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

Optimal. Leaf size=196 \[ \frac {2 \cot (e+f x) \Pi \left (\frac {b (c+d)}{(a+b) d};\text {ArcSin}\left (\frac {\sqrt {\frac {a+b}{c+d}} \sqrt {c+d \sec (e+f x)}}{\sqrt {a+b \sec (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sqrt {-\frac {(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sec (e+f x))}{(c-d) (a+b \sec (e+f x))}} (a+b \sec (e+f x))}{d \sqrt {\frac {a+b}{c+d}} f} \]

[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]
time = 0.13, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {4067} \begin {gather*} \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))}} \Pi \left (\frac {b (c+d)}{(a+b) d};\text {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}}} \end {gather*}

Antiderivative was successfully verified.

[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*} \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) \Pi \left (\frac {b (c+d)}{(a+b) d};\sin ^{-1}\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] Result contains complex when optimal does not.
time = 35.52, size = 44216, normalized size = 225.59 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[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]
time = 4.16, size = 351, normalized size = 1.79

method	result	size
default	\(\frac {2 \left (\EllipticF \left (\frac {\left (-1+\cos \left (f x +e \right )\right ) \sqrt {\frac {a -b}{a +b}}}{\sin \left (f x +e \right )}, \sqrt {\frac {\left (a +b \right ) \left (c -d \right )}{\left (a -b \right ) \left (c +d \right )}}\right ) a -\EllipticF \left (\frac {\left (-1+\cos \left (f x +e \right )\right ) \sqrt {\frac {a -b}{a +b}}}{\sin \left (f x +e \right )}, \sqrt {\frac {\left (a +b \right ) \left (c -d \right )}{\left (a -b \right ) \left (c +d \right )}}\right ) b +2 \EllipticPi \left (\frac {\left (-1+\cos \left (f x +e \right )\right ) \sqrt {\frac {a -b}{a +b}}}{\sin \left (f x +e \right )}, \frac {a +b}{a -b}, \frac {\sqrt {\frac {c -d}{c +d}}}{\sqrt {\frac {a -b}{a +b}}}\right ) b \right ) \cos \left (f x +e \right ) \sqrt {\frac {d +c \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right ) \left (c +d \right )}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (\cos \left (f x +e \right )+1\right ) \left (a +b \right )}}\, \left (\sin ^{2}\left (f x +e \right )\right ) \sqrt {\frac {a \cos \left (f x +e \right )+b}{\cos \left (f x +e \right )}}\, \sqrt {\frac {d +c \cos \left (f x +e \right )}{\cos \left (f x +e \right )}}}{f \left (-1+\cos \left (f x +e \right )\right ) \left (d +c \cos \left (f x +e \right )\right ) \left (a \cos \left (f x +e \right )+b \right ) \sqrt {\frac {a -b}{a +b}}}\)	\(351\)

Verification of antiderivative is not currently implemented for this CAS.

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________