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

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

[Out]

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

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Rubi [A]  time = 0.0450141, 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 = {} \[ \int \frac{\sqrt{d+e x^2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 26.4411, size = 0, normalized size = 0. \[ \int \frac{\sqrt{d+e x^2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.29, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{2}}\sqrt{e{x}^{2}+d}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (c^{3} x^{3} +{\left (c^{2} x^{2} - 1\right )} \sqrt{c x + 1} \sqrt{c x - 1} - c x\right )} \sqrt{e x^{2} + d}}{a b c^{3} x^{2} + \sqrt{c x + 1} \sqrt{c x - 1} a b c^{2} x - a b c +{\left (b^{2} c^{3} x^{2} + \sqrt{c x + 1} \sqrt{c x - 1} b^{2} c^{2} x - b^{2} c\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )} + \int \frac{{\left (2 \, c^{5} e x^{6} +{\left (c^{5} d - 4 \, c^{3} e\right )} x^{4} +{\left (2 \, c^{3} e x^{4} + c^{3} d x^{2} + c d\right )}{\left (c x + 1\right )}{\left (c x - 1\right )} - 2 \,{\left (c^{3} d - c e\right )} x^{2} +{\left (4 \, c^{4} e x^{5} + 2 \,{\left (c^{4} d - 2 \, c^{2} e\right )} x^{3} -{\left (c^{2} d - e\right )} x\right )} \sqrt{c x + 1} \sqrt{c x - 1} + c d\right )} \sqrt{e x^{2} + d}}{a b c^{5} e x^{6} +{\left (c^{5} d - 2 \, c^{3} e\right )} a b x^{4} -{\left (2 \, c^{3} d - c e\right )} a b x^{2} + a b c d +{\left (a b c^{3} e x^{4} + a b c^{3} d x^{2}\right )}{\left (c x + 1\right )}{\left (c x - 1\right )} + 2 \,{\left (a b c^{4} e x^{5} - a b c^{2} d x +{\left (c^{4} d - c^{2} e\right )} a b x^{3}\right )} \sqrt{c x + 1} \sqrt{c x - 1} +{\left (b^{2} c^{5} e x^{6} +{\left (c^{5} d - 2 \, c^{3} e\right )} b^{2} x^{4} -{\left (2 \, c^{3} d - c e\right )} b^{2} x^{2} + b^{2} c d +{\left (b^{2} c^{3} e x^{4} + b^{2} c^{3} d x^{2}\right )}{\left (c x + 1\right )}{\left (c x - 1\right )} + 2 \,{\left (b^{2} c^{4} e x^{5} - b^{2} c^{2} d x +{\left (c^{4} d - c^{2} e\right )} b^{2} x^{3}\right )} \sqrt{c x + 1} \sqrt{c x - 1}\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c^3*x^3 + (c^2*x^2 - 1)*sqrt(c*x + 1)*sqrt(c*x - 1) - c*x)*sqrt(e*x^2 + d)/(a*b*c^3*x^2 + sqrt(c*x + 1)*sqrt
(c*x - 1)*a*b*c^2*x - a*b*c + (b^2*c^3*x^2 + sqrt(c*x + 1)*sqrt(c*x - 1)*b^2*c^2*x - b^2*c)*log(c*x + sqrt(c*x
+ 1)*sqrt(c*x - 1))) + integrate((2*c^5*e*x^6 + (c^5*d - 4*c^3*e)*x^4 + (2*c^3*e*x^4 + c^3*d*x^2 + c*d)*(c*x
+ 1)*(c*x - 1) - 2*(c^3*d - c*e)*x^2 + (4*c^4*e*x^5 + 2*(c^4*d - 2*c^2*e)*x^3 - (c^2*d - e)*x)*sqrt(c*x + 1)*s
qrt(c*x - 1) + c*d)*sqrt(e*x^2 + d)/(a*b*c^5*e*x^6 + (c^5*d - 2*c^3*e)*a*b*x^4 - (2*c^3*d - c*e)*a*b*x^2 + a*b
*c*d + (a*b*c^3*e*x^4 + a*b*c^3*d*x^2)*(c*x + 1)*(c*x - 1) + 2*(a*b*c^4*e*x^5 - a*b*c^2*d*x + (c^4*d - c^2*e)*
a*b*x^3)*sqrt(c*x + 1)*sqrt(c*x - 1) + (b^2*c^5*e*x^6 + (c^5*d - 2*c^3*e)*b^2*x^4 - (2*c^3*d - c*e)*b^2*x^2 +
b^2*c*d + (b^2*c^3*e*x^4 + b^2*c^3*d*x^2)*(c*x + 1)*(c*x - 1) + 2*(b^2*c^4*e*x^5 - b^2*c^2*d*x + (c^4*d - c^2*
e)*b^2*x^3)*sqrt(c*x + 1)*sqrt(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))), 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}}{b^{2} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname{arcosh}\left (c x\right ) + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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