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

Optimal. Leaf size=37 \[ \frac{2 c^5 (1-a x)^5}{5 a}-\frac{c^5 (1-a x)^6}{6 a} \]

[Out]

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

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Rubi [A]  time = 0.0597232, antiderivative size = 37, 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} \[ \frac{2 c^5 (1-a x)^5}{5 a}-\frac{c^5 (1-a x)^6}{6 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0191041, size = 23, normalized size = 0.62 \[ -\frac{c^5 (a x-1)^5 (5 a x+7)}{30 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^5*(-1 + a*x)^5*(7 + 5*a*x))/(30*a)

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Maple [A]  time = 0.039, size = 47, normalized size = 1.3 \begin{align*}{c}^{5} \left ( -{\frac{{x}^{6}{a}^{5}}{6}}+{\frac{3\,{x}^{5}{a}^{4}}{5}}-{\frac{{x}^{4}{a}^{3}}{2}}-{\frac{2\,{x}^{3}{a}^{2}}{3}}+{\frac{3\,a{x}^{2}}{2}}-x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0412, size = 81, normalized size = 2.19 \begin{align*} -\frac{1}{6} \, a^{5} c^{5} x^{6} + \frac{3}{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{3}{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)*(a*x+1)*(-a*c*x+c)^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.51856, size = 130, normalized size = 3.51 \begin{align*} -\frac{1}{6} \, a^{5} c^{5} x^{6} + \frac{3}{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{3}{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)*(a*x+1)*(-a*c*x+c)^5,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.096712, size = 66, normalized size = 1.78 \begin{align*} - \frac{a^{5} c^{5} x^{6}}{6} + \frac{3 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{3 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)*(a*x+1)*(-a*c*x+c)**5,x)

[Out]

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

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Giac [A]  time = 1.14329, size = 81, normalized size = 2.19 \begin{align*} -\frac{1}{6} \, a^{5} c^{5} x^{6} + \frac{3}{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{3}{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)*(a*x+1)*(-a*c*x+c)^5,x, algorithm="giac")

[Out]

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