\(\int e^{2 \coth ^{-1}(a x)} (c-a c x)^5 \, dx\) [168]

Optimal result

Integrand size = 18, antiderivative size = 37 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=\frac {2 c^5 (1-a x)^5}{5 a}-\frac {c^5 (1-a x)^6}{6 a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6302, 6264, 45} \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=\frac {2 c^5 (1-a x)^5}{5 a}-\frac {c^5 (1-a x)^6}{6 a} \]

[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 45

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])

Rule 6264

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 6302

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*} \text {integral}& = -\int e^{2 \text {arctanh}(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] (verified)

Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.62 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=-\frac {c^5 (-1+a x)^5 (7+5 a x)}{30 a} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.22

method	result	size
gosper	\(-\frac {\left (5 a^{5} x^{5}-18 a^{4} x^{4}+15 a^{3} x^{3}+20 a^{2} x^{2}-45 a x +30\right ) c^{5} x}{30}\)	\(45\)
default	\(c^{5} \left (-\frac {1}{6} a^{5} x^{6}+\frac {3}{5} a^{4} x^{5}-\frac {1}{2} a^{3} x^{4}-\frac {2}{3} a^{2} x^{3}+\frac {3}{2} a \,x^{2}-x \right )\)	\(47\)
norman	\(-c^{5} x +\frac {3}{2} a \,c^{5} x^{2}-\frac {1}{2} a^{3} c^{5} x^{4}+\frac {3}{5} a^{4} c^{5} x^{5}-\frac {1}{6} a^{5} c^{5} x^{6}-\frac {2}{3} c^{5} a^{2} x^{3}\)	\(61\)
risch	\(-c^{5} x +\frac {3}{2} a \,c^{5} x^{2}-\frac {1}{2} a^{3} c^{5} x^{4}+\frac {3}{5} a^{4} c^{5} x^{5}-\frac {1}{6} a^{5} c^{5} x^{6}-\frac {2}{3} c^{5} a^{2} x^{3}\)	\(61\)
parallelrisch	\(-c^{5} x +\frac {3}{2} a \,c^{5} x^{2}-\frac {1}{2} a^{3} c^{5} x^{4}+\frac {3}{5} a^{4} c^{5} x^{5}-\frac {1}{6} a^{5} c^{5} x^{6}-\frac {2}{3} c^{5} a^{2} x^{3}\)	\(61\)
meijerg	\(-\frac {c^{5} \left (\frac {a x \left (70 a^{5} x^{5}+84 a^{4} x^{4}+105 a^{3} x^{3}+140 a^{2} x^{2}+210 a x +420\right )}{420}+\ln \left (-a x +1\right )\right )}{a}-\frac {4 c^{5} \left (-\frac {a x \left (12 a^{4} x^{4}+15 a^{3} x^{3}+20 a^{2} x^{2}+30 a x +60\right )}{60}-\ln \left (-a x +1\right )\right )}{a}-\frac {5 c^{5} \left (\frac {a x \left (15 a^{3} x^{3}+20 a^{2} x^{2}+30 a x +60\right )}{60}+\ln \left (-a x +1\right )\right )}{a}+\frac {5 c^{5} \left (\frac {a x \left (3 a x +6\right )}{6}+\ln \left (-a x +1\right )\right )}{a}+\frac {4 c^{5} \left (-a x -\ln \left (-a x +1\right )\right )}{a}+\frac {c^{5} \ln \left (-a x +1\right )}{a}\)	\(216\)

[In]

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

[Out]

-1/30*(5*a^5*x^5-18*a^4*x^4+15*a^3*x^3+20*a^2*x^2-45*a*x+30)*c^5*x

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.62 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=-\frac {1}{6} \, a^{5} c^{5} x^{6} + \frac {3}{5} \, a^{4} c^{5} x^{5} - \frac {1}{2} \, a^{3} c^{5} x^{4} - \frac {2}{3} \, a^{2} c^{5} x^{3} + \frac {3}{2} \, a c^{5} x^{2} - c^{5} x \]

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (27) = 54\).

Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.78 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=- \frac {a^{5} c^{5} x^{6}}{6} + \frac {3 a^{4} c^{5} x^{5}}{5} - \frac {a^{3} c^{5} x^{4}}{2} - \frac {2 a^{2} c^{5} x^{3}}{3} + \frac {3 a c^{5} x^{2}}{2} - c^{5} x \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.62 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=-\frac {1}{6} \, a^{5} c^{5} x^{6} + \frac {3}{5} \, a^{4} c^{5} x^{5} - \frac {1}{2} \, a^{3} c^{5} x^{4} - \frac {2}{3} \, a^{2} c^{5} x^{3} + \frac {3}{2} \, a c^{5} x^{2} - c^{5} x \]

[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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.62 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=-\frac {1}{6} \, a^{5} c^{5} x^{6} + \frac {3}{5} \, a^{4} c^{5} x^{5} - \frac {1}{2} \, a^{3} c^{5} x^{4} - \frac {2}{3} \, a^{2} c^{5} x^{3} + \frac {3}{2} \, a c^{5} x^{2} - c^{5} x \]

[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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.62 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=-\frac {a^5\,c^5\,x^6}{6}+\frac {3\,a^4\,c^5\,x^5}{5}-\frac {a^3\,c^5\,x^4}{2}-\frac {2\,a^2\,c^5\,x^3}{3}+\frac {3\,a\,c^5\,x^2}{2}-c^5\,x \]

[In]

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

[Out]

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