3.114 $$\int \cosh (a+b x) \text{csch}^{1+n}(a+b x) \, dx$$

Optimal. Leaf size=16 $-\frac{\text{csch}^n(a+b x)}{b n}$

[Out]

-(Csch[a + b*x]^n/(b*n))

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Rubi [A]  time = 0.0279535, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.118, Rules used = {2621, 30} $-\frac{\text{csch}^n(a+b x)}{b n}$

Antiderivative was successfully veriﬁed.

[In]

Int[Cosh[a + b*x]*Csch[a + b*x]^(1 + n),x]

[Out]

-(Csch[a + b*x]^n/(b*n))

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 (a+b x) \text{csch}^{1+n}(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int x^{-1+n} \, dx,x,\text{csch}(a+b x)\right )}{b}\\ &=-\frac{\text{csch}^n(a+b x)}{b n}\\ \end{align*}

Mathematica [A]  time = 0.0198964, size = 16, normalized size = 1. $-\frac{\text{csch}^n(a+b x)}{b n}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Cosh[a + b*x]*Csch[a + b*x]^(1 + n),x]

[Out]

-(Csch[a + b*x]^n/(b*n))

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Maple [A]  time = 0.006, size = 17, normalized size = 1.1 \begin{align*} -{\frac{ \left ({\rm csch} \left (bx+a\right ) \right ) ^{n}}{bn}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(coth(b*x+a)*csch(b*x+a)^n,x)

[Out]

-csch(b*x+a)^n/b/n

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Maxima [B]  time = 1.90992, size = 72, normalized size = 4.5 \begin{align*} -\frac{2^{n} e^{\left (-{\left (b x + a\right )} n - n \log \left (e^{\left (-b x - a\right )} + 1\right ) - n \log \left (-e^{\left (-b x - a\right )} + 1\right )\right )}}{b n} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(b*x+a)*csch(b*x+a)^n,x, algorithm="maxima")

[Out]

-2^n*e^(-(b*x + a)*n - n*log(e^(-b*x - a) + 1) - n*log(-e^(-b*x - a) + 1))/(b*n)

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Fricas [B]  time = 1.89532, size = 338, normalized size = 21.12 \begin{align*} -\frac{\cosh \left (n \log \left (\frac{2 \,{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1}\right )\right ) + \sinh \left (n \log \left (\frac{2 \,{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1}\right )\right )}{b n} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(b*x+a)*csch(b*x+a)^n,x, algorithm="fricas")

[Out]

-(cosh(n*log(2*(cosh(b*x + a) + sinh(b*x + a))/(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a
)^2 - 1))) + sinh(n*log(2*(cosh(b*x + a) + sinh(b*x + a))/(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + s
inh(b*x + a)^2 - 1))))/(b*n)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(b*x+a)*csch(b*x+a)**n,x)

[Out]

Integral(coth(a + b*x)*csch(a + b*x)**n, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{csch}\left (b x + a\right )^{n} \coth \left (b x + a\right )\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(b*x+a)*csch(b*x+a)^n,x, algorithm="giac")

[Out]

integrate(csch(b*x + a)^n*coth(b*x + a), x)