3.3.49 \(\int \sqrt {a+b \sec (e+f x)} \, dx\) [249]

Optimal. Leaf size=125 \[ -\frac {2 \cot (e+f x) \Pi \left (\frac {a}{a+b};\text {ArcSin}\left (\frac {\sqrt {a+b}}{\sqrt {a+b \sec (e+f x)}}\right )|\frac {a-b}{a+b}\right ) \sqrt {-\frac {b (1-\sec (e+f x))}{a+b \sec (e+f x)}} \sqrt {\frac {b (1+\sec (e+f x))}{a+b \sec (e+f x)}} (a+b \sec (e+f x))}{\sqrt {a+b} f} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3865} \begin {gather*} -\frac {2 \cot (e+f x) \sqrt {-\frac {b (1-\sec (e+f x))}{a+b \sec (e+f x)}} \sqrt {\frac {b (\sec (e+f x)+1)}{a+b \sec (e+f x)}} (a+b \sec (e+f x)) \Pi \left (\frac {a}{a+b};\text {ArcSin}\left (\frac {\sqrt {a+b}}{\sqrt {a+b \sec (e+f x)}}\right )|\frac {a-b}{a+b}\right )}{f \sqrt {a+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sec[e + f*x]],x]

[Out]

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

Rule 3865

Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[2*((a + b*Csc[c + d*x])/(d*Rt[a + b, 2]*Cot[
c + d*x]))*Sqrt[b*((1 + Csc[c + d*x])/(a + b*Csc[c + d*x]))]*Sqrt[(-b)*((1 - Csc[c + d*x])/(a + b*Csc[c + d*x]
))]*EllipticPi[a/(a + b), ArcSin[Rt[a + b, 2]/Sqrt[a + b*Csc[c + d*x]]], (a - b)/(a + b)], x] /; FreeQ[{a, b,
c, d}, x] && NeQ[a^2 - b^2, 0]

Rubi steps

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

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Mathematica [A]
time = 0.19, size = 151, normalized size = 1.21 \begin {gather*} \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 ((-a+b) F\left (\text {ArcSin}\left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a-b}{a+b}\right )+2 a \Pi \left (-1;\text {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)}}{f (b+a \cos (e+f x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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Maple [A]
time = 1.46, size = 215, normalized size = 1.72

method	result	size
default	\(-\frac {2 \sqrt {\frac {a \cos \left (f x +e \right )+b}{\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (\cos \left (f x +e \right )+1\right ) \left (a +b \right )}}\, \left (\cos \left (f x +e \right )+1\right )^{2} \left (-1+\cos \left (f x +e \right )\right ) \left (\EllipticF \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) a -\EllipticF \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) b -2 a \EllipticPi \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, -1, \sqrt {\frac {a -b}{a +b}}\right )\right )}{f \left (a \cos \left (f x +e \right )+b \right ) \sin \left (f x +e \right )^{2}}\)	\(215\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((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*(-1+cos(f*x+e))*(EllipticF((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*a-Ellip
ticF((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*b-2*a*EllipticPi((-1+cos(f*x+e))/sin(f*x+e),-1,((a-b)/(a+
b))^(1/2)))/(a*cos(f*x+e)+b)/sin(f*x+e)^2

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*sec(e + f*x)), x)

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {a+\frac {b}{\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),x)

[Out]

int((a + b/cos(e + f*x))^(1/2), x)

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