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

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

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3675, 203} \[ \frac {\tan ^{-1}\left (\sqrt {2} \tanh (3 x+2)\right )}{3 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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])

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 {\tan ^{-1}\left (\sqrt {2} \tanh (2+3 x)\right )}{3 \sqrt {2}}\\ \end {align*}

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Mathematica [B]  time = 0.08, size = 47, normalized size = 2.14 \[ \frac {\tan ^{-1}\left (\frac {\left (3+2 e^4+3 e^8\right ) \tanh (3 x)+3 \left (e^8-1\right )}{4 \sqrt {2} e^4}\right )}{3 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.39, size = 47, normalized size = 2.14 \[ -\frac {1}{6} \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} \cosh \left (3 \, x + 2\right ) + 2 \, \sqrt {2} \sinh \left (3 \, x + 2\right )}{2 \, {\left (\cosh \left (3 \, x + 2\right ) - \sinh \left (3 \, x + 2\right )\right )}}\right ) \]

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]

-1/6*sqrt(2)*arctan(-1/2*(sqrt(2)*cosh(3*x + 2) + 2*sqrt(2)*sinh(3*x + 2))/(cosh(3*x + 2) - sinh(3*x + 2)))

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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 >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 190.94Not invertible Error: Bad Argument Value

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maple [B]  time = 0.52, size = 156, normalized size = 7.09 \[ \frac {\sqrt {6}\, \arctan \left (\frac {2 \tanh \left (1+\frac {3 x}{2}\right )}{2 \sqrt {3}-2 \sqrt {2}}\right )}{6 \sqrt {3}-6 \sqrt {2}}-\frac {2 \arctan \left (\frac {2 \tanh \left (1+\frac {3 x}{2}\right )}{2 \sqrt {3}-2 \sqrt {2}}\right )}{3 \left (2 \sqrt {3}-2 \sqrt {2}\right )}-\frac {\sqrt {6}\, \arctan \left (\frac {2 \tanh \left (1+\frac {3 x}{2}\right )}{2 \sqrt {3}+2 \sqrt {2}}\right )}{3 \left (2 \sqrt {3}+2 \sqrt {2}\right )}-\frac {2 \arctan \left (\frac {2 \tanh \left (1+\frac {3 x}{2}\right )}{2 \sqrt {3}+2 \sqrt {2}}\right )}{3 \left (2 \sqrt {3}+2 \sqrt {2}\right )} \]

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]

1/3*6^(1/2)/(2*3^(1/2)-2*2^(1/2))*arctan(2*tanh(1+3/2*x)/(2*3^(1/2)-2*2^(1/2)))-2/3/(2*3^(1/2)-2*2^(1/2))*arct
an(2*tanh(1+3/2*x)/(2*3^(1/2)-2*2^(1/2)))-1/3*6^(1/2)/(2*3^(1/2)+2*2^(1/2))*arctan(2*tanh(1+3/2*x)/(2*3^(1/2)+
2*2^(1/2)))-2/3/(2*3^(1/2)+2*2^(1/2))*arctan(2*tanh(1+3/2*x)/(2*3^(1/2)+2*2^(1/2)))

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maxima [A]  time = 0.69, size = 21, normalized size = 0.95 \[ -\frac {1}{6} \, \sqrt {2} \arctan \left (\frac {1}{4} \, \sqrt {2} {\left (3 \, e^{\left (-6 \, x - 4\right )} - 1\right )}\right ) \]

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]

-1/6*sqrt(2)*arctan(1/4*sqrt(2)*(3*e^(-6*x - 4) - 1))

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mupad [B]  time = 1.53, size = 21, normalized size = 0.95 \[ \frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\left (3\,{\mathrm {e}}^{6\,x+4}-1\right )}{4}\right )}{6} \]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}^{2}{\left (3 x + 2 \right )}}{\coth ^{2}{\left (3 x + 2 \right )} + 2}\, dx \]

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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