3.2.15 \(\int x^m \cosh ^{-1}(a x)^4 \, dx\) [115]

Optimal. Leaf size=59 \[ \frac {x^{1+m} \cosh ^{-1}(a x)^4}{1+m}-\frac {4 a \text {Int}\left (\frac {x^{1+m} \cosh ^{-1}(a x)^3}{\sqrt {-1+a x} \sqrt {1+a x}},x\right )}{1+m} \]

[Out]

x^(1+m)*arccosh(a*x)^4/(1+m)-4*a*Unintegrable(x^(1+m)*arccosh(a*x)^3/(a*x-1)^(1/2)/(a*x+1)^(1/2),x)/(1+m)

________________________________________________________________________________________

Rubi [A]
time = 0.17, 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 = {} \begin {gather*} \int x^m \cosh ^{-1}(a x)^4 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[x^m*ArcCosh[a*x]^4,x]

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 1.77, size = 0, normalized size = 0.00 \begin {gather*} \int x^m \cosh ^{-1}(a x)^4 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[x^m*ArcCosh[a*x]^4,x]

[Out]

Integrate[x^m*ArcCosh[a*x]^4, x]

________________________________________________________________________________________

Maple [A]
time = 5.30, size = 0, normalized size = 0.00 \[\int x^{m} \mathrm {arccosh}\left (a x \right )^{4}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*arccosh(a*x)^4,x)

[Out]

int(x^m*arccosh(a*x)^4,x)

________________________________________________________________________________________

Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*arccosh(a*x)^4,x, algorithm="maxima")

[Out]

x*x^m*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^4/(m + 1) - integrate(4*(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))^3/(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)

________________________________________________________________________________________

Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*arccosh(a*x)^4,x, algorithm="fricas")

[Out]

integral(x^m*arccosh(a*x)^4, x)

________________________________________________________________________________________

Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{m} \operatorname {acosh}^{4}{\left (a x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*acosh(a*x)**4,x)

[Out]

Integral(x**m*acosh(a*x)**4, x)

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*arccosh(a*x)^4,x, algorithm="giac")

[Out]

integrate(x^m*arccosh(a*x)^4, x)

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x^m\,{\mathrm {acosh}\left (a\,x\right )}^4 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*acosh(a*x)^4,x)

[Out]

int(x^m*acosh(a*x)^4, x)

________________________________________________________________________________________