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*arctanh(coth(x)/(1+I*csch(x))^(1/2))*3^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.74, size = 46, normalized size = 2.00 \[ \frac {2 \sqrt {3} \coth (x) \tan ^{-1}\left (\sqrt {-1+i \text {csch}(x)}\right )}{\sqrt {-1+i \text {csch}(x)} \sqrt {1+i \text {csch}(x)}} \]

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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fricas [B]  time = 0.46, size = 218, normalized size = 9.48 \[ \frac {1}{2} \, \sqrt {3} \log \left ({\left (\frac {2 \, \sqrt {3} {\left (\sqrt {3} e^{\left (2 \, x\right )} - \sqrt {3}\right )}}{\sqrt {e^{\left (2 \, x\right )} - 1}} + 6 \, e^{x} + 6 i\right )} e^{\left (-x\right )}\right ) - \frac {1}{2} \, \sqrt {3} \log \left (-{\left (\frac {2 \, \sqrt {3} {\left (\sqrt {3} e^{\left (2 \, x\right )} - \sqrt {3}\right )}}{\sqrt {e^{\left (2 \, x\right )} - 1}} - 6 \, e^{x} - 6 i\right )} e^{\left (-x\right )}\right ) + \frac {1}{2} \, \sqrt {3} \log \left ({\left (6 \, \sqrt {3} e^{\left (2 \, x\right )} - 6 i \, \sqrt {3} e^{x} + \frac {\sqrt {3} {\left (6 \, e^{\left (3 \, x\right )} - 12 i \, e^{\left (2 \, x\right )} - 6 \, e^{x} + 12 i\right )}}{\sqrt {e^{\left (2 \, x\right )} - 1}} - 12 \, \sqrt {3}\right )} e^{\left (-2 \, x\right )}\right ) - \frac {1}{2} \, \sqrt {3} \log \left (-{\left (6 \, \sqrt {3} e^{\left (2 \, x\right )} - 6 i \, \sqrt {3} e^{x} - \frac {\sqrt {3} {\left (6 \, e^{\left (3 \, x\right )} - 12 i \, e^{\left (2 \, x\right )} - 6 \, e^{x} + 12 i\right )}}{\sqrt {e^{\left (2 \, x\right )} - 1}} - 12 \, \sqrt {3}\right )} e^{\left (-2 \, x\right )}\right ) \]

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((2*sqrt(3)*(sqrt(3)*e^(2*x) - sqrt(3))/sqrt(e^(2*x) - 1) + 6*e^x + 6*I)*e^(-x)) - 1/2*sqrt(3)*
log(-(2*sqrt(3)*(sqrt(3)*e^(2*x) - sqrt(3))/sqrt(e^(2*x) - 1) - 6*e^x - 6*I)*e^(-x)) + 1/2*sqrt(3)*log((6*sqrt
(3)*e^(2*x) - 6*I*sqrt(3)*e^x + sqrt(3)*(6*e^(3*x) - 12*I*e^(2*x) - 6*e^x + 12*I)/sqrt(e^(2*x) - 1) - 12*sqrt(
3))*e^(-2*x)) - 1/2*sqrt(3)*log(-(6*sqrt(3)*e^(2*x) - 6*I*sqrt(3)*e^x - sqrt(3)*(6*e^(3*x) - 12*I*e^(2*x) - 6*
e^x + 12*I)/sqrt(e^(2*x) - 1) - 12*sqrt(3))*e^(-2*x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {3 i \, \operatorname {csch}\relax (x) + 3}\,{d x} \]

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)

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maple [F]  time = 0.83, size = 0, normalized size = 0.00 \[ \int \sqrt {3+3 i \mathrm {csch}\relax (x )}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {3 i \, \operatorname {csch}\relax (x) + 3}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \sqrt {3+\frac {3{}\mathrm {i}}{\mathrm {sinh}\relax (x)}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3i/sinh(x) + 3)^(1/2),x)

[Out]

int((3i/sinh(x) + 3)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \sqrt {3} \int \sqrt {i \operatorname {csch}{\relax (x )} + 1}\, dx \]

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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