3.118 \(\int x^m \cosh ^{-1}(a x) \, dx\)

Optimal. Leaf size=91 \[ \frac{x^{m+1} \cosh ^{-1}(a x)}{m+1}-\frac{a \sqrt{1-a^2 x^2} x^{m+2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},a^2 x^2\right )}{\left (m^2+3 m+2\right ) \sqrt{a x-1} \sqrt{a x+1}} \]

[Out]

(x^(1 + m)*ArcCosh[a*x])/(1 + m) - (a*x^(2 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2,
a^2*x^2])/((2 + 3*m + m^2)*Sqrt[-1 + a*x]*Sqrt[1 + a*x])

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Rubi [A]  time = 0.054826, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5662, 126, 365, 364} \[ \frac{x^{m+1} \cosh ^{-1}(a x)}{m+1}-\frac{a \sqrt{1-a^2 x^2} x^{m+2} \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};a^2 x^2\right )}{\left (m^2+3 m+2\right ) \sqrt{a x-1} \sqrt{a x+1}} \]

Antiderivative was successfully verified.

[In]

Int[x^m*ArcCosh[a*x],x]

[Out]

(x^(1 + m)*ArcCosh[a*x])/(1 + m) - (a*x^(2 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2,
a^2*x^2])/((2 + 3*m + m^2)*Sqrt[-1 + a*x]*Sqrt[1 + a*x])

Rule 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 126

Int[((f_.)*(x_))^(p_.)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Dist[((a + b*x)^Fra
cPart[m]*(c + d*x)^FracPart[m])/(a*c + b*d*x^2)^FracPart[m], Int[(a*c + b*d*x^2)^m*(f*x)^p, x], x] /; FreeQ[{a
, b, c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[m - n, 0]

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int x^m \cosh ^{-1}(a x) \, dx &=\frac{x^{1+m} \cosh ^{-1}(a x)}{1+m}-\frac{a \int \frac{x^{1+m}}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{1+m}\\ &=\frac{x^{1+m} \cosh ^{-1}(a x)}{1+m}-\frac{\left (a \sqrt{-1+a^2 x^2}\right ) \int \frac{x^{1+m}}{\sqrt{-1+a^2 x^2}} \, dx}{(1+m) \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{x^{1+m} \cosh ^{-1}(a x)}{1+m}-\frac{\left (a \sqrt{1-a^2 x^2}\right ) \int \frac{x^{1+m}}{\sqrt{1-a^2 x^2}} \, dx}{(1+m) \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{x^{1+m} \cosh ^{-1}(a x)}{1+m}-\frac{a x^{2+m} \sqrt{1-a^2 x^2} \, _2F_1\left (\frac{1}{2},\frac{2+m}{2};\frac{4+m}{2};a^2 x^2\right )}{\left (2+3 m+m^2\right ) \sqrt{-1+a x} \sqrt{1+a x}}\\ \end{align*}

Mathematica [A]  time = 0.0954662, size = 82, normalized size = 0.9 \[ \frac{x^{m+1} \left (\cosh ^{-1}(a x)-\frac{a x \sqrt{1-a^2 x^2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},a^2 x^2\right )}{(m+2) \sqrt{a x-1} \sqrt{a x+1}}\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*ArcCosh[a*x],x]

[Out]

(x^(1 + m)*(ArcCosh[a*x] - (a*x*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/((2 +
m)*Sqrt[-1 + a*x]*Sqrt[1 + a*x])))/(1 + m)

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Maple [F]  time = 0.676, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}{\rm arccosh} \left (ax\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*arccosh(a*x),x)

[Out]

int(x^m*arccosh(a*x),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*arccosh(a*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{m} \operatorname{arcosh}\left (a x\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*arccosh(a*x),x, algorithm="fricas")

[Out]

integral(x^m*arccosh(a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*acosh(a*x),x)

[Out]

Integral(x**m*acosh(a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*arccosh(a*x),x, algorithm="giac")

[Out]

integrate(x^m*arccosh(a*x), x)