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

Optimal result

Integrand size = 25, antiderivative size = 129 \[ \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {-3+2 \sec (c+d x)}} \, dx=\frac {4 \sqrt {2-3 \cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+\pi +d x),\frac {6}{5}\right ) \sqrt {\sec (c+d x)}}{3 \sqrt {5} d \sqrt {-3+2 \sec (c+d x)}}-\frac {2 \sqrt {5} E\left (\frac {1}{2} (c+\pi +d x)|\frac {6}{5}\right ) \sqrt {-3+2 \sec (c+d x)}}{3 d \sqrt {2-3 \cos (c+d x)} \sqrt {\sec (c+d x)}} \]

[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)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3947, 3941, 2733, 3943, 2741} \[ \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {-3+2 \sec (c+d x)}} \, dx=\frac {4 \sqrt {2-3 \cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x+\pi ),\frac {6}{5}\right )}{3 \sqrt {5} d \sqrt {2 \sec (c+d x)-3}}-\frac {2 \sqrt {5} \sqrt {2 \sec (c+d x)-3} E\left (\frac {1}{2} (c+d x+\pi )|\frac {6}{5}\right )}{3 d \sqrt {2-3 \cos (c+d x)} \sqrt {\sec (c+d x)}} \]

[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*} \text {integral}& = -\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)} \operatorname {EllipticF}\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*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.56 \[ \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {-3+2 \sec (c+d x)}} \, dx=\frac {\sqrt {-2+3 \cos (c+d x)} \left (2 E\left (\left .\frac {1}{2} (c+d x)\right |6\right )+4 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),6\right )\right ) \sqrt {\sec (c+d x)}}{3 d \sqrt {-3+2 \sec (c+d x)}} \]

[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]])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 7.57 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.57

method	result	size
default	\(-\frac {2 \sqrt {-\frac {5 \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}\, \left (3 i \operatorname {EllipticF}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i \sqrt {5}\right ) \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1}\, \sqrt {-5 \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1}-i \operatorname {EllipticE}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i \sqrt {5}\right ) \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1}\, \sqrt {-5 \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1}+5 \left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}{3 d \left (5 \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right ) \sqrt {-\frac {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}}\)	\(332\)
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}\, \operatorname {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 ) \operatorname {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}}}\)	\(576\)

[In]

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

[Out]

-2/3/d*(-(5*(1-cos(d*x+c))^2*csc(d*x+c)^2-1)/((1-cos(d*x+c))^2*csc(d*x+c)^2-1))^(1/2)*(3*I*EllipticF(I*(-cot(d
*x+c)+csc(d*x+c)),I*5^(1/2))*((1-cos(d*x+c))^2*csc(d*x+c)^2+1)^(1/2)*(-5*(1-cos(d*x+c))^2*csc(d*x+c)^2+1)^(1/2
)-I*EllipticE(I*(-cot(d*x+c)+csc(d*x+c)),I*5^(1/2))*((1-cos(d*x+c))^2*csc(d*x+c)^2+1)^(1/2)*(-5*(1-cos(d*x+c))
^2*csc(d*x+c)^2+1)^(1/2)+5*(1-cos(d*x+c))^3*csc(d*x+c)^3+cot(d*x+c)-csc(d*x+c))/(5*(1-cos(d*x+c))^2*csc(d*x+c)
^2-1)/(-((1-cos(d*x+c))^2*csc(d*x+c)^2+1)/((1-cos(d*x+c))^2*csc(d*x+c)^2-1))^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.10 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {-3+2 \sec (c+d x)}} \, dx=-\frac {4 \, \sqrt {6} {\rm weierstrassPInverse}\left (-\frac {44}{27}, -\frac {784}{729}, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {4}{9}\right ) + 4 \, \sqrt {6} {\rm weierstrassPInverse}\left (-\frac {44}{27}, -\frac {784}{729}, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {4}{9}\right ) - 9 \, \sqrt {6} {\rm weierstrassZeta}\left (-\frac {44}{27}, -\frac {784}{729}, {\rm weierstrassPInverse}\left (-\frac {44}{27}, -\frac {784}{729}, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {4}{9}\right )\right ) - 9 \, \sqrt {6} {\rm weierstrassZeta}\left (-\frac {44}{27}, -\frac {784}{729}, {\rm weierstrassPInverse}\left (-\frac {44}{27}, -\frac {784}{729}, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {4}{9}\right )\right )}{27 \, d} \]

[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

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {-3+2 \sec (c+d x)}} \, dx=\int \frac {1}{\sqrt {\frac {2}{\cos \left (c+d\,x\right )}-3}\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \]

[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)