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

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {d+e x^2}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 11.42, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {d+e x^2}} \, dx \]

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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fricas [A]  time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname {arcosh}\left (c x\right ) + a^{2}}{\sqrt {e x^{2} + d}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{\sqrt {e x^{2} + d}}\,{d x} \]

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)

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maple [A]  time = 0.47, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}{\sqrt {e \,x^{2}+d}}\, dx \]

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 [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a^{2} \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {e}} + \int \frac {b^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2}}{\sqrt {e x^{2} + d}} + \frac {2 \, a b \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{\sqrt {e x^{2} + d}}\,{d x} \]

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]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{\sqrt {e\,x^2+d}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\sqrt {d + e x^{2}}}\, dx \]

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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