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

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{1}{\left (d+e x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )^{3/2}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0694119, 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 = {} \[ \int \frac{1}{\left (d+e x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Defer[Int][1/((d + e*x^2)*(a + b*ArcCosh[c*x])^(3/2)), x]

Rubi steps

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

Mathematica [A]  time = 0.164607, size = 0, normalized size = 0. \[ \int \frac{1}{\left (d+e x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.25, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{e{x}^{2}+d} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x**2+d)/(a+b*acosh(c*x))**(3/2),x)

[Out]

Integral(1/((a + b*acosh(c*x))**(3/2)*(d + e*x**2)), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x