\(\int \frac {c+d \sec (e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx\) [203]

Optimal result

Integrand size = 25, antiderivative size = 208 \[ \int \frac {c+d \sec (e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx=\frac {2 \sqrt {a+b} d \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{b f}-\frac {2 \sqrt {a+b} c \cot (e+f x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{a f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {4006, 3869, 3917} \[ \int \frac {c+d \sec (e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx=\frac {2 d \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b f}-\frac {2 c \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a f} \]

[In]

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

[Out]

(2*Sqrt[a + b]*d*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))])/(b*f) - (2*Sqrt[a + b]*c*Cot[e + f*x]*E
llipticPi[(a + b)/a, 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))])/(a*f)

Rule 3869

Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[2*(Rt[a + b, 2]/(a*d*Cot[c + d*x]))*Sqrt[b
*((1 - Csc[c + d*x])/(a + b))]*Sqrt[(-b)*((1 + Csc[c + d*x])/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b
*Csc[c + d*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]

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 4006

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[c, In
t[1/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[d, Int[Csc[e + f*x]/Sqrt[a + b*Csc[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}& = c \int \frac {1}{\sqrt {a+b \sec (e+f x)}} \, dx+d \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx \\ & = \frac {2 \sqrt {a+b} d \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}}}{b f}-\frac {2 \sqrt {a+b} c \cot (e+f x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\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}}}{a f} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.78 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.70 \[ \int \frac {c+d \sec (e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx=\frac {4 \cos ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}} \sqrt {\frac {b+a \cos (e+f x)}{(a+b) (1+\cos (e+f x))}} \left ((-c+d) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right )+2 c \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right )\right ) \sec (e+f x)}{f \sqrt {a+b \sec (e+f x)}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 17.24 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.88

method	result	size
default	\(\frac {2 \left (\cos \left (f x +e \right )+1\right ) \left (\operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) c -\operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) d -2 \operatorname {EllipticPi}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), -1, \sqrt {\frac {a -b}{a +b}}\right ) c \right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {a +b \sec \left (f x +e \right )}}{f \left (b +a \cos \left (f x +e \right )\right )}\)	\(182\)
parts	\(\frac {2 c \left (\cos \left (f x +e \right )+1\right ) \left (\operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right )-2 \operatorname {EllipticPi}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), -1, \sqrt {\frac {a -b}{a +b}}\right )\right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {a +b \sec \left (f x +e \right )}}{f \left (b +a \cos \left (f x +e \right )\right )}-\frac {2 d \left (\cos \left (f x +e \right )+1\right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {a +b \sec \left (f x +e \right )}}{f \left (b +a \cos \left (f x +e \right )\right )}\)	\(263\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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