3.7.75 \(\int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {-3+2 \sec (c+d x)}} \, dx\) [675]

Optimal. Leaf size=129 \[ \frac {4 \sqrt {2-3 \cos (c+d x)} F\left (\frac {1}{2} (c+\pi +d x)|\frac {6}{5}\right ) \sqrt {\sec (c+d x)}}{3 \sqrt {5} d \sqrt {-3+2 \sec (c+d x)}}-\frac {2 \sqrt {5} E\left (\frac {1}{2} (c+\pi +d x)|\frac {6}{5}\right ) \sqrt {-3+2 \sec (c+d x)}}{3 d \sqrt {2-3 \cos (c+d x)} \sqrt {\sec (c+d x)}} \]

[Out]

-4/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)*EllipticF(cos(1/2*d*x+1/2*c),1/5*30^(1/2))*(2-3*cos(d*x+
c))^(1/2)*sec(d*x+c)^(1/2)/d*5^(1/2)/(-3+2*sec(d*x+c))^(1/2)+2/3*(sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*
c)*EllipticE(cos(1/2*d*x+1/2*c),1/5*30^(1/2))*5^(1/2)*(-3+2*sec(d*x+c))^(1/2)/d/(2-3*cos(d*x+c))^(1/2)/sec(d*x
+c)^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3947, 3941, 2733, 3943, 2741} \begin {gather*} \frac {4 \sqrt {2-3 \cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\frac {1}{2} (c+d x+\pi )|\frac {6}{5}\right )}{3 \sqrt {5} d \sqrt {2 \sec (c+d x)-3}}-\frac {2 \sqrt {5} \sqrt {2 \sec (c+d x)-3} E\left (\frac {1}{2} (c+d x+\pi )|\frac {6}{5}\right )}{3 d \sqrt {2-3 \cos (c+d x)} \sqrt {\sec (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[Sec[c + d*x]]*Sqrt[-3 + 2*Sec[c + d*x]]),x]

[Out]

(4*Sqrt[2 - 3*Cos[c + d*x]]*EllipticF[(c + Pi + d*x)/2, 6/5]*Sqrt[Sec[c + d*x]])/(3*Sqrt[5]*d*Sqrt[-3 + 2*Sec[
c + d*x]]) - (2*Sqrt[5]*EllipticE[(c + Pi + d*x)/2, 6/5]*Sqrt[-3 + 2*Sec[c + d*x]])/(3*d*Sqrt[2 - 3*Cos[c + d*
x]]*Sqrt[Sec[c + d*x]])

Rule 2733

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a - b]/d)*EllipticE[(1/2)*(c + Pi/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 2741

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 3941

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

Rule 3947

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {-3+2 \sec (c+d x)}} \, dx &=-\left (\frac {1}{3} \int \frac {\sqrt {-3+2 \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx\right )+\frac {2}{3} \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-3+2 \sec (c+d x)}} \, dx\\ &=\frac {\left (2 \sqrt {2-3 \cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {2-3 \cos (c+d x)}} \, dx}{3 \sqrt {-3+2 \sec (c+d x)}}-\frac {\sqrt {-3+2 \sec (c+d x)} \int \sqrt {2-3 \cos (c+d x)} \, dx}{3 \sqrt {2-3 \cos (c+d x)} \sqrt {\sec (c+d x)}}\\ &=\frac {4 \sqrt {2-3 \cos (c+d x)} F\left (\frac {1}{2} (c+\pi +d x)|\frac {6}{5}\right ) \sqrt {\sec (c+d x)}}{3 \sqrt {5} d \sqrt {-3+2 \sec (c+d x)}}-\frac {2 \sqrt {5} E\left (\frac {1}{2} (c+\pi +d x)|\frac {6}{5}\right ) \sqrt {-3+2 \sec (c+d x)}}{3 d \sqrt {2-3 \cos (c+d x)} \sqrt {\sec (c+d x)}}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 72, normalized size = 0.56 \begin {gather*} \frac {\sqrt {-2+3 \cos (c+d x)} \left (2 E\left (\left .\frac {1}{2} (c+d x)\right |6\right )+4 F\left (\left .\frac {1}{2} (c+d x)\right |6\right )\right ) \sqrt {\sec (c+d x)}}{3 d \sqrt {-3+2 \sec (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[Sec[c + d*x]]*Sqrt[-3 + 2*Sec[c + d*x]]),x]

[Out]

(Sqrt[-2 + 3*Cos[c + d*x]]*(2*EllipticE[(c + d*x)/2, 6] + 4*EllipticF[(c + d*x)/2, 6])*Sqrt[Sec[c + d*x]])/(3*
d*Sqrt[-3 + 2*Sec[c + d*x]])

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Maple [C] Result contains complex when optimal does not.
time = 0.41, size = 370, normalized size = 2.87

method	result	size
default	\(-\frac {2 \left (3 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \cos \left (d x +c \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i \sqrt {5}\right )-i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \cos \left (d x +c \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i \sqrt {5}\right )+3 i \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i \sqrt {5}\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )-i \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i \sqrt {5}\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )-3 \left (\cos ^{2}\left (d x +c \right )\right )+5 \cos \left (d x +c \right )-2\right ) \sqrt {-\frac {-2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )}}}{3 d \sqrt {\frac {1}{\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (-2+3 \cos \left (d x +c \right )\right )}\)	\(370\)
risch	\(-\frac {i \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}-4 \,{\mathrm e}^{i \left (d x +c \right )}+3\right ) \sqrt {2}}{3 d \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {-\frac {3 \,{\mathrm e}^{2 i \left (d x +c \right )}-4 \,{\mathrm e}^{i \left (d x +c \right )}+3}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}-\frac {i \left (\frac {-2 \,{\mathrm e}^{2 i \left (d x +c \right )}+\frac {8 \,{\mathrm e}^{i \left (d x +c \right )}}{3}-2}{\sqrt {\left (-3 \,{\mathrm e}^{2 i \left (d x +c \right )}+4 \,{\mathrm e}^{i \left (d x +c \right )}-3\right ) {\mathrm e}^{i \left (d x +c \right )}}}+\frac {\left (-\frac {2}{3}+\frac {i \sqrt {5}}{3}\right ) \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}-\frac {2}{3}+\frac {i \sqrt {5}}{3}}{-\frac {2}{3}+\frac {i \sqrt {5}}{3}}}\, \sqrt {30}\, \sqrt {i \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {2}{3}-\frac {i \sqrt {5}}{3}\right ) \sqrt {5}}\, \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{\frac {2}{3}-\frac {i \sqrt {5}}{3}}}\, \left (-\frac {2 i \sqrt {5}\, \EllipticE \left (\sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}-\frac {2}{3}+\frac {i \sqrt {5}}{3}}{-\frac {2}{3}+\frac {i \sqrt {5}}{3}}}, \frac {\sqrt {30}\, \sqrt {i \left (\frac {2}{3}-\frac {i \sqrt {5}}{3}\right ) \sqrt {5}}}{10}\right )}{3}+\left (\frac {2}{3}+\frac {i \sqrt {5}}{3}\right ) \EllipticF \left (\sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}-\frac {2}{3}+\frac {i \sqrt {5}}{3}}{-\frac {2}{3}+\frac {i \sqrt {5}}{3}}}, \frac {\sqrt {30}\, \sqrt {i \left (\frac {2}{3}-\frac {i \sqrt {5}}{3}\right ) \sqrt {5}}}{10}\right )\right )}{5 \sqrt {-3 \,{\mathrm e}^{3 i \left (d x +c \right )}+4 \,{\mathrm e}^{2 i \left (d x +c \right )}-3 \,{\mathrm e}^{i \left (d x +c \right )}}}\right ) \sqrt {2}\, \sqrt {-{\mathrm e}^{i \left (d x +c \right )} \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}-4 \,{\mathrm e}^{i \left (d x +c \right )}+3\right )}}{d \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {-\frac {3 \,{\mathrm e}^{2 i \left (d x +c \right )}-4 \,{\mathrm e}^{i \left (d x +c \right )}+3}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}\)	\(576\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.93, size = 108, normalized size = 0.84 \begin {gather*} -\frac {4 \, \sqrt {6} {\rm weierstrassPInverse}\left (-\frac {44}{27}, -\frac {784}{729}, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {4}{9}\right ) + 4 \, \sqrt {6} {\rm weierstrassPInverse}\left (-\frac {44}{27}, -\frac {784}{729}, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {4}{9}\right ) - 9 \, \sqrt {6} {\rm weierstrassZeta}\left (-\frac {44}{27}, -\frac {784}{729}, {\rm weierstrassPInverse}\left (-\frac {44}{27}, -\frac {784}{729}, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {4}{9}\right )\right ) - 9 \, \sqrt {6} {\rm weierstrassZeta}\left (-\frac {44}{27}, -\frac {784}{729}, {\rm weierstrassPInverse}\left (-\frac {44}{27}, -\frac {784}{729}, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {4}{9}\right )\right )}{27 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*(4*sqrt(6)*weierstrassPInverse(-44/27, -784/729, cos(d*x + c) + I*sin(d*x + c) - 4/9) + 4*sqrt(6)*weiers
trassPInverse(-44/27, -784/729, cos(d*x + c) - I*sin(d*x + c) - 4/9) - 9*sqrt(6)*weierstrassZeta(-44/27, -784/
729, weierstrassPInverse(-44/27, -784/729, cos(d*x + c) + I*sin(d*x + c) - 4/9)) - 9*sqrt(6)*weierstrassZeta(-
44/27, -784/729, weierstrassPInverse(-44/27, -784/729, cos(d*x + c) - I*sin(d*x + c) - 4/9)))/d

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {2 \sec {\left (c + d x \right )} - 3} \sqrt {\sec {\left (c + d x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {\frac {2}{\cos \left (c+d\,x\right )}-3}\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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