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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.06, 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 = {} \[ \int \frac {\sqrt {a+b \cosh ^{-1}(c x)}}{d+e x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 4.89, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a+b \cosh ^{-1}(c x)}}{d+e x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.65, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a +b \,\mathrm {arccosh}\left (c x \right )}}{e \,x^{2}+d}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________