3.120 \(\int \frac {x^m}{\cosh ^{-1}(a x)^2} \, dx\)

Optimal. Leaf size=13 \[ \text {Int}\left (\frac {x^m}{\cosh ^{-1}(a x)^2},x\right ) \]

[Out]

Unintegrable(x^m/arccosh(a*x)^2,x)

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Rubi [A]  time = 0.02, 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 = {} \[ \int \frac {x^m}{\cosh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^m/ArcCosh[a*x]^2,x]

[Out]

Defer[Int][x^m/ArcCosh[a*x]^2, x]

Rubi steps

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

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Mathematica [A]  time = 1.34, size = 0, normalized size = 0.00 \[ \int \frac {x^m}{\cosh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^m/ArcCosh[a*x]^2,x]

[Out]

Integrate[x^m/ArcCosh[a*x]^2, x]

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fricas [A]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m}}{\operatorname {arcosh}\left (a x\right )^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/arccosh(a*x)^2,x, algorithm="fricas")

[Out]

integral(x^m/arccosh(a*x)^2, x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\operatorname {arcosh}\left (a x\right )^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/arccosh(a*x)^2,x, algorithm="giac")

[Out]

integrate(x^m/arccosh(a*x)^2, x)

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maple [A]  time = 0.87, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\mathrm {arccosh}\left (a x \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/arccosh(a*x)^2,x)

[Out]

int(x^m/arccosh(a*x)^2,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (a^{2} x^{2} - 1\right )} \sqrt {a x + 1} \sqrt {a x - 1} x^{m} + {\left (a^{3} x^{3} - a x\right )} x^{m}}{{\left (a^{3} x^{2} + \sqrt {a x + 1} \sqrt {a x - 1} a^{2} x - a\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )} + \int \frac {{\left (a^{3} {\left (m + 1\right )} x^{3} - a {\left (m - 1\right )} x\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} x^{m} + {\left (2 \, a^{4} {\left (m + 1\right )} x^{4} - a^{2} {\left (3 \, m + 1\right )} x^{2} + m\right )} \sqrt {a x + 1} \sqrt {a x - 1} x^{m} + {\left (a^{5} {\left (m + 1\right )} x^{5} - 2 \, a^{3} {\left (m + 1\right )} x^{3} + a {\left (m + 1\right )} x\right )} x^{m}}{{\left (a^{5} x^{5} + {\left (a x + 1\right )} {\left (a x - 1\right )} a^{3} x^{3} - 2 \, a^{3} x^{3} + 2 \, {\left (a^{4} x^{4} - a^{2} x^{2}\right )} \sqrt {a x + 1} \sqrt {a x - 1} + a x\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/arccosh(a*x)^2,x, algorithm="maxima")

[Out]

-((a^2*x^2 - 1)*sqrt(a*x + 1)*sqrt(a*x - 1)*x^m + (a^3*x^3 - a*x)*x^m)/((a^3*x^2 + sqrt(a*x + 1)*sqrt(a*x - 1)
*a^2*x - a)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))) + integrate(((a^3*(m + 1)*x^3 - a*(m - 1)*x)*(a*x + 1)*(a*
x - 1)*x^m + (2*a^4*(m + 1)*x^4 - a^2*(3*m + 1)*x^2 + m)*sqrt(a*x + 1)*sqrt(a*x - 1)*x^m + (a^5*(m + 1)*x^5 -
2*a^3*(m + 1)*x^3 + a*(m + 1)*x)*x^m)/((a^5*x^5 + (a*x + 1)*(a*x - 1)*a^3*x^3 - 2*a^3*x^3 + 2*(a^4*x^4 - a^2*x
^2)*sqrt(a*x + 1)*sqrt(a*x - 1) + a*x)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.08 \[ \int \frac {x^m}{{\mathrm {acosh}\left (a\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/acosh(a*x)^2,x)

[Out]

int(x^m/acosh(a*x)^2, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m/acosh(a*x)**2,x)

[Out]

Integral(x**m/acosh(a*x)**2, x)

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