3.69 \(\int \cosh (x) \text {csch}^{\frac {7}{3}}(x) \, dx\)

Optimal. Leaf size=10 \[ -\frac {3}{4} \text {csch}^{\frac {4}{3}}(x) \]

[Out]

-3/4*csch(x)^(4/3)

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Rubi [A]  time = 0.03, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2621, 30} \[ -\frac {3}{4} \text {csch}^{\frac {4}{3}}(x) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]*Csch[x]^(7/3),x]

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2621

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(f*a^n)^(-1), Subst
[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && Integer
Q[(n + 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rubi steps

\begin {align*} \int \cosh (x) \text {csch}^{\frac {7}{3}}(x) \, dx &=-\operatorname {Subst}\left (\int \sqrt [3]{x} \, dx,x,\text {csch}(x)\right )\\ &=-\frac {3}{4} \text {csch}^{\frac {4}{3}}(x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 10, normalized size = 1.00 \[ -\frac {3}{4} \text {csch}^{\frac {4}{3}}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]*Csch[x]^(7/3),x]

[Out]

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

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fricas [B]  time = 0.46, size = 54, normalized size = 5.40 \[ -\frac {3 \cdot 2^{\frac {1}{3}} \left (\frac {\cosh \relax (x) + \sinh \relax (x)}{\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - 1}\right )^{\frac {1}{3}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )}}{2 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*csch(x)^(7/3),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.13, size = 17, normalized size = 1.70 \[ -\frac {3 \cdot 2^{\frac {1}{3}} e^{\left (\frac {4}{3} \, x\right )}}{2 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{\frac {4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*csch(x)^(7/3),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.08, size = 7, normalized size = 0.70 \[ -\frac {3 \mathrm {csch}\relax (x )^{\frac {4}{3}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*csch(x)^(7/3),x)

[Out]

-3/4*csch(x)^(4/3)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh \relax (x) \operatorname {csch}\relax (x)^{\frac {7}{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*csch(x)^(7/3),x, algorithm="maxima")

[Out]

integrate(cosh(x)*csch(x)^(7/3), x)

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mupad [B]  time = 1.46, size = 31, normalized size = 3.10 \[ -\frac {3\,{\mathrm {e}}^x\,{\left (-\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}}\right )}^{1/3}}{2\,\left ({\mathrm {e}}^{2\,x}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*(1/sinh(x))^(7/3),x)

[Out]

-(3*exp(x)*(-1/(exp(-x)/2 - exp(x)/2))^(1/3))/(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(cosh(x)*csch(x)**(7/3),x)

[Out]

Timed out

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