3.1.5 \(\int \frac {\text {csch}^2(2+3 x)}{2-\coth ^2(2+3 x)} \, dx\) [5]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 = {3756, 212} \begin {gather*} -\frac {\tanh ^{-1}\left (\sqrt {2} \tanh (3 x+2)\right )}{3 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*ArcTanh[Sqrt[2]*Tanh[2 + 3*x]]/Sqrt[2]

Rule 212

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

Rule 3756

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])

Rubi steps

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

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Mathematica [A]
time = 0.08, size = 42, normalized size = 1.91 \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {-1+e^8+\left (1+6 e^4+e^8\right ) \tanh (3 x)}{4 \sqrt {2} e^4}\right )}{3 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(43\) vs. \(2(16)=32\).
time = 2.50, size = 44, normalized size = 2.00

method	result	size
risch	\(\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{4+6 x}-3-2 \sqrt {2}\right )}{12}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{4+6 x}-3+2 \sqrt {2}\right )}{12}\)	\(40\)
derivativedivides	\(-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \left (1+\frac {3 x}{2}\right )+2\right ) \sqrt {2}}{4}\right )}{6}-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \left (1+\frac {3 x}{2}\right )-2\right ) \sqrt {2}}{4}\right )}{6}\)	\(44\)
default	\(-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \left (1+\frac {3 x}{2}\right )+2\right ) \sqrt {2}}{4}\right )}{6}-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \left (1+\frac {3 x}{2}\right )-2\right ) \sqrt {2}}{4}\right )}{6}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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] Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (16) = 32\).
time = 0.48, size = 69, normalized size = 3.14 \begin {gather*} -\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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (16) = 32\).
time = 0.38, size = 89, normalized size = 4.05 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {\operatorname {csch}^{2}{\left (3 x + 2 \right )}}{\coth ^{2}{\left (3 x + 2 \right )} - 2}\, dx \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (16) = 32\).
time = 0.39, size = 51, normalized size = 2.32 \begin {gather*} \frac {1}{12} \, \sqrt {2} \log \left (\frac {{\left | -4 \, \sqrt {2} e^{4} - 6 \, e^{4} + 2 \, e^{\left (6 \, x + 8\right )} \right |}}{{\left | 4 \, \sqrt {2} e^{4} - 6 \, e^{4} + 2 \, e^{\left (6 \, x + 8\right )} \right |}}\right ) \end {gather*}

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]

1/12*sqrt(2)*log(abs(-4*sqrt(2)*e^4 - 6*e^4 + 2*e^(6*x + 8))/abs(4*sqrt(2)*e^4 - 6*e^4 + 2*e^(6*x + 8)))

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Mupad [B]
time = 0.50, size = 57, normalized size = 2.59 \begin {gather*} \frac {\sqrt {2}\,\left (\ln \left (\sqrt {2}\,\left (3\,{\mathrm {e}}^{6\,x+4}-1\right )-4\,{\mathrm {e}}^{6\,x+4}\right )-\ln \left (-4\,{\mathrm {e}}^{6\,x+4}-\sqrt {2}\,\left (3\,{\mathrm {e}}^{6\,x+4}-1\right )\right )\right )}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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