3.283 \(\int \frac{(1-c^2 x^2)^{3/2}}{x^4 (a+b \cosh ^{-1}(c x))} \, dx\)

Optimal. Leaf size=30 \[ \text{Unintegrable}\left (\frac{\left (1-c^2 x^2\right )^{3/2}}{x^4 \left (a+b \cosh ^{-1}(c x)\right )},x\right ) \]

[Out]

Unintegrable[(1 - c^2*x^2)^(3/2)/(x^4*(a + b*ArcCosh[c*x])), x]

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Rubi [A]  time = 0.5394, 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 = {} \[ \int \frac{\left (1-c^2 x^2\right )^{3/2}}{x^4 \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(1 - c^2*x^2)^(3/2)/(x^4*(a + b*ArcCosh[c*x])),x]

[Out]

-((Sqrt[1 - c^2*x^2]*Defer[Int][((-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(x^4*(a + b*ArcCosh[c*x])), x])/(Sqrt[-1 +
c*x]*Sqrt[1 + c*x]))

Rubi steps

\begin{align*} \int \frac{\left (1-c^2 x^2\right )^{3/2}}{x^4 \left (a+b \cosh ^{-1}(c x)\right )} \, dx &=-\frac{\sqrt{1-c^2 x^2} \int \frac{(-1+c x)^{3/2} (1+c x)^{3/2}}{x^4 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 0.877244, size = 0, normalized size = 0. \[ \int \frac{\left (1-c^2 x^2\right )^{3/2}}{x^4 \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(1 - c^2*x^2)^(3/2)/(x^4*(a + b*ArcCosh[c*x])),x]

[Out]

Integrate[(1 - c^2*x^2)^(3/2)/(x^4*(a + b*ArcCosh[c*x])), x]

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Maple [A]  time = 0.382, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4} \left ( a+b{\rm arccosh} \left (cx\right ) \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((-c^2*x^2+1)^(3/2)/x^4/(a+b*arccosh(c*x)),x)

[Out]

int((-c^2*x^2+1)^(3/2)/x^4/(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}}}{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*x^2+1)^(3/2)/x^4/(a+b*arccosh(c*x)),x, algorithm="maxima")

[Out]

integrate((-c^2*x^2 + 1)^(3/2)/((b*arccosh(c*x) + a)*x^4), 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 )}^{\frac{3}{2}}}{b x^{4} \operatorname{arcosh}\left (c x\right ) + a x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*x^2+1)^(3/2)/x^4/(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

integral((-c^2*x^2 + 1)^(3/2)/(b*x^4*arccosh(c*x) + a*x^4), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{2}}}{x^{4} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*x**2+1)**(3/2)/x**4/(a+b*acosh(c*x)),x)

[Out]

Integral((-(c*x - 1)*(c*x + 1))**(3/2)/(x**4*(a + b*acosh(c*x))), x)

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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}}}{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*x^2+1)^(3/2)/x^4/(a+b*arccosh(c*x)),x, algorithm="giac")

[Out]

integrate((-c^2*x^2 + 1)^(3/2)/((b*arccosh(c*x) + a)*x^4), x)