3.7 \(\int \frac{\text{csch}^2(2+3 x)}{1-2 \coth ^2(2+3 x)} \, dx\)

Optimal. Leaf size=22 \[ -\frac{\tanh ^{-1}\left (\frac{\tanh (3 x+2)}{\sqrt{2}}\right )}{3 \sqrt{2}} \]

[Out]

-ArcTanh[Tanh[2 + 3*x]/Sqrt[2]]/(3*Sqrt[2])

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Rubi [A]  time = 0.0394867, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {3675, 206} \[ -\frac{\tanh ^{-1}\left (\frac{\tanh (3 x+2)}{\sqrt{2}}\right )}{3 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[Csch[2 + 3*x]^2/(1 - 2*Coth[2 + 3*x]^2),x]

[Out]

-ArcTanh[Tanh[2 + 3*x]/Sqrt[2]]/(3*Sqrt[2])

Rule 3675

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/(c^(m - 1)*f), Subst[Int[(c^2 + ff^2*x^2)^(m/2 - 1)*(a + b*(ff*x)
^n)^p, x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2] && (IntegersQ[n, p
] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rule 206

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

Rubi steps

\begin{align*} \int \frac{\text{csch}^2(2+3 x)}{1-2 \coth ^2(2+3 x)} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1-2 x^2} \, dx,x,\coth (2+3 x)\right )\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{\tanh (2+3 x)}{\sqrt{2}}\right )}{3 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.104643, size = 42, normalized size = 1.91 \[ \frac{\tanh ^{-1}\left (\frac{\left (1-6 e^4+e^8\right ) \tanh (3 x)+e^8-1}{4 \sqrt{2} e^4}\right )}{3 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[2 + 3*x]^2/(1 - 2*Coth[2 + 3*x]^2),x]

[Out]

ArcTanh[(-1 + E^8 + (1 - 6*E^4 + E^8)*Tanh[3*x])/(4*Sqrt[2]*E^4)]/(3*Sqrt[2])

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Maple [B]  time = 0.054, size = 102, normalized size = 4.6 \begin{align*} -{\frac{\sqrt{2}}{24}\ln \left ({ \left ( \left ( \tanh \left ( 1+{\frac{3\,x}{2}} \right ) \right ) ^{2}+\sqrt{2}\tanh \left ( 1+{\frac{3\,x}{2}} \right ) +1 \right ) \left ( \left ( \tanh \left ( 1+{\frac{3\,x}{2}} \right ) \right ) ^{2}-\sqrt{2}\tanh \left ( 1+{\frac{3\,x}{2}} \right ) +1 \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}}{24}\ln \left ({ \left ( \left ( \tanh \left ( 1+{\frac{3\,x}{2}} \right ) \right ) ^{2}-\sqrt{2}\tanh \left ( 1+{\frac{3\,x}{2}} \right ) +1 \right ) \left ( \left ( \tanh \left ( 1+{\frac{3\,x}{2}} \right ) \right ) ^{2}+\sqrt{2}\tanh \left ( 1+{\frac{3\,x}{2}} \right ) +1 \right ) ^{-1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*2^(1/2)*ln((tanh(1+3/2*x)^2+2^(1/2)*tanh(1+3/2*x)+1)/(tanh(1+3/2*x)^2-2^(1/2)*tanh(1+3/2*x)+1))+1/24*2^(
1/2)*ln((tanh(1+3/2*x)^2-2^(1/2)*tanh(1+3/2*x)+1)/(tanh(1+3/2*x)^2+2^(1/2)*tanh(1+3/2*x)+1))

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Maxima [B]  time = 1.67463, size = 51, normalized size = 2.32 \begin{align*} \frac{1}{12} \, \sqrt{2} \log \left (-\frac{2 \, \sqrt{2} - e^{\left (-6 \, x - 4\right )} - 3}{2 \, \sqrt{2} + e^{\left (-6 \, x - 4\right )} + 3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sqrt(2)*log(-(2*sqrt(2) - e^(-6*x - 4) - 3)/(2*sqrt(2) + e^(-6*x - 4) + 3))

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Fricas [B]  time = 2.09315, size = 263, normalized size = 11.95 \begin{align*} \frac{1}{12} \, \sqrt{2} \log \left (\frac{3 \,{\left (2 \, \sqrt{2} + 3\right )} \cosh \left (3 \, x + 2\right )^{2} - 4 \,{\left (3 \, \sqrt{2} + 4\right )} \cosh \left (3 \, x + 2\right ) \sinh \left (3 \, x + 2\right ) + 3 \,{\left (2 \, \sqrt{2} + 3\right )} \sinh \left (3 \, x + 2\right )^{2} + 2 \, \sqrt{2} + 3}{\cosh \left (3 \, x + 2\right )^{2} + \sinh \left (3 \, x + 2\right )^{2} + 3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sqrt(2)*log((3*(2*sqrt(2) + 3)*cosh(3*x + 2)^2 - 4*(3*sqrt(2) + 4)*cosh(3*x + 2)*sinh(3*x + 2) + 3*(2*sqr
t(2) + 3)*sinh(3*x + 2)^2 + 2*sqrt(2) + 3)/(cosh(3*x + 2)^2 + sinh(3*x + 2)^2 + 3))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{\operatorname{csch}^{2}{\left (3 x + 2 \right )}}{2 \coth ^{2}{\left (3 x + 2 \right )} - 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(csch(3*x + 2)**2/(2*coth(3*x + 2)**2 - 1), x)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out