3.2.85 \(\int \frac {1}{(c e+d e x) (a+b \cosh ^{-1}(c+d x))^{3/2}} \, dx\) [185]

Optimal. Leaf size=29 \[ \frac {\text {Int}\left (\frac {1}{(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}},x\right )}{e} \]

[Out]

Unintegrable(1/(d*x+c)/(a+b*arccosh(d*x+c))^(3/2),x)/e

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Rubi [A]
time = 0.07, 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 = {} \begin {gather*} \int \frac {1}{(c e+d e x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/((c*e + d*e*x)*(a + b*ArcCosh[c + d*x])^(3/2)),x]

[Out]

Defer[Subst][Defer[Int][1/(x*(a + b*ArcCosh[x])^(3/2)), x], x, c + d*x]/(d*e)

Rubi steps

\begin {align*} \int \frac {1}{(c e+d e x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{e x \left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x \left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{d e}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(c e+d e x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[1/((c*e + d*e*x)*(a + b*ArcCosh[c + d*x])^(3/2)),x]

[Out]

Integrate[1/((c*e + d*e*x)*(a + b*ArcCosh[c + d*x])^(3/2)), x]

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Maple [A]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (d e x +c e \right ) \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*e*x+c*e)/(a+b*arccosh(d*x+c))^(3/2),x)

[Out]

int(1/(d*e*x+c*e)/(a+b*arccosh(d*x+c))^(3/2),x)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)/(a+b*arccosh(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((d*x*e + c*e)*(b*arccosh(d*x + c) + a)^(3/2)), x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)/(a+b*arccosh(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)/(a+b*acosh(d*x+c))**(3/2),x)

[Out]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)/(a+b*arccosh(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate(1/((d*e*x + c*e)*(b*arccosh(d*x + c) + a)^(3/2)), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{\left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((c*e + d*e*x)*(a + b*acosh(c + d*x))^(3/2)),x)

[Out]

int(1/((c*e + d*e*x)*(a + b*acosh(c + d*x))^(3/2)), x)

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