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

Optimal. Leaf size=2 \[ \tanh (x) \]

[Out]

Tanh[x]

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Rubi [A]  time = 0.0163758, antiderivative size = 2, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {3175, 3767, 8} \[ \tanh (x) \]

Antiderivative was successfully verified.

[In]

Int[(1 + Sinh[x]^2)^(-1),x]

[Out]

Tanh[x]

Rule 3175

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Dist[a^p, Int[ActivateTrig[u*cos[e + f*x
]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.0036714, size = 2, normalized size = 1. \[ \tanh (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Sinh[x]^2)^(-1),x]

[Out]

Tanh[x]

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Maple [B]  time = 0.016, size = 17, normalized size = 8.5 \begin{align*} 2\,{\frac{\tanh \left ( x/2 \right ) }{ \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.02647, size = 14, normalized size = 7. \begin{align*} \frac{2}{e^{\left (-2 \, x\right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(e^(-2*x) + 1)

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Fricas [B]  time = 1.75827, size = 70, normalized size = 35. \begin{align*} -\frac{2}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)

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Sympy [B]  time = 0.894588, size = 14, normalized size = 7. \begin{align*} \frac{2 \tanh{\left (\frac{x}{2} \right )}}{\tanh ^{2}{\left (\frac{x}{2} \right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*tanh(x/2)/(tanh(x/2)**2 + 1)

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Giac [B]  time = 1.29848, size = 14, normalized size = 7. \begin{align*} -\frac{2}{e^{\left (2 \, x\right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(e^(2*x) + 1)