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]

-4*sech(x)/csch(x)^(3/2)+2*x/csch(x)^(1/2)

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Rubi [A]  time = 0.17, antiderivative size = 20, normalized size of antiderivative = 1.00, 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 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]

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 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 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 6742

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

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*}

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Mathematica [A]  time = 1.14, 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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (x \cosh \relax (x) - 4 \, \operatorname {sech}\relax (x) \tanh \relax (x)\right )} \sqrt {\operatorname {csch}\relax (x)}\,{d x} \]

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)

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maple [F]  time = 0.84, size = 0, normalized size = 0.00 \[ \int \sqrt {\mathrm {csch}\relax (x )}\, \left (x \cosh \relax (x )-4 \,\mathrm {sech}\relax (x ) \tanh \relax (x )\right )\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (x \cosh \relax (x) - 4 \, \operatorname {sech}\relax (x) \tanh \relax (x)\right )} \sqrt {\operatorname {csch}\relax (x)}\,{d x} \]

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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mupad [B]  time = 1.90, size = 51, normalized size = 2.55 \[ \frac {{\mathrm {e}}^{-x}\,\sqrt {-\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}}}\,\left ({\mathrm {e}}^{2\,x}-1\right )\,\left (x-2\,{\mathrm {e}}^{2\,x}+x\,{\mathrm {e}}^{2\,x}+2\right )}{{\mathrm {e}}^{2\,x}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(exp(-x)*(-1/(exp(-x)/2 - exp(x)/2))^(1/2)*(exp(2*x) - 1)*(x - 2*exp(2*x) + x*exp(2*x) + 2))/(exp(2*x) + 1)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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