3.7.78 \(\int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-2+3 \sec (c+d x)}} \, dx\) [678]

Optimal. Leaf size=54 \[ \frac {2 \sqrt {3-2 \cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |-4\right ) \sqrt {\sec (c+d x)}}{d \sqrt {-2+3 \sec (c+d x)}} \]

[Out]

2*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2*I)*(3-2*cos(d*x+c))^(1/2)*sec
(d*x+c)^(1/2)/d/(-2+3*sec(d*x+c))^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {3943, 2740} \begin {gather*} \frac {2 \sqrt {3-2 \cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |-4\right )}{d \sqrt {3 \sec (c+d x)-2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[Sec[c + d*x]]/Sqrt[-2 + 3*Sec[c + d*x]],x]

[Out]

(2*Sqrt[3 - 2*Cos[c + d*x]]*EllipticF[(c + d*x)/2, -4]*Sqrt[Sec[c + d*x]])/(d*Sqrt[-2 + 3*Sec[c + d*x]])

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 3943

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[Sqrt[d*C
sc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/Sqrt[a + b*Csc[e + f*x]]), Int[1/Sqrt[b + a*Sin[e + f*x]], x], x] /; Fr
eeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-2+3 \sec (c+d x)}} \, dx &=\frac {\left (\sqrt {3-2 \cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {3-2 \cos (c+d x)}} \, dx}{\sqrt {-2+3 \sec (c+d x)}}\\ &=\frac {2 \sqrt {3-2 \cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |-4\right ) \sqrt {\sec (c+d x)}}{d \sqrt {-2+3 \sec (c+d x)}}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 54, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {3-2 \cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |-4\right ) \sqrt {\sec (c+d x)}}{d \sqrt {-2+3 \sec (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Sec[c + d*x]]/Sqrt[-2 + 3*Sec[c + d*x]],x]

[Out]

(2*Sqrt[3 - 2*Cos[c + d*x]]*EllipticF[(c + d*x)/2, -4]*Sqrt[Sec[c + d*x]])/(d*Sqrt[-2 + 3*Sec[c + d*x]])

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Maple [A]
time = 0.26, size = 137, normalized size = 2.54

method	result	size
default	\(\frac {i \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, \sqrt {5}\right ) \sqrt {2}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {-\frac {2 \left (2 \cos \left (d x +c \right )-3\right )}{1+\cos \left (d x +c \right )}}\, \left (\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )}}\, \sqrt {-\frac {2 \cos \left (d x +c \right )-3}{\cos \left (d x +c \right )}}}{d \left (2 \left (\cos ^{2}\left (d x +c \right )\right )-5 \cos \left (d x +c \right )+3\right )}\)	\(137\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^(1/2)/(-2+3*sec(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

I/d*EllipticF(I*(-1+cos(d*x+c))/sin(d*x+c),5^(1/2))*2^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*(-2*(2*cos(d*x+c)-3)/(1+c
os(d*x+c)))^(1/2)*sin(d*x+c)^2*cos(d*x+c)*(1/cos(d*x+c))^(1/2)*(-(2*cos(d*x+c)-3)/cos(d*x+c))^(1/2)/(2*cos(d*x
+c)^2-5*cos(d*x+c)+3)

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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(sec(d*x+c)^(1/2)/(-2+3*sec(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(sec(d*x + c))/sqrt(3*sec(d*x + c) - 2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.69, size = 44, normalized size = 0.81 \begin {gather*} -\frac {{\rm weierstrassPInverse}\left (8, 4, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - 1\right ) + {\rm weierstrassPInverse}\left (8, 4, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - 1\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^(1/2)/(-2+3*sec(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

-(weierstrassPInverse(8, 4, cos(d*x + c) + I*sin(d*x + c) - 1) + weierstrassPInverse(8, 4, cos(d*x + c) - I*si
n(d*x + c) - 1))/d

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\sec {\left (c + d x \right )}}}{\sqrt {3 \sec {\left (c + d x \right )} - 2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**(1/2)/(-2+3*sec(d*x+c))**(1/2),x)

[Out]

Integral(sqrt(sec(c + d*x))/sqrt(3*sec(c + d*x) - 2), 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(sec(d*x+c)^(1/2)/(-2+3*sec(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(sec(d*x + c))/sqrt(3*sec(d*x + c) - 2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}}{\sqrt {\frac {3}{\cos \left (c+d\,x\right )}-2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/cos(c + d*x))^(1/2)/(3/cos(c + d*x) - 2)^(1/2),x)

[Out]

int((1/cos(c + d*x))^(1/2)/(3/cos(c + d*x) - 2)^(1/2), x)

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