3.58 \(\int \sqrt{3+3 i \text{csch}(x)} \, dx\)

Optimal. Leaf size=23 \[ 2 \sqrt{3} \tanh ^{-1}\left (\frac{\coth (x)}{\sqrt{1+i \text{csch}(x)}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.017324, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3774, 203} \[ 2 \sqrt{3} \tanh ^{-1}\left (\frac{\coth (x)}{\sqrt{1+i \text{csch}(x)}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[3 + (3*I)*Csch[x]],x]

[Out]

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

Rule 3774

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]

Rule 203

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

Rubi steps

\begin{align*} \int \sqrt{3+3 i \text{csch}(x)} \, dx &=-\left (6 i \operatorname{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} \tanh ^{-1}\left (\frac{\coth (x)}{\sqrt{1+i \text{csch}(x)}}\right )\\ \end{align*}

Mathematica [A]  time = 0.685641, size = 46, normalized size = 2. \[ \frac{2 \sqrt{3} \coth (x) \tan ^{-1}\left (\sqrt{-1+i \text{csch}(x)}\right )}{\sqrt{-1+i \text{csch}(x)} \sqrt{1+i \text{csch}(x)}} \]

Antiderivative was successfully verified.

[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]])

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Maple [F]  time = 0.261, size = 0, normalized size = 0. \begin{align*} \int \sqrt{3+3\,i{\rm csch} \left (x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{3 i \, \operatorname{csch}\left (x\right ) + 3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(3*I*csch(x) + 3), x)

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Fricas [B]  time = 2.12005, size = 581, normalized size = 25.26 \begin{align*} -\frac{1}{2} \, \sqrt{3} \log \left (\frac{\sqrt{\frac{3 \, e^{\left (2 \, x\right )} + 6 i \, e^{x} - 3}{e^{\left (2 \, x\right )} - 1}}{\left (\left (i - 4\right ) \, e^{\left (3 \, x\right )} + \left (4 i + 1\right ) \, e^{\left (2 \, x\right )} - \left (i - 4\right ) \, e^{x} - 4 i - 1\right )} - \left (i - 4\right ) \, \sqrt{3} e^{\left (3 \, x\right )} - \left (4 i + 1\right ) \, \sqrt{3}}{\left (10 i + 24\right ) \, e^{\left (2 \, x\right )} + \left (24 i - 10\right ) \, e^{x}}\right ) + \frac{1}{2} \, \sqrt{3} \log \left (\frac{\sqrt{\frac{3 \, e^{\left (2 \, x\right )} + 6 i \, e^{x} - 3}{e^{\left (2 \, x\right )} - 1}}{\left (\left (i - 4\right ) \, e^{\left (3 \, x\right )} + \left (4 i + 1\right ) \, e^{\left (2 \, x\right )} - \left (i - 4\right ) \, e^{x} - 4 i - 1\right )} + \left (i - 4\right ) \, \sqrt{3} e^{\left (3 \, x\right )} + \left (4 i + 1\right ) \, \sqrt{3}}{\left (10 i + 24\right ) \, e^{\left (2 \, x\right )} + \left (24 i - 10\right ) \, e^{x}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(3)*log((sqrt((3*e^(2*x) + 6*I*e^x - 3)/(e^(2*x) - 1))*((I - 4)*e^(3*x) + (4*I + 1)*e^(2*x) - (I - 4)
*e^x - 4*I - 1) - (I - 4)*sqrt(3)*e^(3*x) - (4*I + 1)*sqrt(3))/((10*I + 24)*e^(2*x) + (24*I - 10)*e^x)) + 1/2*
sqrt(3)*log((sqrt((3*e^(2*x) + 6*I*e^x - 3)/(e^(2*x) - 1))*((I - 4)*e^(3*x) + (4*I + 1)*e^(2*x) - (I - 4)*e^x
- 4*I - 1) + (I - 4)*sqrt(3)*e^(3*x) + (4*I + 1)*sqrt(3))/((10*I + 24)*e^(2*x) + (24*I - 10)*e^x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \sqrt{3} \int \sqrt{i \operatorname{csch}{\left (x \right )} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{3 i \, \operatorname{csch}\left (x\right ) + 3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(3*I*csch(x) + 3), x)