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

Optimal. Leaf size=29 \[ \frac {\text {Int}\left (\frac {1}{(c+d x) \sqrt {a+b \cosh ^{-1}(c+d x)}},x\right )}{e} \]

[Out]

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

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Rubi [A]  time = 0.09, 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 = {} \[ \int \frac {1}{(c e+d e x) \sqrt {a+b \cosh ^{-1}(c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c e+d e x) \sqrt {a+b \cosh ^{-1}(c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d e x +c e \right ) \sqrt {a +b \,\mathrm {arccosh}\left (d x +c \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\left (c\,e+d\,e\,x\right )\,\sqrt {a+b\,\mathrm {acosh}\left (c+d\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{c \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + d x \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}}}\, dx}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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