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

Optimal. Leaf size=15 \[ \frac{\tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )}{\sqrt{2}} \]

[Out]

ArcTanh[Sqrt[2]*Tanh[x]]/Sqrt[2]

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Rubi [A]  time = 0.0127272, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3181, 206} \[ \frac{\tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - Sinh[x]^2)^(-1),x]

[Out]

ArcTanh[Sqrt[2]*Tanh[x]]/Sqrt[2]

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]

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])

Rubi steps

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

Mathematica [F]  time = 0.0215643, size = 0, normalized size = 0. \[ \int \frac{1}{1-\sinh ^2(x)} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(1 - Sinh[x]^2)^(-1),x]

[Out]

Integrate[(1 - Sinh[x]^2)^(-1), x]

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Maple [B]  time = 0.016, size = 40, normalized size = 2.7 \begin{align*}{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\frac{\sqrt{2}}{4} \left ( 2\,\tanh \left ( x/2 \right ) -2 \right ) } \right ) }+{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\frac{\sqrt{2}}{4} \left ( 2\,\tanh \left ( x/2 \right ) +2 \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.53046, size = 82, normalized size = 5.47 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{\left (-x\right )} + 1}{\sqrt{2} + e^{\left (-x\right )} - 1}\right ) - \frac{1}{4} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{\left (-x\right )} - 1}{\sqrt{2} + e^{\left (-x\right )} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(2)*log(-(sqrt(2) - e^(-x) + 1)/(sqrt(2) + e^(-x) - 1)) - 1/4*sqrt(2)*log(-(sqrt(2) - e^(-x) - 1)/(sqr
t(2) + e^(-x) + 1))

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Fricas [B]  time = 1.84293, size = 215, normalized size = 14.33 \begin{align*} \frac{1}{4} \, \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-sinh(x)^2),x, algorithm="fricas")

[Out]

1/4*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))

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Sympy [B]  time = 3.56741, size = 209, normalized size = 13.93 \begin{align*} \frac{12 \sqrt{2} \log{\left (\tanh{\left (\frac{x}{2} \right )} - 1 + \sqrt{2} \right )}}{48 + 34 \sqrt{2}} + \frac{17 \log{\left (\tanh{\left (\frac{x}{2} \right )} - 1 + \sqrt{2} \right )}}{48 + 34 \sqrt{2}} + \frac{12 \sqrt{2} \log{\left (\tanh{\left (\frac{x}{2} \right )} + 1 + \sqrt{2} \right )}}{48 + 34 \sqrt{2}} + \frac{17 \log{\left (\tanh{\left (\frac{x}{2} \right )} + 1 + \sqrt{2} \right )}}{48 + 34 \sqrt{2}} - \frac{17 \log{\left (\tanh{\left (\frac{x}{2} \right )} - \sqrt{2} - 1 \right )}}{48 + 34 \sqrt{2}} - \frac{12 \sqrt{2} \log{\left (\tanh{\left (\frac{x}{2} \right )} - \sqrt{2} - 1 \right )}}{48 + 34 \sqrt{2}} - \frac{17 \log{\left (\tanh{\left (\frac{x}{2} \right )} - \sqrt{2} + 1 \right )}}{48 + 34 \sqrt{2}} - \frac{12 \sqrt{2} \log{\left (\tanh{\left (\frac{x}{2} \right )} - \sqrt{2} + 1 \right )}}{48 + 34 \sqrt{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

12*sqrt(2)*log(tanh(x/2) - 1 + sqrt(2))/(48 + 34*sqrt(2)) + 17*log(tanh(x/2) - 1 + sqrt(2))/(48 + 34*sqrt(2))
+ 12*sqrt(2)*log(tanh(x/2) + 1 + sqrt(2))/(48 + 34*sqrt(2)) + 17*log(tanh(x/2) + 1 + sqrt(2))/(48 + 34*sqrt(2)
) - 17*log(tanh(x/2) - sqrt(2) - 1)/(48 + 34*sqrt(2)) - 12*sqrt(2)*log(tanh(x/2) - sqrt(2) - 1)/(48 + 34*sqrt(
2)) - 17*log(tanh(x/2) - sqrt(2) + 1)/(48 + 34*sqrt(2)) - 12*sqrt(2)*log(tanh(x/2) - sqrt(2) + 1)/(48 + 34*sqr
t(2))

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Giac [B]  time = 1.26601, size = 50, normalized size = 3.33 \begin{align*} -\frac{1}{4} \, \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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