3.116 \(\int x^m \cosh ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=58 \[ \frac{x^{m+1} \cosh ^{-1}(a x)^3}{m+1}-\frac{3 a \text{Unintegrable}\left (\frac{x^{m+1} \cosh ^{-1}(a x)^2}{\sqrt{a x-1} \sqrt{a x+1}},x\right )}{m+1} \]

[Out]

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

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Rubi [A]  time = 0.298019, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int x^m \cosh ^{-1}(a x)^3 \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^m*ArcCosh[a*x]^3,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.73522, size = 0, normalized size = 0. \[ \int x^m \cosh ^{-1}(a x)^3 \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^m*ArcCosh[a*x]^3,x]

[Out]

Integrate[x^m*ArcCosh[a*x]^3, x]

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Maple [A]  time = 0.569, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*arccosh(a*x)^3,x)

[Out]

int(x^m*arccosh(a*x)^3,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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{m} \operatorname{arcosh}\left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*arccosh(a*x)^3,x, algorithm="fricas")

[Out]

integral(x^m*arccosh(a*x)^3, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \operatorname{acosh}^{3}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*acosh(a*x)**3,x)

[Out]

Integral(x**m*acosh(a*x)**3, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \operatorname{arcosh}\left (a x\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*arccosh(a*x)^3,x, algorithm="giac")

[Out]

integrate(x^m*arccosh(a*x)^3, x)