\(\int \sqrt {3+3 i \text {csch}(x)} \, dx\) [58]

Optimal result

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

[Out]

2*arctanh(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, 209} \[ \int \sqrt {3+3 i \text {csch}(x)} \, dx=2 \sqrt {3} \text {arctanh}\left (\frac {\coth (x)}{\sqrt {1+i \text {csch}(x)}}\right ) \]

[In]

Int[Sqrt[3 + (3*I)*Csch[x]],x]

[Out]

2*Sqrt[3]*ArcTanh[Coth[x]/Sqrt[1 + I*Csch[x]]]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[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}& = -\left (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 )\right ) \\ & = 2 \sqrt {3} \text {arctanh}\left (\frac {\coth (x)}{\sqrt {1+i \text {csch}(x)}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.00 \[ \int \sqrt {3+3 i \text {csch}(x)} \, dx=\frac {2 \sqrt {3} \arctan \left (\sqrt {-1+i \text {csch}(x)}\right ) \coth (x)}{\sqrt {-1+i \text {csch}(x)} \sqrt {1+i \text {csch}(x)}} \]

[In]

Integrate[Sqrt[3 + (3*I)*Csch[x]],x]

[Out]

(2*Sqrt[3]*ArcTan[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. 212 vs. \(2 (17) = 34\).

Time = 0.27 (sec) , antiderivative size = 212, normalized size of antiderivative = 9.22 \[ \int \sqrt {3+3 i \text {csch}(x)} \, dx=\frac {1}{2} \, \sqrt {3} \log \left (2 \, {\left (\frac {\sqrt {3} {\left (\sqrt {3} e^{\left (2 \, x\right )} - \sqrt {3}\right )}}{\sqrt {e^{\left (2 \, x\right )} - 1}} + 3 \, e^{x} + 3 i\right )} e^{\left (-x\right )}\right ) - \frac {1}{2} \, \sqrt {3} \log \left (-2 \, {\left (\frac {\sqrt {3} {\left (\sqrt {3} e^{\left (2 \, x\right )} - \sqrt {3}\right )}}{\sqrt {e^{\left (2 \, x\right )} - 1}} - 3 \, e^{x} - 3 i\right )} e^{\left (-x\right )}\right ) + \frac {1}{2} \, \sqrt {3} \log \left (6 \, {\left (\sqrt {3} e^{\left (2 \, x\right )} - i \, \sqrt {3} e^{x} + \frac {\sqrt {3} {\left (e^{\left (3 \, x\right )} - 2 i \, e^{\left (2 \, x\right )} - e^{x} + 2 i\right )}}{\sqrt {e^{\left (2 \, x\right )} - 1}} - 2 \, \sqrt {3}\right )} e^{\left (-2 \, x\right )}\right ) - \frac {1}{2} \, \sqrt {3} \log \left (-6 \, {\left (\sqrt {3} e^{\left (2 \, x\right )} - i \, \sqrt {3} e^{x} - \frac {\sqrt {3} {\left (e^{\left (3 \, x\right )} - 2 i \, e^{\left (2 \, x\right )} - e^{x} + 2 i\right )}}{\sqrt {e^{\left (2 \, x\right )} - 1}} - 2 \, \sqrt {3}\right )} e^{\left (-2 \, x\right )}\right ) \]

[In]

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

[Out]

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