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

Optimal. Leaf size=11 \[ \tanh (x)-\frac{\tanh ^3(x)}{3} \]

[Out]

Tanh[x] - Tanh[x]^3/3

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Rubi [A]  time = 0.0190848, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3175, 3767} \[ \tanh (x)-\frac{\tanh ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Sinh[x]^2)^(-2),x]

[Out]

Tanh[x] - Tanh[x]^3/3

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]

Rubi steps

\begin{align*} \int \frac{1}{\left (1+\sinh ^2(x)\right )^2} \, dx &=\int \text{sech}^4(x) \, dx\\ &=i \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \tanh (x)\right )\\ &=\tanh (x)-\frac{\tanh ^3(x)}{3}\\ \end{align*}

Mathematica [A]  time = 0.0027172, size = 17, normalized size = 1.55 \[ \frac{2 \tanh (x)}{3}+\frac{1}{3} \tanh (x) \text{sech}^2(x) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Sinh[x]^2)^(-2),x]

[Out]

(2*Tanh[x])/3 + (Sech[x]^2*Tanh[x])/3

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Maple [B]  time = 0.015, size = 36, normalized size = 3.3 \begin{align*} -2\,{\frac{- \left ( \tanh \left ( x/2 \right ) \right ) ^{5}-2/3\, \left ( \tanh \left ( x/2 \right ) \right ) ^{3}-\tanh \left ( x/2 \right ) }{ \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(-tanh(1/2*x)^5-2/3*tanh(1/2*x)^3-tanh(1/2*x))/(tanh(1/2*x)^2+1)^3

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Maxima [B]  time = 1.03861, size = 66, normalized size = 6. \begin{align*} \frac{4 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1} + \frac{4}{3 \,{\left (3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*e^(-2*x)/(3*e^(-2*x) + 3*e^(-4*x) + e^(-6*x) + 1) + 4/3/(3*e^(-2*x) + 3*e^(-4*x) + e^(-6*x) + 1)

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Fricas [B]  time = 1.79279, size = 286, normalized size = 26. \begin{align*} -\frac{8 \,{\left (2 \, \cosh \left (x\right ) + \sinh \left (x\right )\right )}}{3 \,{\left (\cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5} +{\left (10 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{3} +{\left (10 \, \cosh \left (x\right )^{3} + 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} +{\left (5 \, \cosh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{2} + 2\right )} \sinh \left (x\right ) + 4 \, \cosh \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/3*(2*cosh(x) + sinh(x))/(cosh(x)^5 + 5*cosh(x)*sinh(x)^4 + sinh(x)^5 + (10*cosh(x)^2 + 3)*sinh(x)^3 + 3*cos
h(x)^3 + (10*cosh(x)^3 + 9*cosh(x))*sinh(x)^2 + (5*cosh(x)^4 + 9*cosh(x)^2 + 2)*sinh(x) + 4*cosh(x))

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Sympy [B]  time = 4.90601, size = 104, normalized size = 9.45 \begin{align*} \frac{6 \tanh ^{5}{\left (\frac{x}{2} \right )}}{3 \tanh ^{6}{\left (\frac{x}{2} \right )} + 9 \tanh ^{4}{\left (\frac{x}{2} \right )} + 9 \tanh ^{2}{\left (\frac{x}{2} \right )} + 3} + \frac{4 \tanh ^{3}{\left (\frac{x}{2} \right )}}{3 \tanh ^{6}{\left (\frac{x}{2} \right )} + 9 \tanh ^{4}{\left (\frac{x}{2} \right )} + 9 \tanh ^{2}{\left (\frac{x}{2} \right )} + 3} + \frac{6 \tanh{\left (\frac{x}{2} \right )}}{3 \tanh ^{6}{\left (\frac{x}{2} \right )} + 9 \tanh ^{4}{\left (\frac{x}{2} \right )} + 9 \tanh ^{2}{\left (\frac{x}{2} \right )} + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*tanh(x/2)**5/(3*tanh(x/2)**6 + 9*tanh(x/2)**4 + 9*tanh(x/2)**2 + 3) + 4*tanh(x/2)**3/(3*tanh(x/2)**6 + 9*tan
h(x/2)**4 + 9*tanh(x/2)**2 + 3) + 6*tanh(x/2)/(3*tanh(x/2)**6 + 9*tanh(x/2)**4 + 9*tanh(x/2)**2 + 3)

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Giac [A]  time = 1.27509, size = 24, normalized size = 2.18 \begin{align*} -\frac{4 \,{\left (3 \, e^{\left (2 \, x\right )} + 1\right )}}{3 \,{\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*(3*e^(2*x) + 1)/(e^(2*x) + 1)^3