\(\int \frac {\sec (e+f x) \sqrt {a+b \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx\) [265]

Optimal result

Integrand size = 35, antiderivative size = 196 \[ \int \frac {\sec (e+f x) \sqrt {a+b \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx=\frac {2 \cot (e+f x) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {\frac {a+b}{c+d}} \sqrt {c+d \sec (e+f x)}}{\sqrt {a+b \sec (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sqrt {-\frac {(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sec (e+f x))}{(c-d) (a+b \sec (e+f x))}} (a+b \sec (e+f x))}{d \sqrt {\frac {a+b}{c+d}} f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {4067} \[ \int \frac {\sec (e+f x) \sqrt {a+b \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx=\frac {2 \cot (e+f x) (a+b \sec (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt {\frac {(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {\frac {a+b}{c+d}} \sqrt {c+d \sec (e+f x)}}{\sqrt {a+b \sec (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{d f \sqrt {\frac {a+b}{c+d}}} \]

[In]

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

[Out]

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

Rule 4067

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

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 37.86 (sec) , antiderivative size = 44664, normalized size of antiderivative = 227.88 \[ \int \frac {\sec (e+f x) \sqrt {a+b \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

Result too large to show

Maple [A] (verified)

Time = 12.84 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.64

method	result	size
default	\(\frac {2 \sqrt {\frac {d +c \cos \left (f x +e \right )}{\left (c +d \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {a +b \sec \left (f x +e \right )}\, \sqrt {c +d \sec \left (f x +e \right )}\, \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \left (\operatorname {EllipticF}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), \sqrt {\frac {\left (a +b \right ) \left (c -d \right )}{\left (a -b \right ) \left (c +d \right )}}\right ) a -\operatorname {EllipticF}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), \sqrt {\frac {\left (a +b \right ) \left (c -d \right )}{\left (a -b \right ) \left (c +d \right )}}\right ) b +2 \operatorname {EllipticPi}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), \frac {a +b}{a -b}, \frac {\sqrt {\frac {c -d}{c +d}}}{\sqrt {\frac {a -b}{a +b}}}\right ) b \right ) \left (\cos \left (f x +e \right )^{2}+\cos \left (f x +e \right )\right )}{f \sqrt {\frac {a -b}{a +b}}\, \left (d +c \cos \left (f x +e \right )\right ) \left (b +a \cos \left (f x +e \right )\right )}\)	\(321\)

[In]

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

[Out]

2/f/((a-b)/(a+b))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(cos(f*x+e)+1))^(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)*(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))*a-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))*b+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))*b)/(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. \[ \int \frac {\sec (e+f x) \sqrt {a+b \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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