\(\int \frac {1}{\text {sech}^2(x)-\tanh ^2(x)} \, dx\) [817]

Optimal result

Integrand size = 13, antiderivative size = 19 \[ \int \frac {1}{\text {sech}^2(x)-\tanh ^2(x)} \, dx=-x+\sqrt {2} \text {arctanh}\left (\sqrt {2} \tanh (x)\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1107, 213} \[ \int \frac {1}{\text {sech}^2(x)-\tanh ^2(x)} \, dx=\sqrt {2} \text {arctanh}\left (\sqrt {2} \tanh (x)\right )-x \]

[In]

Int[(Sech[x]^2 - Tanh[x]^2)^(-1),x]

[Out]

-x + Sqrt[2]*ArcTanh[Sqrt[2]*Tanh[x]]

Rule 213

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

Rule 1107

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.67 (sec) , antiderivative size = 529, normalized size of antiderivative = 27.84 \[ \int \frac {1}{\text {sech}^2(x)-\tanh ^2(x)} \, dx=\frac {\text {sech}(x) \left (\sqrt {2+2 i} \arcsin \left (\frac {1}{2} \sqrt {1+i} \sqrt {(-1-i) (i+\sinh (x))}\right ) \sqrt {(-1-i) (i+\sinh (x))}-2 \text {arctanh}\left (\frac {\sqrt {(-1-i) (i+\sinh (x))}}{\sqrt {(1+i)-(1-i) \sinh (x)}}\right ) \sqrt {(1+i)-(1-i) \sinh (x)} \sqrt {1+i \sinh (x)} \sqrt {(-1-i) (i+\sinh (x))}+2 (-1)^{3/4} \text {arctanh}\left (\frac {(-1)^{3/4} \sqrt {(-1-i) (i+\sinh (x))}}{\sqrt {(1+i)-(1-i) \sinh (x)}}\right ) \sqrt {(1+i)-(1-i) \sinh (x)} \sqrt {1+i \sinh (x)} \sqrt {(-1-i) (i+\sinh (x))}+i \sqrt {2+2 i} \arcsin \left (\frac {1}{2} \sqrt {1+i} \sqrt {(-1-i) (i+\sinh (x))}\right ) \sinh (x) \sqrt {(-1-i) (i+\sinh (x))}+\sqrt {-2+2 i} \text {arcsinh}\left (\frac {1}{2} \sqrt {-1+i} \sqrt {(1-i) (i+\sinh (x))}\right ) \sqrt {(1-i) (i+\sinh (x))}+i \sqrt {-2+2 i} \text {arcsinh}\left (\frac {1}{2} \sqrt {-1+i} \sqrt {(1-i) (i+\sinh (x))}\right ) \sinh (x) \sqrt {(1-i) (i+\sinh (x))}+2 (-1)^{3/4} \arctan \left (\frac {(-1)^{3/4} \sqrt {(1-i) (i+\sinh (x))}}{\sqrt {(1+i) (-i+\sinh (x))}}\right ) \sqrt {1+i \sinh (x)} \sqrt {(1+i) (-i+\sinh (x))} \sqrt {(1-i) (i+\sinh (x))}+2 \text {arctanh}\left (\frac {\sqrt {(1-i) (i+\sinh (x))}}{\sqrt {(1+i) (-i+\sinh (x))}}\right ) \sqrt {1+i \sinh (x)} \sqrt {(1+i) (-i+\sinh (x))} \sqrt {(1-i) (i+\sinh (x))}\right )}{2 \sqrt {1+i \sinh (x)}} \]

[In]

Integrate[(Sech[x]^2 - Tanh[x]^2)^(-1),x]

[Out]

(Sech[x]*(Sqrt[2 + 2*I]*ArcSin[(Sqrt[1 + I]*Sqrt[(-1 - I)*(I + Sinh[x])])/2]*Sqrt[(-1 - I)*(I + Sinh[x])] - 2*
ArcTanh[Sqrt[(-1 - I)*(I + Sinh[x])]/Sqrt[(1 + I) - (1 - I)*Sinh[x]]]*Sqrt[(1 + I) - (1 - I)*Sinh[x]]*Sqrt[1 +
I*Sinh[x]]*Sqrt[(-1 - I)*(I + Sinh[x])] + 2*(-1)^(3/4)*ArcTanh[((-1)^(3/4)*Sqrt[(-1 - I)*(I + Sinh[x])])/Sqrt
[(1 + I) - (1 - I)*Sinh[x]]]*Sqrt[(1 + I) - (1 - I)*Sinh[x]]*Sqrt[1 + I*Sinh[x]]*Sqrt[(-1 - I)*(I + Sinh[x])]
+ I*Sqrt[2 + 2*I]*ArcSin[(Sqrt[1 + I]*Sqrt[(-1 - I)*(I + Sinh[x])])/2]*Sinh[x]*Sqrt[(-1 - I)*(I + Sinh[x])] +
Sqrt[-2 + 2*I]*ArcSinh[(Sqrt[-1 + I]*Sqrt[(1 - I)*(I + Sinh[x])])/2]*Sqrt[(1 - I)*(I + Sinh[x])] + I*Sqrt[-2 +
2*I]*ArcSinh[(Sqrt[-1 + I]*Sqrt[(1 - I)*(I + Sinh[x])])/2]*Sinh[x]*Sqrt[(1 - I)*(I + Sinh[x])] + 2*(-1)^(3/4)
*ArcTan[((-1)^(3/4)*Sqrt[(1 - I)*(I + Sinh[x])])/Sqrt[(1 + I)*(-I + Sinh[x])]]*Sqrt[1 + I*Sinh[x]]*Sqrt[(1 + I
)*(-I + Sinh[x])]*Sqrt[(1 - I)*(I + Sinh[x])] + 2*ArcTanh[Sqrt[(1 - I)*(I + Sinh[x])]/Sqrt[(1 + I)*(-I + Sinh[
x])]]*Sqrt[1 + I*Sinh[x]]*Sqrt[(1 + I)*(-I + Sinh[x])]*Sqrt[(1 - I)*(I + Sinh[x])]))/(2*Sqrt[1 + I*Sinh[x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(38\) vs. \(2(15)=30\).

Time = 0.46 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.05

method	result	size
risch	\(-x +\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+2 \sqrt {2}-3\right )}{2}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-3-2 \sqrt {2}\right )}{2}\)	\(39\)
default	\(-\ln \left (1+\tanh \left (\frac {x}{2}\right )\right )+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )+2\right ) \sqrt {2}}{4}\right )+\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )-2\right ) \sqrt {2}}{4}\right )\)	\(54\)

[In]

int(1/(sech(x)^2-tanh(x)^2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (15) = 30\).

Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.68 \[ \int \frac {1}{\text {sech}^2(x)-\tanh ^2(x)} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (-\frac {3 \, {\left (2 \, \sqrt {2} - 3\right )} \cosh \left (x\right )^{2} - 4 \, {\left (3 \, \sqrt {2} - 4\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 3 \, {\left (2 \, \sqrt {2} - 3\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt {2} + 3}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}\right ) - x \]

[In]

integrate(1/(sech(x)^2-tanh(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

Sympy [F]

\[ \int \frac {1}{\text {sech}^2(x)-\tanh ^2(x)} \, dx=\int \frac {1}{\left (- \tanh {\left (x \right )} + \operatorname {sech}{\left (x \right )}\right ) \left (\tanh {\left (x \right )} + \operatorname {sech}{\left (x \right )}\right )}\, dx \]

[In]

integrate(1/(sech(x)**2-tanh(x)**2),x)

[Out]

Integral(1/((-tanh(x) + sech(x))*(tanh(x) + sech(x))), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (15) = 30\).

Time = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 3.37 \[ \int \frac {1}{\text {sech}^2(x)-\tanh ^2(x)} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} + 1}{\sqrt {2} + e^{\left (-x\right )} - 1}\right ) - \frac {1}{2} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} - 1}{\sqrt {2} + e^{\left (-x\right )} + 1}\right ) - x \]

[In]

integrate(1/(sech(x)^2-tanh(x)^2),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (15) = 30\).

Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.16 \[ \int \frac {1}{\text {sech}^2(x)-\tanh ^2(x)} \, dx=-\frac {1}{2} \, \sqrt {2} \log \left (\frac {{\left | -4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}{{\left | 4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}\right ) - x \]

[In]

integrate(1/(sech(x)^2-tanh(x)^2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.95 \[ \int \frac {1}{\text {sech}^2(x)-\tanh ^2(x)} \, dx=\frac {\sqrt {2}\,\ln \left (8\,{\mathrm {e}}^{2\,x}+\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}-4\right )}{2}\right )}{2}-\frac {\sqrt {2}\,\ln \left (8\,{\mathrm {e}}^{2\,x}-\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}-4\right )}{2}\right )}{2}-x \]

[In]

int(1/(1/cosh(x)^2 - tanh(x)^2),x)

[Out]

(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 - x