3.293 \(\int x^{-1+n} \cosh ^{-1}(a+b x^n) \, dx\)

Optimal. Leaf size=55 \[ \frac{\left (a+b x^n\right ) \cosh ^{-1}\left (a+b x^n\right )}{b n}-\frac{\sqrt{a+b x^n-1} \sqrt{a+b x^n+1}}{b n} \]

[Out]

-((Sqrt[-1 + a + b*x^n]*Sqrt[1 + a + b*x^n])/(b*n)) + ((a + b*x^n)*ArcCosh[a + b*x^n])/(b*n)

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Rubi [A]  time = 0.0544168, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {6715, 5864, 5654, 74} \[ \frac{\left (a+b x^n\right ) \cosh ^{-1}\left (a+b x^n\right )}{b n}-\frac{\sqrt{a+b x^n-1} \sqrt{a+b x^n+1}}{b n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + n)*ArcCosh[a + b*x^n],x]

[Out]

-((Sqrt[-1 + a + b*x^n]*Sqrt[1 + a + b*x^n])/(b*n)) + ((a + b*x^n)*ArcCosh[a + b*x^n])/(b*n)

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 5864

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcCosh[x])^n, x
], x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]

Rule 5654

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rubi steps

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

Mathematica [A]  time = 0.0460324, size = 50, normalized size = 0.91 \[ \frac{\left (a+b x^n\right ) \cosh ^{-1}\left (a+b x^n\right )-\sqrt{a+b x^n-1} \sqrt{a+b x^n+1}}{b n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + n)*ArcCosh[a + b*x^n],x]

[Out]

(-(Sqrt[-1 + a + b*x^n]*Sqrt[1 + a + b*x^n]) + (a + b*x^n)*ArcCosh[a + b*x^n])/(b*n)

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \begin{align*} \int{x}^{n-1}{\rm arccosh} \left (a+b{x}^{n}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(n-1)*arccosh(a+b*x^n),x)

[Out]

int(x^(n-1)*arccosh(a+b*x^n),x)

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Maxima [A]  time = 0.971817, size = 53, normalized size = 0.96 \begin{align*} \frac{{\left (b x^{n} + a\right )} \operatorname{arcosh}\left (b x^{n} + a\right ) - \sqrt{{\left (b x^{n} + a\right )}^{2} - 1}}{b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*arccosh(a+b*x^n),x, algorithm="maxima")

[Out]

((b*x^n + a)*arccosh(b*x^n + a) - sqrt((b*x^n + a)^2 - 1))/(b*n)

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Fricas [B]  time = 2.19117, size = 443, normalized size = 8.05 \begin{align*} \frac{{\left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right )} \log \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a + \sqrt{\frac{2 \, a b +{\left (a^{2} + b^{2} - 1\right )} \cosh \left (n \log \left (x\right )\right ) -{\left (a^{2} - b^{2} - 1\right )} \sinh \left (n \log \left (x\right )\right )}{\cosh \left (n \log \left (x\right )\right ) - \sinh \left (n \log \left (x\right )\right )}}\right ) - \sqrt{\frac{2 \, a b +{\left (a^{2} + b^{2} - 1\right )} \cosh \left (n \log \left (x\right )\right ) -{\left (a^{2} - b^{2} - 1\right )} \sinh \left (n \log \left (x\right )\right )}{\cosh \left (n \log \left (x\right )\right ) - \sinh \left (n \log \left (x\right )\right )}}}{b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*arccosh(a+b*x^n),x, algorithm="fricas")

[Out]

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

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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(x**(-1+n)*acosh(a+b*x**n),x)

[Out]

Timed out

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Giac [B]  time = 1.18431, size = 154, normalized size = 2.8 \begin{align*} -\frac{b{\left (\frac{a \log \left ({\left | -a b -{\left (x^{n}{\left | b \right |} - \sqrt{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} - 1}\right )}{\left | b \right |} \right |}\right )}{b{\left | b \right |}} + \frac{\sqrt{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} - 1}}{b^{2}}\right )} - x^{n} \log \left (b x^{n} + a + \sqrt{{\left (b x^{n} + a\right )}^{2} - 1}\right )}{n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*arccosh(a+b*x^n),x, algorithm="giac")

[Out]

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