3.239 \(\int \frac{(f x)^m \cosh ^{-1}(c x)^2}{\sqrt{1-c^2 x^2}} \, dx\)

Optimal. Leaf size=28 \[ \text{Unintegrable}\left (\frac{\cosh ^{-1}(c x)^2 (f x)^m}{\sqrt{1-c^2 x^2}},x\right ) \]

[Out]

Unintegrable[((f*x)^m*ArcCosh[c*x]^2)/Sqrt[1 - c^2*x^2], x]

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Rubi [A]  time = 0.369915, 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 = {} \[ \int \frac{(f x)^m \cosh ^{-1}(c x)^2}{\sqrt{1-c^2 x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[((f*x)^m*ArcCosh[c*x]^2)/Sqrt[1 - c^2*x^2],x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.738518, size = 0, normalized size = 0. \[ \int \frac{(f x)^m \cosh ^{-1}(c x)^2}{\sqrt{1-c^2 x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[((f*x)^m*ArcCosh[c*x]^2)/Sqrt[1 - c^2*x^2],x]

[Out]

Integrate[((f*x)^m*ArcCosh[c*x]^2)/Sqrt[1 - c^2*x^2], x]

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Maple [A]  time = 0.341, size = 0, normalized size = 0. \begin{align*} \int{ \left ( fx \right ) ^{m} \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}{\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^m*arccosh(c*x)^2/(-c^2*x^2+1)^(1/2),x)

[Out]

int((f*x)^m*arccosh(c*x)^2/(-c^2*x^2+1)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((f*x)^m*arccosh(c*x)^2/sqrt(-c^2*x^2 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (f x\right )^{m} \operatorname{acosh}^{2}{\left (c x \right )}}{\sqrt{- \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)**m*acosh(c*x)**2/(-c**2*x**2+1)**(1/2),x)

[Out]

Integral((f*x)**m*acosh(c*x)**2/sqrt(-(c*x - 1)*(c*x + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((f*x)^m*arccosh(c*x)^2/sqrt(-c^2*x^2 + 1), x)