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

Optimal. Leaf size=30 \[ \text{Unintegrable}\left (\frac{\left (1-c^2 x^2\right )^{3/2} x^m}{a+b \cosh ^{-1}(c x)},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.539941, 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 = {} \[ \int \frac{x^m \left (1-c^2 x^2\right )^{3/2}}{a+b \cosh ^{-1}(c x)} \, dx \]

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

Rubi steps

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

Mathematica [A]  time = 1.03556, size = 0, normalized size = 0. \[ \int \frac{x^m \left (1-c^2 x^2\right )^{3/2}}{a+b \cosh ^{-1}(c x)} \, dx \]

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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Maple [A]  time = 0.753, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{a+b{\rm arccosh} \left (cx\right )} \left ( -{c}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{m}}{b \operatorname{arcosh}\left (c x\right ) + a}\,{d x} \end{align*}

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (c^{2} x^{2} - 1\right )} \sqrt{-c^{2} x^{2} + 1} x^{m}}{b \operatorname{arcosh}\left (c x\right ) + a}, x\right ) \end{align*}

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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

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