### 3.681 $$\int (-\cosh (x)+\text{sech}(x))^2 \, dx$$

Optimal. Leaf size=22 $-\frac{3 x}{2}+\frac{3 \tanh (x)}{2}+\frac{1}{2} \sinh ^2(x) \tanh (x)$

[Out]

(-3*x)/2 + (3*Tanh[x])/2 + (Sinh[x]^2*Tanh[x])/2

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Rubi [A]  time = 0.0249294, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 9, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.333, Rules used = {288, 321, 206} $-\frac{3 x}{2}+\frac{3 \tanh (x)}{2}+\frac{1}{2} \sinh ^2(x) \tanh (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[(-Cosh[x] + Sech[x])^2,x]

[Out]

(-3*x)/2 + (3*Tanh[x])/2 + (Sinh[x]^2*Tanh[x])/2

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, 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 (-\cosh (x)+\text{sech}(x))^2 \, dx &=\operatorname{Subst}\left (\int \frac{x^4}{\left (1-x^2\right )^2} \, dx,x,\tanh (x)\right )\\ &=\frac{1}{2} \sinh ^2(x) \tanh (x)-\frac{3}{2} \operatorname{Subst}\left (\int \frac{x^2}{1-x^2} \, dx,x,\tanh (x)\right )\\ &=\frac{3 \tanh (x)}{2}+\frac{1}{2} \sinh ^2(x) \tanh (x)-\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (x)\right )\\ &=-\frac{3 x}{2}+\frac{3 \tanh (x)}{2}+\frac{1}{2} \sinh ^2(x) \tanh (x)\\ \end{align*}

Mathematica [A]  time = 0.0274479, size = 16, normalized size = 0.73 $-\frac{3 x}{2}+\frac{1}{4} \sinh (2 x)+\tanh (x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-Cosh[x] + Sech[x])^2,x]

[Out]

(-3*x)/2 + Sinh[2*x]/4 + Tanh[x]

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Maple [A]  time = 0.015, size = 13, normalized size = 0.6 \begin{align*}{\frac{\cosh \left ( x \right ) \sinh \left ( x \right ) }{2}}-{\frac{3\,x}{2}}+\tanh \left ( x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((-cosh(x)+sech(x))^2,x)

[Out]

1/2*cosh(x)*sinh(x)-3/2*x+tanh(x)

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Maxima [A]  time = 1.1983, size = 35, normalized size = 1.59 \begin{align*} -\frac{3}{2} \, x + \frac{2}{e^{\left (-2 \, x\right )} + 1} + \frac{1}{8} \, e^{\left (2 \, x\right )} - \frac{1}{8} \, e^{\left (-2 \, x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cosh(x)+sech(x))^2,x, algorithm="maxima")

[Out]

-3/2*x + 2/(e^(-2*x) + 1) + 1/8*e^(2*x) - 1/8*e^(-2*x)

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Fricas [A]  time = 1.9106, size = 101, normalized size = 4.59 \begin{align*} \frac{\sinh \left (x\right )^{3} - 4 \,{\left (3 \, x + 2\right )} \cosh \left (x\right ) + 3 \,{\left (\cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )}{8 \, \cosh \left (x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cosh(x)+sech(x))^2,x, algorithm="fricas")

[Out]

1/8*(sinh(x)^3 - 4*(3*x + 2)*cosh(x) + 3*(cosh(x)^2 + 3)*sinh(x))/cosh(x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (- \cosh{\left (x \right )} + \operatorname{sech}{\left (x \right )}\right )^{2}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cosh(x)+sech(x))**2,x)

[Out]

Integral((-cosh(x) + sech(x))**2, x)

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Giac [B]  time = 1.14546, size = 50, normalized size = 2.27 \begin{align*} -\frac{3}{2} \, x + \frac{3 \, e^{\left (4 \, x\right )} - 14 \, e^{\left (2 \, x\right )} - 1}{8 \,{\left (e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}\right )}} + \frac{1}{8} \, e^{\left (2 \, x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cosh(x)+sech(x))^2,x, algorithm="giac")

[Out]

-3/2*x + 1/8*(3*e^(4*x) - 14*e^(2*x) - 1)/(e^(4*x) + e^(2*x)) + 1/8*e^(2*x)