### 3.5 $$\int \frac{\text{csch}^2(2+3 x)}{2-\coth ^2(2+3 x)} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0386269, 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} $-\frac{\tanh ^{-1}\left (\sqrt{2} \tanh (3 x+2)\right )}{3 \sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-ArcTanh[Sqrt[2]*Tanh[2 + 3*x]]/(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)}{2-\coth ^2(2+3 x)} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\coth (2+3 x)\right )\right )\\ &=-\frac{\tanh ^{-1}\left (\sqrt{2} \tanh (2+3 x)\right )}{3 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.100644, 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/(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 = 0.052, size = 44, normalized size = 2. \begin{align*} -{\frac{\sqrt{2}}{6}{\it Artanh} \left ({\frac{\sqrt{2}}{4} \left ( 2\,\tanh \left ( 1+3/2\,x \right ) -2 \right ) } \right ) }-{\frac{\sqrt{2}}{6}{\it Artanh} \left ({\frac{\sqrt{2}}{4} \left ( 2\,\tanh \left ( 1+3/2\,x \right ) +2 \right ) } \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*2^(1/2)*arctanh(1/4*(2*tanh(1+3/2*x)-2)*2^(1/2))-1/6*2^(1/2)*arctanh(1/4*(2*tanh(1+3/2*x)+2)*2^(1/2))

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Maxima [B]  time = 1.60329, size = 93, normalized size = 4.23 \begin{align*} -\frac{1}{12} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{\left (-3 \, x - 2\right )} + 1}{\sqrt{2} + e^{\left (-3 \, x - 2\right )} - 1}\right ) + \frac{1}{12} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{\left (-3 \, x - 2\right )} - 1}{\sqrt{2} + e^{\left (-3 \, x - 2\right )} + 1}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.11168, 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/(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 )}}{\coth ^{2}{\left (3 x + 2 \right )} - 2}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError