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

Optimal. Leaf size=12 \[ \text{Unintegrable}\left (\frac{x^m}{\cosh ^{-1}(a x)^3},x\right ) \]

[Out]

Unintegrable[x^m/ArcCosh[a*x]^3, x]

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Rubi [A]  time = 0.0196945, 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 = {} \[ \int \frac{x^m}{\cosh ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Defer[Int][x^m/ArcCosh[a*x]^3, x]

Rubi steps

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

Mathematica [A]  time = 1.32557, size = 0, normalized size = 0. \[ \int \frac{x^m}{\cosh ^{-1}(a x)^3} \, dx \]

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 = 0.457, size = 0, normalized size = 0. \begin{align*} \int{\frac{{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 [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((a^5*x^5 - a^3*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2)*x^m + (3*a^6*x^6 - 5*a^4*x^4 + 2*a^2*x^2)*(a*x + 1)*
(a*x - 1)*x^m + (3*a^7*x^7 - 7*a^5*x^5 + 5*a^3*x^3 - a*x)*sqrt(a*x + 1)*sqrt(a*x - 1)*x^m + (a^8*x^8 - 3*a^6*x
^6 + 3*a^4*x^4 - a^2*x^2)*x^m + ((a^5*(m + 1)*x^5 - 2*a^3*m*x^3 + a*(m - 1)*x)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2)
*x^m + (3*a^6*(m + 1)*x^6 - a^4*(7*m + 3)*x^4 + 5*a^2*m*x^2 - m)*(a*x + 1)*(a*x - 1)*x^m + (3*a^7*(m + 1)*x^7
- 2*a^5*(4*m + 3)*x^5 + a^3*(7*m + 4)*x^3 - a*(2*m + 1)*x)*sqrt(a*x + 1)*sqrt(a*x - 1)*x^m + (a^8*(m + 1)*x^8
- 3*a^6*(m + 1)*x^6 + 3*a^4*(m + 1)*x^4 - a^2*(m + 1)*x^2)*x^m)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1)))/((a^8*
x^7 + (a*x + 1)^(3/2)*(a*x - 1)^(3/2)*a^5*x^4 - 3*a^6*x^5 + 3*a^4*x^3 + 3*(a^6*x^5 - a^4*x^3)*(a*x + 1)*(a*x -
1) - a^2*x + 3*(a^7*x^6 - 2*a^5*x^4 + a^3*x^2)*sqrt(a*x + 1)*sqrt(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x
- 1))^2) + integrate(1/2*(((m^2 + 2*m + 1)*a^6*x^6 - 2*(m^2 - m)*a^4*x^4 + (m^2 - 4*m + 3)*a^2*x^2)*(a*x + 1)^
2*(a*x - 1)^2*x^m + (4*(m^2 + 2*m + 1)*a^7*x^7 - 2*(5*m^2 + m + 2)*a^5*x^5 + (8*m^2 - 11*m + 3)*a^3*x^3 - (2*m
^2 - 5*m)*a*x)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2)*x^m + (6*(m^2 + 2*m + 1)*a^8*x^8 - 6*(3*m^2 + 3*m + 2)*a^6*x^6
+ (19*m^2 + 2*m + 3)*a^4*x^4 - (8*m^2 - 5*m - 3)*a^2*x^2 + m^2 - m)*(a*x + 1)*(a*x - 1)*x^m + (4*(m^2 + 2*m +
1)*a^9*x^9 - 2*(7*m^2 + 11*m + 6)*a^7*x^7 + 3*(6*m^2 + 7*m + 3)*a^5*x^5 - (10*m^2 + 8*m + 1)*a^3*x^3 + (2*m^2
+ m)*a*x)*sqrt(a*x + 1)*sqrt(a*x - 1)*x^m + ((m^2 + 2*m + 1)*a^10*x^10 - 4*(m^2 + 2*m + 1)*a^8*x^8 + 6*(m^2 +
2*m + 1)*a^6*x^6 - 4*(m^2 + 2*m + 1)*a^4*x^4 + (m^2 + 2*m + 1)*a^2*x^2)*x^m)/((a^10*x^10 + (a*x + 1)^2*(a*x -
1)^2*a^6*x^6 - 4*a^8*x^8 + 6*a^6*x^6 - 4*a^4*x^4 + 4*(a^7*x^7 - a^5*x^5)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + a^2
*x^2 + 6*(a^8*x^8 - 2*a^6*x^6 + a^4*x^4)*(a*x + 1)*(a*x - 1) + 4*(a^9*x^9 - 3*a^7*x^7 + 3*a^5*x^5 - a^3*x^3)*s
qrt(a*x + 1)*sqrt(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{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 \frac{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 \frac{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)