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

Optimal. Leaf size=18 \[ x-\sqrt {2} \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right ) \]

[Out]

x-arctanh(1/2*2^(1/2)*tanh(x))*2^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1130, 207} \[ x-\sqrt {2} \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(Coth[x]^2 + Csch[x]^2)^(-1),x]

[Out]

x - Sqrt[2]*ArcTanh[Tanh[x]/Sqrt[2]]

Rule 207

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

Rule 1130

Int[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(
d^2*(b/q + 1))/2, Int[(d*x)^(m - 2)/(b/2 + q/2 + c*x^2), x], x] - Dist[(d^2*(b/q - 1))/2, Int[(d*x)^(m - 2)/(b
/2 - q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 2]

Rubi steps

\begin {align*} \int \frac {1}{\coth ^2(x)+\text {csch}^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {x^2}{2-3 x^2+x^4} \, dx,x,\tanh (x)\right )\\ &=2 \operatorname {Subst}\left (\int \frac {1}{-2+x^2} \, dx,x,\tanh (x)\right )-\operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tanh (x)\right )\\ &=x-\sqrt {2} \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 18, normalized size = 1.00 \[ x-\sqrt {2} \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x - Sqrt[2]*ArcTanh[Tanh[x]/Sqrt[2]]

________________________________________________________________________________________

fricas [B]  time = 0.45, size = 67, normalized size = 3.72 \[ \frac {1}{2} \, \sqrt {2} \log \left (\frac {3 \, {\left (2 \, \sqrt {2} + 3\right )} \cosh \relax (x)^{2} - 4 \, {\left (3 \, \sqrt {2} + 4\right )} \cosh \relax (x) \sinh \relax (x) + 3 \, {\left (2 \, \sqrt {2} + 3\right )} \sinh \relax (x)^{2} + 2 \, \sqrt {2} + 3}{\cosh \relax (x)^{2} + \sinh \relax (x)^{2} + 3}\right ) + x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [B]  time = 0.15, size = 36, normalized size = 2.00 \[ -\frac {1}{2} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (2 \, x\right )} + 3}\right ) + x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.21, size = 102, normalized size = 5.67 \[ -\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-\frac {\sqrt {2}\, \ln \left (\frac {\tanh ^{2}\left (\frac {x}{2}\right )+\sqrt {2}\, \tanh \left (\frac {x}{2}\right )+1}{\tanh ^{2}\left (\frac {x}{2}\right )-\sqrt {2}\, \tanh \left (\frac {x}{2}\right )+1}\right )}{4}+\frac {\sqrt {2}\, \ln \left (\frac {\tanh ^{2}\left (\frac {x}{2}\right )-\sqrt {2}\, \tanh \left (\frac {x}{2}\right )+1}{\tanh ^{2}\left (\frac {x}{2}\right )+\sqrt {2}\, \tanh \left (\frac {x}{2}\right )+1}\right )}{4}+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(tanh(1/2*x)-1)-1/4*2^(1/2)*ln((tanh(1/2*x)^2+2^(1/2)*tanh(1/2*x)+1)/(tanh(1/2*x)^2-2^(1/2)*tanh(1/2*x)+1))
+1/4*2^(1/2)*ln((tanh(1/2*x)^2-2^(1/2)*tanh(1/2*x)+1)/(tanh(1/2*x)^2+2^(1/2)*tanh(1/2*x)+1))+ln(tanh(1/2*x)+1)

________________________________________________________________________________________

maxima [B]  time = 0.41, size = 36, normalized size = 2.00 \[ \frac {1}{2} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (-2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (-2 \, x\right )} + 3}\right ) + x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.15, size = 54, normalized size = 3.00 \[ x+\frac {\sqrt {2}\,\ln \left (8\,{\mathrm {e}}^{2\,x}-\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}+4\right )}{2}\right )}{2}-\frac {\sqrt {2}\,\ln \left (8\,{\mathrm {e}}^{2\,x}+\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}+4\right )}{2}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(coth(x)^2 + 1/sinh(x)^2),x)

[Out]

x + (2^(1/2)*log(8*exp(2*x) - (2^(1/2)*(12*exp(2*x) + 4))/2))/2 - (2^(1/2)*log(8*exp(2*x) + (2^(1/2)*(12*exp(2
*x) + 4))/2))/2

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\coth ^{2}{\relax (x )} + \operatorname {csch}^{2}{\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________