3.1042 $$\int \sqrt{\text{csch}(x)} (x \cosh (x)-4 \text{sech}(x) \tanh (x)) \, dx$$

Optimal. Leaf size=20 $\frac{2 x}{\sqrt{\text{csch}(x)}}-\frac{4 \text{sech}(x)}{\text{csch}^{\frac{3}{2}}(x)}$

[Out]

(2*x)/Sqrt[Csch[x]] - (4*Sech[x])/Csch[x]^(3/2)

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Rubi [A]  time = 0.166912, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.278, Rules used = {6742, 5445, 3771, 2639, 2626} $\frac{2 x}{\sqrt{\text{csch}(x)}}-\frac{4 \text{sech}(x)}{\text{csch}^{\frac{3}{2}}(x)}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[Csch[x]]*(x*Cosh[x] - 4*Sech[x]*Tanh[x]),x]

[Out]

(2*x)/Sqrt[Csch[x]] - (4*Sech[x])/Csch[x]^(3/2)

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 5445

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]*Csch[(a_.) + (b_.)*(x_)^(n_.)]^(p_)*(x_)^(m_.), x_Symbol] :> -Simp[(x^(m -
n + 1)*Csch[a + b*x^n]^(p - 1))/(b*n*(p - 1)), x] + Dist[(m - n + 1)/(b*n*(p - 1)), Int[x^(m - n)*Csch[a + b*x
^n]^(p - 1), x], x] /; FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m - n, 0] && NeQ[p, 1]

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rule 2626

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*b*(a*Csc[
e + f*x])^(m - 1)*(b*Sec[e + f*x])^(n - 1))/(f*(n - 1)), x] + Dist[(b^2*(m + n - 2))/(n - 1), Int[(a*Csc[e + f
*x])^m*(b*Sec[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && IntegersQ[2*m, 2*n]

Rubi steps

\begin{align*} \int \sqrt{\text{csch}(x)} (x \cosh (x)-4 \text{sech}(x) \tanh (x)) \, dx &=\int \left (x \cosh (x) \sqrt{\text{csch}(x)}-\frac{4 \text{sech}^2(x)}{\sqrt{\text{csch}(x)}}\right ) \, dx\\ &=-\left (4 \int \frac{\text{sech}^2(x)}{\sqrt{\text{csch}(x)}} \, dx\right )+\int x \cosh (x) \sqrt{\text{csch}(x)} \, dx\\ &=\frac{2 x}{\sqrt{\text{csch}(x)}}-\frac{4 \text{sech}(x)}{\text{csch}^{\frac{3}{2}}(x)}\\ \end{align*}

Mathematica [A]  time = 1.10116, size = 17, normalized size = 0.85 $\frac{2 (x \text{csch}(x)-2 \text{sech}(x))}{\text{csch}^{\frac{3}{2}}(x)}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sqrt[Csch[x]]*(x*Cosh[x] - 4*Sech[x]*Tanh[x]),x]

[Out]

(2*(x*Csch[x] - 2*Sech[x]))/Csch[x]^(3/2)

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Maple [F]  time = 0.158, size = 0, normalized size = 0. \begin{align*} \int \sqrt{{\rm csch} \left (x\right )} \left ( x\cosh \left ( x \right ) -4\,{\rm sech} \left (x\right )\tanh \left ( x \right ) \right ) \, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)^(1/2)*(x*cosh(x)-4*sech(x)*tanh(x)),x)

[Out]

int(csch(x)^(1/2)*(x*cosh(x)-4*sech(x)*tanh(x)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (x \cosh \left (x\right ) - 4 \, \operatorname{sech}\left (x\right ) \tanh \left (x\right )\right )} \sqrt{\operatorname{csch}\left (x\right )}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^(1/2)*(x*cosh(x)-4*sech(x)*tanh(x)),x, algorithm="maxima")

[Out]

integrate((x*cosh(x) - 4*sech(x)*tanh(x))*sqrt(csch(x)), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^(1/2)*(x*cosh(x)-4*sech(x)*tanh(x)),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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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(csch(x)**(1/2)*(x*cosh(x)-4*sech(x)*tanh(x)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (x \cosh \left (x\right ) - 4 \, \operatorname{sech}\left (x\right ) \tanh \left (x\right )\right )} \sqrt{\operatorname{csch}\left (x\right )}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^(1/2)*(x*cosh(x)-4*sech(x)*tanh(x)),x, algorithm="giac")

[Out]

integrate((x*cosh(x) - 4*sech(x)*tanh(x))*sqrt(csch(x)), x)