\(\int \sqrt {-3-3 i \text {csch}(x)} \, dx\) [61]

Optimal result

Integrand size = 12, antiderivative size = 23 \[ \int \sqrt {-3-3 i \text {csch}(x)} \, dx=-2 \sqrt {3} \arctan \left (\frac {\coth (x)}{\sqrt {-1-i \text {csch}(x)}}\right ) \]

[Out]

-2*arctan(coth(x)/(-1-I*csch(x))^(1/2))*3^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3859, 213} \[ \int \sqrt {-3-3 i \text {csch}(x)} \, dx=-2 \sqrt {3} \arctan \left (\frac {\coth (x)}{\sqrt {-1-i \text {csch}(x)}}\right ) \]

[In]

Int[Sqrt[-3 - (3*I)*Csch[x]],x]

[Out]

-2*Sqrt[3]*ArcTan[Coth[x]/Sqrt[-1 - I*Csch[x]]]

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 3859

Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[-2*(b/d), Subst[Int[1/(a + x^2), x], x, b*(C
ot[c + d*x]/Sqrt[a + b*Csc[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = 6 i \text {Subst}\left (\int \frac {1}{-3+x^2} \, dx,x,-\frac {3 i \coth (x)}{\sqrt {-3-3 i \text {csch}(x)}}\right ) \\ & = -2 \sqrt {3} \arctan \left (\frac {\coth (x)}{\sqrt {-1-i \text {csch}(x)}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.00 \[ \int \sqrt {-3-3 i \text {csch}(x)} \, dx=-\frac {2 \sqrt {3} \text {arctanh}\left (\sqrt {1-i \text {csch}(x)}\right ) \coth (x)}{\sqrt {-1-i \text {csch}(x)} \sqrt {1-i \text {csch}(x)}} \]

[In]

Integrate[Sqrt[-3 - (3*I)*Csch[x]],x]

[Out]

(-2*Sqrt[3]*ArcTanh[Sqrt[1 - I*Csch[x]]]*Coth[x])/(Sqrt[-1 - I*Csch[x]]*Sqrt[1 - I*Csch[x]])

Maple [F]

\[\int \sqrt {-3-3 i \operatorname {csch}\left (x \right )}d x\]

[In]

int((-3-3*I*csch(x))^(1/2),x)

[Out]

int((-3-3*I*csch(x))^(1/2),x)

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (17) = 34\).

Time = 0.30 (sec) , antiderivative size = 218, normalized size of antiderivative = 9.48 \[ \int \sqrt {-3-3 i \text {csch}(x)} \, dx=-\frac {1}{2} i \, \sqrt {3} \log \left (-2 \, {\left (\frac {\sqrt {3} {\left (i \, \sqrt {3} e^{\left (2 \, x\right )} - i \, \sqrt {3}\right )}}{\sqrt {e^{\left (2 \, x\right )} - 1}} - 3 i \, e^{x} + 3\right )} e^{\left (-x\right )}\right ) + \frac {1}{2} i \, \sqrt {3} \log \left (-2 \, {\left (\frac {\sqrt {3} {\left (-i \, \sqrt {3} e^{\left (2 \, x\right )} + i \, \sqrt {3}\right )}}{\sqrt {e^{\left (2 \, x\right )} - 1}} - 3 i \, e^{x} + 3\right )} e^{\left (-x\right )}\right ) - \frac {1}{2} i \, \sqrt {3} \log \left (-6 \, {\left (i \, \sqrt {3} e^{\left (2 \, x\right )} + \sqrt {3} e^{x} + \frac {\sqrt {3} {\left (-i \, e^{\left (3 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + i \, e^{x} + 2\right )}}{\sqrt {e^{\left (2 \, x\right )} - 1}} - 2 i \, \sqrt {3}\right )} e^{\left (-2 \, x\right )}\right ) + \frac {1}{2} i \, \sqrt {3} \log \left (-6 \, {\left (-i \, \sqrt {3} e^{\left (2 \, x\right )} - \sqrt {3} e^{x} + \frac {\sqrt {3} {\left (-i \, e^{\left (3 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + i \, e^{x} + 2\right )}}{\sqrt {e^{\left (2 \, x\right )} - 1}} + 2 i \, \sqrt {3}\right )} e^{\left (-2 \, x\right )}\right ) \]

[In]

integrate((-3-3*I*csch(x))^(1/2),x, algorithm="fricas")

[Out]

-1/2*I*sqrt(3)*log(-2*(sqrt(3)*(I*sqrt(3)*e^(2*x) - I*sqrt(3))/sqrt(e^(2*x) - 1) - 3*I*e^x + 3)*e^(-x)) + 1/2*
I*sqrt(3)*log(-2*(sqrt(3)*(-I*sqrt(3)*e^(2*x) + I*sqrt(3))/sqrt(e^(2*x) - 1) - 3*I*e^x + 3)*e^(-x)) - 1/2*I*sq
rt(3)*log(-6*(I*sqrt(3)*e^(2*x) + sqrt(3)*e^x + sqrt(3)*(-I*e^(3*x) - 2*e^(2*x) + I*e^x + 2)/sqrt(e^(2*x) - 1)
- 2*I*sqrt(3))*e^(-2*x)) + 1/2*I*sqrt(3)*log(-6*(-I*sqrt(3)*e^(2*x) - sqrt(3)*e^x + sqrt(3)*(-I*e^(3*x) - 2*e
^(2*x) + I*e^x + 2)/sqrt(e^(2*x) - 1) + 2*I*sqrt(3))*e^(-2*x))

Sympy [F]

\[ \int \sqrt {-3-3 i \text {csch}(x)} \, dx=\sqrt {3} \int \sqrt {- i \operatorname {csch}{\left (x \right )} - 1}\, dx \]

[In]

integrate((-3-3*I*csch(x))**(1/2),x)

[Out]

sqrt(3)*Integral(sqrt(-I*csch(x) - 1), x)

Maxima [F]

\[ \int \sqrt {-3-3 i \text {csch}(x)} \, dx=\int { \sqrt {-3 i \, \operatorname {csch}\left (x\right ) - 3} \,d x } \]

[In]

integrate((-3-3*I*csch(x))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-3*I*csch(x) - 3), x)

Giac [F]

\[ \int \sqrt {-3-3 i \text {csch}(x)} \, dx=\int { \sqrt {-3 i \, \operatorname {csch}\left (x\right ) - 3} \,d x } \]

[In]

integrate((-3-3*I*csch(x))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(-3*I*csch(x) - 3), x)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {-3-3 i \text {csch}(x)} \, dx=\int \sqrt {-3-\frac {3{}\mathrm {i}}{\mathrm {sinh}\left (x\right )}} \,d x \]

[In]

int((- 3i/sinh(x) - 3)^(1/2),x)

[Out]

int((- 3i/sinh(x) - 3)^(1/2), x)