3.313 \(\int \frac {x^m}{(1-c^2 x^2)^{3/2} (a+b \cosh ^{-1}(c x))} \, dx\)

Optimal. Leaf size=31 \[ \text {Int}\left (\frac {x^m}{\left (1-c^2 x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.57, 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 = {} \[ \int \frac {x^m}{\left (1-c^2 x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.19, size = 0, normalized size = 0.00 \[ \int \frac {x^m}{\left (1-c^2 x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^m/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])),x]

[Out]

Integrate[x^m/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])), x]

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fricas [A]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{m}}{a c^{4} x^{4} - 2 \, a c^{2} x^{2} + {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \operatorname {arcosh}\left (c x\right ) + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

integral(sqrt(-c^2*x^2 + 1)*x^m/(a*c^4*x^4 - 2*a*c^2*x^2 + (b*c^4*x^4 - 2*b*c^2*x^2 + b)*arccosh(c*x) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x)),x, algorithm="giac")

[Out]

integrate(x^m/((-c^2*x^2 + 1)^(3/2)*(b*arccosh(c*x) + a)), x)

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maple [A]  time = 0.71, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x)),x)

[Out]

int(x^m/(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x)),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x)),x, algorithm="maxima")

[Out]

integrate(x^m/((-c^2*x^2 + 1)^(3/2)*(b*arccosh(c*x) + a)), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^m}{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (1-c^2\,x^2\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/((a + b*acosh(c*x))*(1 - c^2*x^2)^(3/2)),x)

[Out]

int(x^m/((a + b*acosh(c*x))*(1 - c^2*x^2)^(3/2)), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m/(-c**2*x**2+1)**(3/2)/(a+b*acosh(c*x)),x)

[Out]

Integral(x**m/((-(c*x - 1)*(c*x + 1))**(3/2)*(a + b*acosh(c*x))), x)

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