3.3.81 \(\int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))} \, dx\) [281]

Optimal. Leaf size=102 \[ \frac {2 \Pi \left (\frac {2 d}{c+d};\text {ArcSin}\left (\frac {\sqrt {1-\sec (e+f x)}}{\sqrt {2}}\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sec (e+f x)}{a+b}} \tan (e+f x)}{(c+d) f \sqrt {a+b \sec (e+f x)} \sqrt {-\tan ^2(e+f x)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {4058} \begin {gather*} \frac {2 \tan (e+f x) \sqrt {\frac {a+b \sec (e+f x)}{a+b}} \Pi \left (\frac {2 d}{c+d};\text {ArcSin}\left (\frac {\sqrt {1-\sec (e+f x)}}{\sqrt {2}}\right )|\frac {2 b}{a+b}\right )}{f (c+d) \sqrt {-\tan ^2(e+f x)} \sqrt {a+b \sec (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*EllipticPi[(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 + d)*f*Sqrt[a + b*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])

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*} \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))} \, dx &=\frac {2 \Pi \left (\frac {2 d}{c+d};\sin ^{-1}\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+d) f \sqrt {a+b \sec (e+f x)} \sqrt {-\tan ^2(e+f x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 6.74, size = 187, normalized size = 1.83 \begin {gather*} \frac {2 \sqrt {\frac {b+a \cos (e+f x)}{(a+b) (1+\cos (e+f x))}} \left ((c+d) F\left (\text {ArcSin}\left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a-b}{a+b}\right )-2 d \Pi \left (\frac {c-d}{c+d};\text {ArcSin}\left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a-b}{a+b}\right )\right ) \sqrt {\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\sec (e+f x)} \sqrt {1+\sec (e+f x)}}{(c-d) (c+d) f \sqrt {\sec ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {a+b \sec (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(235\) vs. \(2(97)=194\).
time = 5.59, size = 236, normalized size = 2.31

method	result	size
default	\(\frac {2 \sqrt {\frac {a \cos \left (f x +e \right )+b}{\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (\cos \left (f x +e \right )+1\right ) \left (a +b \right )}}\, \left (\cos \left (f x +e \right )+1\right )^{2} \left (\EllipticF \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) c +\EllipticF \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) d -2 \EllipticPi \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \frac {c -d}{c +d}, \sqrt {\frac {a -b}{a +b}}\right ) d \right ) \left (-1+\cos \left (f x +e \right )\right )}{f \left (a \cos \left (f x +e \right )+b \right ) \sin \left (f x +e \right )^{2} \left (c -d \right ) \left (c +d \right )}\)	\(236\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sec {\left (e + f x \right )}}{\sqrt {a + b \sec {\left (e + f x \right )}} \left (c + d \sec {\left (e + f x \right )}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\cos \left (e+f\,x\right )\,\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}\,\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________