\(\int \frac {1}{\sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx\) [219]

Optimal result

Integrand size = 29, antiderivative size = 398 \[ \int \frac {1}{\sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx=-\frac {2 \sqrt {c+d} \cot (e+f x) \operatorname {EllipticPi}\left (\frac {a (c+d)}{(a+b) c},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sqrt {-\frac {(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sec (e+f x))}{(c-d) (a+b \sec (e+f x))}} (a+b \sec (e+f x))}{a \sqrt {a+b} c f}-\frac {2 b \sqrt {a+b} \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {\frac {(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sec (e+f x))}{(a-b) (c+d \sec (e+f x))}} (c+d \sec (e+f x))}{a \sqrt {c+d} (b c-a d) f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {4023, 4021, 4069} \[ \int \frac {1}{\sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx=-\frac {2 b \sqrt {a+b} \cot (e+f x) (c+d \sec (e+f x)) \sqrt {\frac {(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt {-\frac {(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{a f \sqrt {c+d} (b c-a d)}-\frac {2 \sqrt {c+d} \cot (e+f x) (a+b \sec (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt {\frac {(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \operatorname {EllipticPi}\left (\frac {a (c+d)}{(a+b) c},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{a c f \sqrt {a+b}} \]

[In]

Int[1/(Sqrt[a + b*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]),x]

[Out]

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

Rule 4021

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)], x_Symbol] :> Simp[2
*((a + b*Csc[e + f*x])/(c*f*Rt[(a + b)/(c + d), 2]*Cot[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Csc[e + f*x])/((c - d
)*(a + b*Csc[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 - Csc[e + f*x])/((c + d)*(a + b*Csc[e + f*x])))]*EllipticPi[
a*((c + d)/(c*(a + b))), ArcSin[Rt[(a + b)/(c + d), 2]*(Sqrt[c + d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]])], (
a - b)*((c + d)/((a + b)*(c - d)))], 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 4023

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

Rule 4069

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.14 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.63 \[ \int \frac {1}{\sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx=\frac {4 i \cos ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {\frac {b+a \cos (e+f x)}{(a+b) (1+\cos (e+f x))}} \sqrt {\frac {d+c \cos (e+f x)}{(c+d) (1+\cos (e+f x))}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {-a+b}{a+b}} \tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )-2 \operatorname {EllipticPi}\left (-\frac {a+b}{a-b},i \text {arcsinh}\left (\sqrt {\frac {-a+b}{a+b}} \tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )\right ) \sec (e+f x)}{\sqrt {\frac {-a+b}{a+b}} f \sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \]

[In]

Integrate[1/(Sqrt[a + b*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]),x]

[Out]

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

Maple [A] (verified)

Time = 13.40 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.66

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

[In]

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

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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