\(\int e^{4 \coth ^{-1}(a x)} (c-a c x)^5 \, dx\) [188]

Optimal result

Integrand size = 18, antiderivative size = 53 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=-\frac {c^5 (1-a x)^4}{a}+\frac {4 c^5 (1-a x)^5}{5 a}-\frac {c^5 (1-a x)^6}{6 a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 53, 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} \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=-\frac {c^5 (1-a x)^6}{6 a}+\frac {4 c^5 (1-a x)^5}{5 a}-\frac {c^5 (1-a x)^4}{a} \]

[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 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^{4 \text {arctanh}(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] (verified)

Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.58 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=-\frac {c^5 (-1+a x)^4 \left (11+14 a x+5 a^2 x^2\right )}{30 a} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.85

method	result	size
gosper	\(-\frac {x \left (5 a^{5} x^{5}-6 a^{4} x^{4}-15 a^{3} x^{3}+20 a^{2} x^{2}+15 a x -30\right ) c^{5}}{30}\)	\(45\)
default	\(c^{5} \left (-\frac {1}{6} a^{5} x^{6}+\frac {1}{5} a^{4} x^{5}+\frac {1}{2} a^{3} x^{4}-\frac {2}{3} a^{2} x^{3}-\frac {1}{2} a \,x^{2}+x \right )\)	\(45\)
risch	\(-\frac {1}{6} a^{5} c^{5} x^{6}+\frac {1}{5} a^{4} c^{5} x^{5}+\frac {1}{2} a^{3} c^{5} x^{4}-\frac {2}{3} c^{5} a^{2} x^{3}-\frac {1}{2} a \,c^{5} x^{2}+c^{5} x\)	\(60\)
parallelrisch	\(-\frac {1}{6} a^{5} c^{5} x^{6}+\frac {1}{5} a^{4} c^{5} x^{5}+\frac {1}{2} a^{3} c^{5} x^{4}-\frac {2}{3} c^{5} a^{2} x^{3}-\frac {1}{2} a \,c^{5} x^{2}+c^{5} x\)	\(60\)
norman	\(\frac {-c^{5} x +\frac {3}{2} a \,c^{5} x^{2}-\frac {7}{6} a^{3} c^{5} x^{4}+\frac {3}{10} a^{4} c^{5} x^{5}+\frac {11}{30} a^{5} c^{5} x^{6}-\frac {1}{6} a^{6} c^{5} x^{7}+\frac {1}{6} c^{5} a^{2} x^{3}}{a x -1}\)	\(80\)
meijerg	\(-\frac {c^{5} \left (\frac {a x \left (-20 a^{6} x^{6}-28 a^{5} x^{5}-42 a^{4} x^{4}-70 a^{3} x^{3}-140 a^{2} x^{2}-420 a x +840\right )}{-120 a x +120}+7 \ln \left (-a x +1\right )\right )}{a}-\frac {3 c^{5} \left (-\frac {a x \left (-14 a^{5} x^{5}-21 a^{4} x^{4}-35 a^{3} x^{3}-70 a^{2} x^{2}-210 a x +420\right )}{70 \left (-a x +1\right )}-6 \ln \left (-a x +1\right )\right )}{a}-\frac {c^{5} \left (\frac {a x \left (-3 a^{4} x^{4}-5 a^{3} x^{3}-10 a^{2} x^{2}-30 a x +60\right )}{-12 a x +12}+5 \ln \left (-a x +1\right )\right )}{a}+\frac {5 c^{5} \left (-\frac {a x \left (-5 a^{3} x^{3}-10 a^{2} x^{2}-30 a x +60\right )}{15 \left (-a x +1\right )}-4 \ln \left (-a x +1\right )\right )}{a}+\frac {5 c^{5} \left (\frac {a x \left (-2 a^{2} x^{2}-6 a x +12\right )}{-4 a x +4}+3 \ln \left (-a x +1\right )\right )}{a}-\frac {c^{5} \left (-\frac {a x \left (-3 a x +6\right )}{3 \left (-a x +1\right )}-2 \ln \left (-a x +1\right )\right )}{a}-\frac {3 c^{5} \left (\frac {a x}{-a x +1}+\ln \left (-a x +1\right )\right )}{a}+\frac {c^{5} x}{-a x +1}\)	\(378\)

[In]

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

[Out]

-1/30*x*(5*a^5*x^5-6*a^4*x^4-15*a^3*x^3+20*a^2*x^2+15*a*x-30)*c^5

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.11 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=-\frac {1}{6} \, a^{5} c^{5} x^{6} + \frac {1}{5} \, a^{4} c^{5} x^{5} + \frac {1}{2} \, a^{3} c^{5} x^{4} - \frac {2}{3} \, a^{2} c^{5} x^{3} - \frac {1}{2} \, a c^{5} x^{2} + c^{5} x \]

[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

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.19 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=- \frac {a^{5} c^{5} x^{6}}{6} + \frac {a^{4} c^{5} x^{5}}{5} + \frac {a^{3} c^{5} x^{4}}{2} - \frac {2 a^{2} c^{5} x^{3}}{3} - \frac {a c^{5} x^{2}}{2} + c^{5} x \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.11 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=-\frac {1}{6} \, a^{5} c^{5} x^{6} + \frac {1}{5} \, a^{4} c^{5} x^{5} + \frac {1}{2} \, a^{3} c^{5} x^{4} - \frac {2}{3} \, a^{2} c^{5} x^{3} - \frac {1}{2} \, a c^{5} x^{2} + c^{5} x \]

[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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.79 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=-\frac {{\left (5 \, c^{5} + \frac {24 \, c^{5}}{a x - 1} + \frac {30 \, c^{5}}{{\left (a x - 1\right )}^{2}}\right )} {\left (a x - 1\right )}^{6}}{30 \, a} \]

[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

Mupad [B] (verification not implemented)

Time = 4.16 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.11 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=-\frac {a^5\,c^5\,x^6}{6}+\frac {a^4\,c^5\,x^5}{5}+\frac {a^3\,c^5\,x^4}{2}-\frac {2\,a^2\,c^5\,x^3}{3}-\frac {a\,c^5\,x^2}{2}+c^5\,x \]

[In]

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

[Out]

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