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

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

[Out]

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

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Rubi [A]  time = 0.0431376, 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 = {} \[ \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{d+e x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 11.0387, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{d+e x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.289, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{2}{\frac{1}{\sqrt{e{x}^{2}+d}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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