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

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Rubi [A]  time = 0.0259891, antiderivative size = 10, normalized size of antiderivative = 1., 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 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])

Rule 30

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

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*}

Mathematica [A]  time = 0.0077568, size = 10, normalized size = 1. $-\frac{3}{4} \text{csch}^{\frac{4}{3}}(x)$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 7, normalized size = 0.7 \begin{align*} -{\frac{3}{4} \left ({\rm csch} \left (x\right ) \right ) ^{{\frac{4}{3}}}} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \cosh \left (x\right ) \operatorname{csch}\left (x\right )^{\frac{7}{3}}\,{d x} \end{align*}

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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Fricas [B]  time = 1.82942, size = 213, normalized size = 21.3 \begin{align*} -\frac{3 \cdot 2^{\frac{1}{3}} \left (\frac{\cosh \left (x\right ) + \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1}\right )^{\frac{1}{3}}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}}{2 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )}} \end{align*}

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh \left (x\right ) \operatorname{csch}\left (x\right )^{\frac{7}{3}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cosh(x)*csch(x)^(7/3), x)