\(\int \frac {\sqrt {a+b \sec (e+f x)}}{c+d \sec (e+f x)} \, dx\) [199]

Optimal result

Integrand size = 27, antiderivative size = 220 \[ \int \frac {\sqrt {a+b \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=-\frac {2 \sqrt {a+b} \cot (e+f x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{c f}+\frac {2 (b c-a d) \operatorname {EllipticPi}\left (\frac {2 d}{c+d},\arcsin \left (\frac {\sqrt {1-\sec (e+f x)}}{\sqrt {2}}\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sec (e+f x)}{a+b}} \tan (e+f x)}{c (c+d) f \sqrt {a+b \sec (e+f x)} \sqrt {-\tan ^2(e+f x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4011, 3869, 4058} \[ \int \frac {\sqrt {a+b \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=\frac {2 (b c-a d) \tan (e+f x) \sqrt {\frac {a+b \sec (e+f x)}{a+b}} \operatorname {EllipticPi}\left (\frac {2 d}{c+d},\arcsin \left (\frac {\sqrt {1-\sec (e+f x)}}{\sqrt {2}}\right ),\frac {2 b}{a+b}\right )}{c f (c+d) \sqrt {-\tan ^2(e+f x)} \sqrt {a+b \sec (e+f x)}}-\frac {2 \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{c f} \]

[In]

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

[Out]

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

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 4011

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Dist[a/c,
Int[1/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[(b*c - a*d)/c, Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*(c + d
*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] && NeQ[c^2 -
d^2, 0]

Rule 4058

Int[csc[(e_.) + (f_.)*(x_)]/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))
), x_Symbol] :> Simp[-2*(Cot[e + f*x]/(f*(c + d)*Sqrt[a + b*Csc[e + f*x]]*Sqrt[-Cot[e + f*x]^2]))*Sqrt[(a + b*
Csc[e + f*x])/(a + b)]*EllipticPi[2*(d/(c + d)), ArcSin[Sqrt[1 - Csc[e + f*x]]/Sqrt[2]], 2*(b/(a + b))], 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]

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

Mathematica [A] (verified)

Time = 8.25 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {a+b \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=\frac {4 \cos ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}} \sqrt {\frac {b+a \cos (e+f x)}{(a+b) (1+\cos (e+f x))}} \left (-\left ((a-b) c (c+d) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right )\right )+2 a \left (c^2-d^2\right ) \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right )+2 d (-b c+a d) \operatorname {EllipticPi}\left (\frac {c-d}{c+d},\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right )\right ) \sqrt {a+b \sec (e+f x)}}{c (c-d) (c+d) f (b+a \cos (e+f x))} \]

[In]

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

Maple [A] (verified)

Time = 6.16 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.82

method	result	size
default	\(\frac {2 \left (\cos \left (f x +e \right )+1\right ) \left (\operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) a \,c^{2}+\operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) a c d -\operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) b \,c^{2}-\operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) b c d -2 \operatorname {EllipticPi}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), -1, \sqrt {\frac {a -b}{a +b}}\right ) a \,c^{2}+2 \operatorname {EllipticPi}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), -1, \sqrt {\frac {a -b}{a +b}}\right ) a \,d^{2}-2 \operatorname {EllipticPi}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \frac {c -d}{c +d}, \sqrt {\frac {a -b}{a +b}}\right ) a \,d^{2}+2 \operatorname {EllipticPi}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \frac {c -d}{c +d}, \sqrt {\frac {a -b}{a +b}}\right ) b c d \right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {a +b \sec \left (f x +e \right )}}{f c \left (c -d \right ) \left (c +d \right ) \left (b +a \cos \left (f x +e \right )\right )}\)	\(400\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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