3.574 \(\int \frac {1+\sinh ^2(x)}{1-\sinh ^2(x)} \, dx\)

Optimal. Leaf size=19 \[ \sqrt {2} \tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )-x \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3171, 3181, 206} \[ \sqrt {2} \tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )-x \]

Antiderivative was successfully verified.

[In]

Int[(1 + Sinh[x]^2)/(1 - Sinh[x]^2),x]

[Out]

-x + Sqrt[2]*ArcTanh[Sqrt[2]*Tanh[x]]

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])

Rule 3171

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(B*x
)/b, x] + Dist[(A*b - a*B)/b, Int[1/(a + b*Sin[e + f*x]^2), x], x] /; FreeQ[{a, b, e, f, A, B}, x]

Rule 3181

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[1/(a + (a + b)*ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 24, normalized size = 1.26 \[ -2 \left (\frac {x}{2}-\frac {\tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{\sqrt {2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Sinh[x]^2)/(1 - Sinh[x]^2),x]

[Out]

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

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fricas [B]  time = 0.42, size = 70, normalized size = 3.68 \[ \frac {1}{2} \, \sqrt {2} \log \left (-\frac {3 \, {\left (2 \, \sqrt {2} - 3\right )} \cosh \relax (x)^{2} - 4 \, {\left (3 \, \sqrt {2} - 4\right )} \cosh \relax (x) \sinh \relax (x) + 3 \, {\left (2 \, \sqrt {2} - 3\right )} \sinh \relax (x)^{2} - 2 \, \sqrt {2} + 3}{\cosh \relax (x)^{2} + \sinh \relax (x)^{2} - 3}\right ) - x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+sinh(x)^2)/(1-sinh(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

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giac [B]  time = 0.14, size = 41, normalized size = 2.16 \[ -\frac {1}{2} \, \sqrt {2} \log \left (\frac {{\left | -4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}{{\left | 4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}\right ) - x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+sinh(x)^2)/(1-sinh(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

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maple [B]  time = 0.16, size = 54, normalized size = 2.84 \[ \sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )-2\right ) \sqrt {2}}{4}\right )+\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )+2\right ) \sqrt {2}}{4}\right )-\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.46, size = 64, normalized size = 3.37 \[ \frac {1}{2} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} + 1}{\sqrt {2} + e^{\left (-x\right )} - 1}\right ) - \frac {1}{2} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} - 1}{\sqrt {2} + e^{\left (-x\right )} + 1}\right ) - x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+sinh(x)^2)/(1-sinh(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

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mupad [B]  time = 0.13, size = 56, normalized size = 2.95 \[ \frac {\sqrt {2}\,\ln \left (8\,{\mathrm {e}}^{2\,x}+\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}-4\right )}{2}\right )}{2}-\frac {\sqrt {2}\,\ln \left (8\,{\mathrm {e}}^{2\,x}-\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}-4\right )}{2}\right )}{2}-x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(sinh(x)^2 + 1)/(sinh(x)^2 - 1),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

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sympy [B]  time = 6.54, size = 238, normalized size = 12.53 \[ - \frac {1331714 x}{941664 \sqrt {2} + 1331714} - \frac {941664 \sqrt {2} x}{941664 \sqrt {2} + 1331714} + \frac {941664 \log {\left (\tanh {\left (\frac {x}{2} \right )} - 1 + \sqrt {2} \right )}}{941664 \sqrt {2} + 1331714} + \frac {665857 \sqrt {2} \log {\left (\tanh {\left (\frac {x}{2} \right )} - 1 + \sqrt {2} \right )}}{941664 \sqrt {2} + 1331714} + \frac {941664 \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 + \sqrt {2} \right )}}{941664 \sqrt {2} + 1331714} + \frac {665857 \sqrt {2} \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 + \sqrt {2} \right )}}{941664 \sqrt {2} + 1331714} - \frac {665857 \sqrt {2} \log {\left (\tanh {\left (\frac {x}{2} \right )} - \sqrt {2} - 1 \right )}}{941664 \sqrt {2} + 1331714} - \frac {941664 \log {\left (\tanh {\left (\frac {x}{2} \right )} - \sqrt {2} - 1 \right )}}{941664 \sqrt {2} + 1331714} - \frac {665857 \sqrt {2} \log {\left (\tanh {\left (\frac {x}{2} \right )} - \sqrt {2} + 1 \right )}}{941664 \sqrt {2} + 1331714} - \frac {941664 \log {\left (\tanh {\left (\frac {x}{2} \right )} - \sqrt {2} + 1 \right )}}{941664 \sqrt {2} + 1331714} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+sinh(x)**2)/(1-sinh(x)**2),x)

[Out]

-1331714*x/(941664*sqrt(2) + 1331714) - 941664*sqrt(2)*x/(941664*sqrt(2) + 1331714) + 941664*log(tanh(x/2) - 1
+ sqrt(2))/(941664*sqrt(2) + 1331714) + 665857*sqrt(2)*log(tanh(x/2) - 1 + sqrt(2))/(941664*sqrt(2) + 1331714
) + 941664*log(tanh(x/2) + 1 + sqrt(2))/(941664*sqrt(2) + 1331714) + 665857*sqrt(2)*log(tanh(x/2) + 1 + sqrt(2
))/(941664*sqrt(2) + 1331714) - 665857*sqrt(2)*log(tanh(x/2) - sqrt(2) - 1)/(941664*sqrt(2) + 1331714) - 94166
4*log(tanh(x/2) - sqrt(2) - 1)/(941664*sqrt(2) + 1331714) - 665857*sqrt(2)*log(tanh(x/2) - sqrt(2) + 1)/(94166
4*sqrt(2) + 1331714) - 941664*log(tanh(x/2) - sqrt(2) + 1)/(941664*sqrt(2) + 1331714)

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