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

Optimal. Leaf size=59 \[ \frac {x^{m+1} \cosh ^{-1}(a x)^3}{m+1}-\frac {3 a \text {Int}\left (\frac {x^{m+1} \cosh ^{-1}(a x)^2}{\sqrt {a x-1} \sqrt {a x+1}},x\right )}{m+1} \]

[Out]

x^(1+m)*arccosh(a*x)^3/(1+m)-3*a*Unintegrable(x^(1+m)*arccosh(a*x)^2/(a*x-1)^(1/2)/(a*x+1)^(1/2),x)/(1+m)

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Rubi [A]  time = 0.30, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 1.79, size = 0, normalized size = 0.00 \[ \int x^m \cosh ^{-1}(a x)^3 \, dx \]

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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fricas [A]  time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \operatorname {arcosh}\left (a x\right )^{3}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \operatorname {arcosh}\left (a x\right )^{3}\,{d x} \]

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)

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maple [A]  time = 0.94, size = 0, normalized size = 0.00 \[ \int x^{m} \mathrm {arccosh}\left (a x \right )^{3}\, dx \]

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 [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {x x^{m} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{3}}{m + 1} - \int \frac {3 \, {\left (\sqrt {a x + 1} \sqrt {a x - 1} a^{2} x^{2} x^{m} + {\left (a^{3} x^{3} - a x\right )} x^{m}\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{2}}{a^{3} {\left (m + 1\right )} x^{3} - a {\left (m + 1\right )} x + {\left (a^{2} {\left (m + 1\right )} x^{2} - m - 1\right )} \sqrt {a x + 1} \sqrt {a x - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*x^m*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^3/(m + 1) - integrate(3*(sqrt(a*x + 1)*sqrt(a*x - 1)*a^2*x^2*x^m
+ (a^3*x^3 - a*x)*x^m)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2/(a^3*(m + 1)*x^3 - a*(m + 1)*x + (a^2*(m + 1)*
x^2 - m - 1)*sqrt(a*x + 1)*sqrt(a*x - 1)), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int x^m\,{\mathrm {acosh}\left (a\,x\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^m*acosh(a*x)^3, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \operatorname {acosh}^{3}{\left (a x \right )}\, dx \]

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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