^(-1))^(3/2) - 8*Sqrt[Sec[e + f*x]]*(Sin
[e + f*x] - Tan[(e + f*x)/2])))/(4*c*f*(1 + Sec[e + f*x]))

Maple [A] (verified)

Time = 7.30 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.29

method	result	size
default	\(\frac {\left (-a -b \right ) \left (\cos \left (f x +e \right )+1\right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {a +b \sec \left (f x +e \right )}}{c f \left (b +a \cos \left (f x +e \right )\right )}\)	\(123\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(sqrt(b*sec(f*x + e) + a)*sec(f*x + e)/(c*sec(f*x + e) + c), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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