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

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

[Out]

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

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Rubi [A]  time = 0.0479128, 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 = {} \[ \int \frac{1}{\sqrt{d+e x^2} \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.04994, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{d+e x^2} \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.256, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{a+b{\rm arccosh} \left (cx\right )}{\frac{1}{\sqrt{e{x}^{2}+d}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(e*x^2 + d)*(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{\sqrt{e x^{2} + d}}{a e x^{2} + a d +{\left (b e x^{2} + b d\right )} \operatorname{arcosh}\left (c x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(e*x^2 + d)/(a*e*x^2 + a*d + (b*e*x^2 + b*d)*arccosh(c*x)), x)

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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 ) \sqrt{d + e x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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