3.54 \(\int \frac{1}{\sqrt{a+i a \text{csch}(c+d x)}} \, dx\)

Optimal. Leaf size=91 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a+i a \text{csch}(c+d x)}}\right )}{\sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{2} \sqrt{a+i a \text{csch}(c+d x)}}\right )}{\sqrt{a} d} \]

[Out]

(2*ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a + I*a*Csch[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Coth[
c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Csch[c + d*x]])])/(Sqrt[a]*d)

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Rubi [A]  time = 0.0873695, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {3776, 3774, 203, 3795} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a+i a \text{csch}(c+d x)}}\right )}{\sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{2} \sqrt{a+i a \text{csch}(c+d x)}}\right )}{\sqrt{a} d} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a + I*a*Csch[c + d*x]],x]

[Out]

(2*ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a + I*a*Csch[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Coth[
c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Csch[c + d*x]])])/(Sqrt[a]*d)

Rule 3776

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

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 3795

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[-2/f, Subst[Int[1/(2
*a + x^2), x], x, (b*Cot[e + f*x])/Sqrt[a + b*Csc[e + f*x]]], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0
]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a+i a \text{csch}(c+d x)}} \, dx &=-\left (i \int \frac{\text{csch}(c+d x)}{\sqrt{a+i a \text{csch}(c+d x)}} \, dx\right )+\frac{\int \sqrt{a+i a \text{csch}(c+d x)} \, dx}{a}\\ &=-\frac{(2 i) \operatorname{Subst}\left (\int \frac{1}{a+x^2} \, dx,x,\frac{i a \coth (c+d x)}{\sqrt{a+i a \text{csch}(c+d x)}}\right )}{d}+\frac{(2 i) \operatorname{Subst}\left (\int \frac{1}{2 a+x^2} \, dx,x,\frac{i a \coth (c+d x)}{\sqrt{a+i a \text{csch}(c+d x)}}\right )}{d}\\ &=\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a+i a \text{csch}(c+d x)}}\right )}{\sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{2} \sqrt{a+i a \text{csch}(c+d x)}}\right )}{\sqrt{a} d}\\ \end{align*}

Mathematica [A]  time = 1.08103, size = 118, normalized size = 1.3 \[ \frac{\sqrt{a} \coth (c+d x) \left (2 \tan ^{-1}\left (\frac{\sqrt{i a (\text{csch}(c+d x)+i)}}{\sqrt{a}}\right )-\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{i a (\text{csch}(c+d x)+i)}}{\sqrt{2} \sqrt{a}}\right )\right )}{d \sqrt{i a (\text{csch}(c+d x)+i)} \sqrt{a+i a \text{csch}(c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a + I*a*Csch[c + d*x]],x]

[Out]

(Sqrt[a]*(2*ArcTan[Sqrt[I*a*(I + Csch[c + d*x])]/Sqrt[a]] - Sqrt[2]*ArcTan[Sqrt[I*a*(I + Csch[c + d*x])]/(Sqrt
[2]*Sqrt[a])])*Coth[c + d*x])/(d*Sqrt[I*a*(I + Csch[c + d*x])]*Sqrt[a + I*a*Csch[c + d*x]])

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Maple [F]  time = 0.324, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{a+ia{\rm csch} \left (dx+c\right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+I*a*csch(d*x+c))^(1/2),x)

[Out]

int(1/(a+I*a*csch(d*x+c))^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{i \, a \operatorname{csch}\left (d x + c\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(I*a*csch(d*x + c) + a), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*csch(d*x+c))**(1/2),x)

[Out]

Integral(1/sqrt(I*a*csch(c + d*x) + a), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{i \, a \operatorname{csch}\left (d x + c\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(I*a*csch(d*x + c) + a), x)