### 3.675 $$\int (\text{csch}(x)+\sinh (x))^2 \, dx$$

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

[Out]

(3*x)/2 - (3*Coth[x])/2 + (Cosh[x]^2*Coth[x])/2

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Rubi [A]  time = 0.0249942, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.429, Rules used = {290, 325, 206} $\frac{3 x}{2}-\frac{3 \coth (x)}{2}+\frac{1}{2} \cosh ^2(x) \coth (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[(Csch[x] + Sinh[x])^2,x]

[Out]

(3*x)/2 - (3*Coth[x])/2 + (Cosh[x]^2*Coth[x])/2

Rule 290

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

Rule 325

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

Mathematica [A]  time = 0.0043097, size = 18, normalized size = 0.82 $\frac{3 x}{2}+\frac{1}{4} \sinh (2 x)-\coth (x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(Csch[x] + Sinh[x])^2,x]

[Out]

(3*x)/2 - Coth[x] + Sinh[2*x]/4

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Maple [A]  time = 0.011, size = 15, normalized size = 0.7 \begin{align*} -{\rm coth} \left (x\right )+{\frac{3\,x}{2}}+{\frac{\cosh \left ( x \right ) \sinh \left ( x \right ) }{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((csch(x)+sinh(x))^2,x)

[Out]

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

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Maxima [A]  time = 1.1756, 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((csch(x)+sinh(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 = 1.76542, size = 109, normalized size = 4.95 \begin{align*} \frac{\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + 4 \,{\left (3 \, x + 2\right )} \sinh \left (x\right ) - 9 \, \cosh \left (x\right )}{8 \, \sinh \left (x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(cosh(x)^3 + 3*cosh(x)*sinh(x)^2 + 4*(3*x + 2)*sinh(x) - 9*cosh(x))/sinh(x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((csch(x)+sinh(x))**2,x)

[Out]

Integral((sinh(x) + csch(x))**2, x)

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Giac [B]  time = 1.14688, size = 53, normalized size = 2.41 \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((csch(x)+sinh(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)