3.387 \(\int e^{2 \coth ^{-1}(a x)} (c-\frac {c}{a x})^5 \, dx\)

Optimal. Leaf size=61 \[ -\frac {c^5}{4 a^5 x^4}+\frac {c^5}{a^4 x^3}-\frac {c^5}{a^3 x^2}-\frac {2 c^5}{a^2 x}-\frac {3 c^5 \log (x)}{a}+c^5 x \]

[Out]

-1/4*c^5/a^5/x^4+c^5/a^4/x^3-c^5/a^3/x^2-2*c^5/a^2/x+c^5*x-3*c^5*ln(x)/a

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Rubi [A]  time = 0.14, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6167, 6131, 6129, 75} \[ -\frac {c^5}{a^3 x^2}+\frac {c^5}{a^4 x^3}-\frac {c^5}{4 a^5 x^4}-\frac {2 c^5}{a^2 x}-\frac {3 c^5 \log (x)}{a}+c^5 x \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcCoth[a*x])*(c - c/(a*x))^5,x]

[Out]

-c^5/(4*a^5*x^4) + c^5/(a^4*x^3) - c^5/(a^3*x^2) - (2*c^5)/(a^2*x) + c^5*x - (3*c^5*Log[x])/a

Rule 75

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rule 6129

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[c^p, Int[(u*(1 + (d*x)/c)
^p*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ
[p] || GtQ[c, 0])

Rule 6131

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> Dist[d^p, Int[(u*(1 + (c*x)/d)
^p*E^(n*ArcTanh[a*x]))/x^p, x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[c^2 - a^2*d^2, 0] && IntegerQ[p]

Rule 6167

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rubi steps

\begin {align*} \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^5 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^5 \, dx\\ &=\frac {c^5 \int \frac {e^{2 \tanh ^{-1}(a x)} (1-a x)^5}{x^5} \, dx}{a^5}\\ &=\frac {c^5 \int \frac {(1-a x)^4 (1+a x)}{x^5} \, dx}{a^5}\\ &=\frac {c^5 \int \left (a^5+\frac {1}{x^5}-\frac {3 a}{x^4}+\frac {2 a^2}{x^3}+\frac {2 a^3}{x^2}-\frac {3 a^4}{x}\right ) \, dx}{a^5}\\ &=-\frac {c^5}{4 a^5 x^4}+\frac {c^5}{a^4 x^3}-\frac {c^5}{a^3 x^2}-\frac {2 c^5}{a^2 x}+c^5 x-\frac {3 c^5 \log (x)}{a}\\ \end {align*}

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Mathematica [A]  time = 0.24, size = 63, normalized size = 1.03 \[ -\frac {c^5}{4 a^5 x^4}+\frac {c^5}{a^4 x^3}-\frac {c^5}{a^3 x^2}-\frac {2 c^5}{a^2 x}-\frac {3 c^5 \log (a x)}{a}+c^5 x \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(2*ArcCoth[a*x])*(c - c/(a*x))^5,x]

[Out]

-1/4*c^5/(a^5*x^4) + c^5/(a^4*x^3) - c^5/(a^3*x^2) - (2*c^5)/(a^2*x) + c^5*x - (3*c^5*Log[a*x])/a

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fricas [A]  time = 0.53, size = 67, normalized size = 1.10 \[ \frac {4 \, a^{5} c^{5} x^{5} - 12 \, a^{4} c^{5} x^{4} \log \relax (x) - 8 \, a^{3} c^{5} x^{3} - 4 \, a^{2} c^{5} x^{2} + 4 \, a c^{5} x - c^{5}}{4 \, a^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)*(a*x+1)*(c-c/a/x)^5,x, algorithm="fricas")

[Out]

1/4*(4*a^5*c^5*x^5 - 12*a^4*c^5*x^4*log(x) - 8*a^3*c^5*x^3 - 4*a^2*c^5*x^2 + 4*a*c^5*x - c^5)/(a^5*x^4)

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giac [A]  time = 0.12, size = 58, normalized size = 0.95 \[ c^{5} x - \frac {3 \, c^{5} \log \left ({\left | x \right |}\right )}{a} - \frac {8 \, a^{3} c^{5} x^{3} + 4 \, a^{2} c^{5} x^{2} - 4 \, a c^{5} x + c^{5}}{4 \, a^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)*(a*x+1)*(c-c/a/x)^5,x, algorithm="giac")

[Out]

c^5*x - 3*c^5*log(abs(x))/a - 1/4*(8*a^3*c^5*x^3 + 4*a^2*c^5*x^2 - 4*a*c^5*x + c^5)/(a^5*x^4)

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maple [A]  time = 0.04, size = 60, normalized size = 0.98 \[ -\frac {c^{5}}{4 a^{5} x^{4}}+\frac {c^{5}}{a^{4} x^{3}}-\frac {c^{5}}{x^{2} a^{3}}-\frac {2 c^{5}}{a^{2} x}+c^{5} x -\frac {3 c^{5} \ln \relax (x )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)/(a*x-1)*(c-c/a/x)^5,x)

[Out]

-1/4*c^5/a^5/x^4+c^5/a^4/x^3-c^5/x^2/a^3-2*c^5/a^2/x+c^5*x-3*c^5*ln(x)/a

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maxima [A]  time = 0.30, size = 57, normalized size = 0.93 \[ c^{5} x - \frac {3 \, c^{5} \log \relax (x)}{a} - \frac {8 \, a^{3} c^{5} x^{3} + 4 \, a^{2} c^{5} x^{2} - 4 \, a c^{5} x + c^{5}}{4 \, a^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)*(a*x+1)*(c-c/a/x)^5,x, algorithm="maxima")

[Out]

c^5*x - 3*c^5*log(x)/a - 1/4*(8*a^3*c^5*x^3 + 4*a^2*c^5*x^2 - 4*a*c^5*x + c^5)/(a^5*x^4)

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mupad [B]  time = 0.07, size = 51, normalized size = 0.84 \[ -\frac {c^5\,\left (4\,a^2\,x^2-4\,a\,x+8\,a^3\,x^3-4\,a^5\,x^5+12\,a^4\,x^4\,\ln \relax (x)+1\right )}{4\,a^5\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c - c/(a*x))^5*(a*x + 1))/(a*x - 1),x)

[Out]

-(c^5*(4*a^2*x^2 - 4*a*x + 8*a^3*x^3 - 4*a^5*x^5 + 12*a^4*x^4*log(x) + 1))/(4*a^5*x^4)

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sympy [A]  time = 0.27, size = 63, normalized size = 1.03 \[ \frac {a^{5} c^{5} x - 3 a^{4} c^{5} \log {\relax (x )} + \frac {- 8 a^{3} c^{5} x^{3} - 4 a^{2} c^{5} x^{2} + 4 a c^{5} x - c^{5}}{4 x^{4}}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)*(a*x+1)*(c-c/a/x)**5,x)

[Out]

(a**5*c**5*x - 3*a**4*c**5*log(x) + (-8*a**3*c**5*x**3 - 4*a**2*c**5*x**2 + 4*a*c**5*x - c**5)/(4*x**4))/a**5

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