3.153 \(\int \frac {1}{(c e+d e x) (a+b \cosh ^{-1}(c+d x))^4} \, dx\)

Optimal. Leaf size=27 \[ \frac {\text {Int}\left (\frac {1}{(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^4},x\right )}{e} \]

[Out]

Unintegrable(1/(d*x+c)/(a+b*arccosh(d*x+c))^4,x)/e

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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 = {} \[ \int \frac {1}{(c e+d e x) \left (a+b \cosh ^{-1}(c+d x)\right )^4} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((c*e + d*e*x)*(a + b*ArcCosh[c + d*x])^4),x]

[Out]

Defer[Subst][Defer[Int][1/(x*(a + b*ArcCosh[x])^4), 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 )^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{e x \left (a+b \cosh ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \left (a+b \cosh ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d e}\\ \end {align*}

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Mathematica [A]  time = 14.55, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c e+d e x) \left (a+b \cosh ^{-1}(c+d x)\right )^4} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((c*e + d*e*x)*(a + b*ArcCosh[c + d*x])^4),x]

[Out]

Integrate[1/((c*e + d*e*x)*(a + b*ArcCosh[c + d*x])^4), x]

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fricas [A]  time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a^{4} d e x + a^{4} c e + {\left (b^{4} d e x + b^{4} c e\right )} \operatorname {arcosh}\left (d x + c\right )^{4} + 4 \, {\left (a b^{3} d e x + a b^{3} c e\right )} \operatorname {arcosh}\left (d x + c\right )^{3} + 6 \, {\left (a^{2} b^{2} d e x + a^{2} b^{2} c e\right )} \operatorname {arcosh}\left (d x + c\right )^{2} + 4 \, {\left (a^{3} b d e x + a^{3} b c e\right )} \operatorname {arcosh}\left (d x + c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)/(a+b*arccosh(d*x+c))^4,x, algorithm="fricas")

[Out]

integral(1/(a^4*d*e*x + a^4*c*e + (b^4*d*e*x + b^4*c*e)*arccosh(d*x + c)^4 + 4*(a*b^3*d*e*x + a*b^3*c*e)*arcco
sh(d*x + c)^3 + 6*(a^2*b^2*d*e*x + a^2*b^2*c*e)*arccosh(d*x + c)^2 + 4*(a^3*b*d*e*x + a^3*b*c*e)*arccosh(d*x +
c)), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (d e x + c e\right )} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)/(a+b*arccosh(d*x+c))^4,x, algorithm="giac")

[Out]

integrate(1/((d*e*x + c*e)*(b*arccosh(d*x + c) + a)^4), x)

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maple [A]  time = 0.26, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d e x +c e \right ) \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*e*x+c*e)/(a+b*arccosh(d*x+c))^4,x)

[Out]

int(1/(d*e*x+c*e)/(a+b*arccosh(d*x+c))^4,x)

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)/(a+b*arccosh(d*x+c))^4,x, algorithm="maxima")

[Out]

Timed out

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{\left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((c*e + d*e*x)*(a + b*acosh(c + d*x))^4),x)

[Out]

int(1/((c*e + d*e*x)*(a + b*acosh(c + d*x))^4), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{a^{4} c + a^{4} d x + 4 a^{3} b c \operatorname {acosh}{\left (c + d x \right )} + 4 a^{3} b d x \operatorname {acosh}{\left (c + d x \right )} + 6 a^{2} b^{2} c \operatorname {acosh}^{2}{\left (c + d x \right )} + 6 a^{2} b^{2} d x \operatorname {acosh}^{2}{\left (c + d x \right )} + 4 a b^{3} c \operatorname {acosh}^{3}{\left (c + d x \right )} + 4 a b^{3} d x \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{4} c \operatorname {acosh}^{4}{\left (c + d x \right )} + b^{4} d x \operatorname {acosh}^{4}{\left (c + d x \right )}}\, dx}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)/(a+b*acosh(d*x+c))**4,x)

[Out]

Integral(1/(a**4*c + a**4*d*x + 4*a**3*b*c*acosh(c + d*x) + 4*a**3*b*d*x*acosh(c + d*x) + 6*a**2*b**2*c*acosh(
c + d*x)**2 + 6*a**2*b**2*d*x*acosh(c + d*x)**2 + 4*a*b**3*c*acosh(c + d*x)**3 + 4*a*b**3*d*x*acosh(c + d*x)**
3 + b**4*c*acosh(c + d*x)**4 + b**4*d*x*acosh(c + d*x)**4), x)/e

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