3.188 \(\int e^{4 \coth ^{-1}(a x)} (c-a c x)^5 \, dx\)

Optimal. Leaf size=53 \[ -\frac{c^5 (1-a x)^6}{6 a}+\frac{4 c^5 (1-a x)^5}{5 a}-\frac{c^5 (1-a x)^4}{a} \]

[Out]

-((c^5*(1 - a*x)^4)/a) + (4*c^5*(1 - a*x)^5)/(5*a) - (c^5*(1 - a*x)^6)/(6*a)

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Rubi [A]  time = 0.0680539, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6167, 6129, 43} \[ -\frac{c^5 (1-a x)^6}{6 a}+\frac{4 c^5 (1-a x)^5}{5 a}-\frac{c^5 (1-a x)^4}{a} \]

Antiderivative was successfully verified.

[In]

Int[E^(4*ArcCoth[a*x])*(c - a*c*x)^5,x]

[Out]

-((c^5*(1 - a*x)^4)/a) + (4*c^5*(1 - a*x)^5)/(5*a) - (c^5*(1 - a*x)^6)/(6*a)

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]

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int e^{4 \coth ^{-1}(a x)} (c-a c x)^5 \, dx &=\int e^{4 \tanh ^{-1}(a x)} (c-a c x)^5 \, dx\\ &=c^5 \int (1-a x)^3 (1+a x)^2 \, dx\\ &=c^5 \int \left (4 (1-a x)^3-4 (1-a x)^4+(1-a x)^5\right ) \, dx\\ &=-\frac{c^5 (1-a x)^4}{a}+\frac{4 c^5 (1-a x)^5}{5 a}-\frac{c^5 (1-a x)^6}{6 a}\\ \end{align*}

Mathematica [A]  time = 0.0208371, size = 31, normalized size = 0.58 \[ -\frac{c^5 (a x-1)^4 \left (5 a^2 x^2+14 a x+11\right )}{30 a} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(4*ArcCoth[a*x])*(c - a*c*x)^5,x]

[Out]

-(c^5*(-1 + a*x)^4*(11 + 14*a*x + 5*a^2*x^2))/(30*a)

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Maple [A]  time = 0.038, size = 45, normalized size = 0.9 \begin{align*}{c}^{5} \left ( -{\frac{{x}^{6}{a}^{5}}{6}}+{\frac{{x}^{5}{a}^{4}}{5}}+{\frac{{x}^{4}{a}^{3}}{2}}-{\frac{2\,{x}^{3}{a}^{2}}{3}}-{\frac{a{x}^{2}}{2}}+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x-1)^2*(a*x+1)^2*(-a*c*x+c)^5,x)

[Out]

c^5*(-1/6*x^6*a^5+1/5*x^5*a^4+1/2*x^4*a^3-2/3*x^3*a^2-1/2*a*x^2+x)

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Maxima [A]  time = 0.992122, size = 80, normalized size = 1.51 \begin{align*} -\frac{1}{6} \, a^{5} c^{5} x^{6} + \frac{1}{5} \, a^{4} c^{5} x^{5} + \frac{1}{2} \, a^{3} c^{5} x^{4} - \frac{2}{3} \, a^{2} c^{5} x^{3} - \frac{1}{2} \, a c^{5} x^{2} + c^{5} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)^2*(a*x+1)^2*(-a*c*x+c)^5,x, algorithm="maxima")

[Out]

-1/6*a^5*c^5*x^6 + 1/5*a^4*c^5*x^5 + 1/2*a^3*c^5*x^4 - 2/3*a^2*c^5*x^3 - 1/2*a*c^5*x^2 + c^5*x

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Fricas [A]  time = 1.47756, size = 130, normalized size = 2.45 \begin{align*} -\frac{1}{6} \, a^{5} c^{5} x^{6} + \frac{1}{5} \, a^{4} c^{5} x^{5} + \frac{1}{2} \, a^{3} c^{5} x^{4} - \frac{2}{3} \, a^{2} c^{5} x^{3} - \frac{1}{2} \, a c^{5} x^{2} + c^{5} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)^2*(a*x+1)^2*(-a*c*x+c)^5,x, algorithm="fricas")

[Out]

-1/6*a^5*c^5*x^6 + 1/5*a^4*c^5*x^5 + 1/2*a^3*c^5*x^4 - 2/3*a^2*c^5*x^3 - 1/2*a*c^5*x^2 + c^5*x

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Sympy [A]  time = 0.19778, size = 63, normalized size = 1.19 \begin{align*} - \frac{a^{5} c^{5} x^{6}}{6} + \frac{a^{4} c^{5} x^{5}}{5} + \frac{a^{3} c^{5} x^{4}}{2} - \frac{2 a^{2} c^{5} x^{3}}{3} - \frac{a c^{5} x^{2}}{2} + c^{5} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)**2*(a*x+1)**2*(-a*c*x+c)**5,x)

[Out]

-a**5*c**5*x**6/6 + a**4*c**5*x**5/5 + a**3*c**5*x**4/2 - 2*a**2*c**5*x**3/3 - a*c**5*x**2/2 + c**5*x

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Giac [A]  time = 1.15922, size = 57, normalized size = 1.08 \begin{align*} -\frac{{\left (5 \, c^{5} + \frac{24 \, c^{5}}{a x - 1} + \frac{30 \, c^{5}}{{\left (a x - 1\right )}^{2}}\right )}{\left (a x - 1\right )}^{6}}{30 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)^2*(a*x+1)^2*(-a*c*x+c)^5,x, algorithm="giac")

[Out]

-1/30*(5*c^5 + 24*c^5/(a*x - 1) + 30*c^5/(a*x - 1)^2)*(a*x - 1)^6/a