### 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 + Sqrt[2]*ArcTanh[Sqrt[2]*Tanh[x]]

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Rubi [A]  time = 0.0415577, antiderivative size = 19, normalized size of antiderivative = 1., 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 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]

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+\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*}

Mathematica [A]  time = 0.031268, 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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Maple [B]  time = 0.02, size = 54, normalized size = 2.8 \begin{align*} -\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) +\sqrt{2}{\it Artanh} \left ({\frac{\sqrt{2}}{4} \left ( 2\,\tanh \left ( x/2 \right ) +2 \right ) } \right ) +\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) +\sqrt{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+sinh(x)^2)/(1-sinh(x)^2),x)

[Out]

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

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Maxima [B]  time = 1.61228, size = 86, normalized size = 4.53 \begin{align*} \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 \end{align*}

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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Fricas [B]  time = 2.4234, size = 220, normalized size = 11.58 \begin{align*} \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 \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.15389, size = 55, normalized size = 2.89 \begin{align*} -\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 \end{align*}

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