3.207 \(\int e^{-2 \coth ^{-1}(a x)} (c-a c x)^4 \, dx\)

Optimal. Leaf size=91 \[ -\frac {c^4 (1-a x)^5}{5 a}-\frac {c^4 (1-a x)^4}{2 a}-\frac {4 c^4 (1-a x)^3}{3 a}-\frac {4 c^4 (1-a x)^2}{a}-\frac {32 c^4 \log (a x+1)}{a}+16 c^4 x \]

[Out]

16*c^4*x-4*c^4*(-a*x+1)^2/a-4/3*c^4*(-a*x+1)^3/a-1/2*c^4*(-a*x+1)^4/a-1/5*c^4*(-a*x+1)^5/a-32*c^4*ln(a*x+1)/a

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 91, normalized size of antiderivative = 1.00, 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} \[ -\frac {c^4 (1-a x)^5}{5 a}-\frac {c^4 (1-a x)^4}{2 a}-\frac {4 c^4 (1-a x)^3}{3 a}-\frac {4 c^4 (1-a x)^2}{a}-\frac {32 c^4 \log (a x+1)}{a}+16 c^4 x \]

Antiderivative was successfully verified.

[In]

Int[(c - a*c*x)^4/E^(2*ArcCoth[a*x]),x]

[Out]

16*c^4*x - (4*c^4*(1 - a*x)^2)/a - (4*c^4*(1 - a*x)^3)/(3*a) - (c^4*(1 - a*x)^4)/(2*a) - (c^4*(1 - a*x)^5)/(5*
a) - (32*c^4*Log[1 + a*x])/a

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])

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 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)} (c-a c x)^4 \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} (c-a c x)^4 \, dx\\ &=-\left (c^4 \int \frac {(1-a x)^5}{1+a x} \, dx\right )\\ &=-\left (c^4 \int \left (-16-8 (1-a x)-4 (1-a x)^2-2 (1-a x)^3-(1-a x)^4+\frac {32}{1+a x}\right ) \, dx\right )\\ &=16 c^4 x-\frac {4 c^4 (1-a x)^2}{a}-\frac {4 c^4 (1-a x)^3}{3 a}-\frac {c^4 (1-a x)^4}{2 a}-\frac {c^4 (1-a x)^5}{5 a}-\frac {32 c^4 \log (1+a x)}{a}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 56, normalized size = 0.62 \[ \frac {c^4 \left (6 a^5 x^5-45 a^4 x^4+160 a^3 x^3-390 a^2 x^2+930 a x-960 \log (a x+1)-181\right )}{30 a} \]

Antiderivative was successfully verified.

[In]

Integrate[(c - a*c*x)^4/E^(2*ArcCoth[a*x]),x]

[Out]

(c^4*(-181 + 930*a*x - 390*a^2*x^2 + 160*a^3*x^3 - 45*a^4*x^4 + 6*a^5*x^5 - 960*Log[1 + a*x]))/(30*a)

________________________________________________________________________________________

fricas [A]  time = 0.81, size = 68, normalized size = 0.75 \[ \frac {6 \, a^{5} c^{4} x^{5} - 45 \, a^{4} c^{4} x^{4} + 160 \, a^{3} c^{4} x^{3} - 390 \, a^{2} c^{4} x^{2} + 930 \, a c^{4} x - 960 \, c^{4} \log \left (a x + 1\right )}{30 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^4*(a*x-1)/(a*x+1),x, algorithm="fricas")

[Out]

1/30*(6*a^5*c^4*x^5 - 45*a^4*c^4*x^4 + 160*a^3*c^4*x^3 - 390*a^2*c^4*x^2 + 930*a*c^4*x - 960*c^4*log(a*x + 1))
/a

________________________________________________________________________________________

giac [A]  time = 0.14, size = 75, normalized size = 0.82 \[ -\frac {32 \, c^{4} \log \left ({\left | a x + 1 \right |}\right )}{a} + \frac {6 \, a^{9} c^{4} x^{5} - 45 \, a^{8} c^{4} x^{4} + 160 \, a^{7} c^{4} x^{3} - 390 \, a^{6} c^{4} x^{2} + 930 \, a^{5} c^{4} x}{30 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^4*(a*x-1)/(a*x+1),x, algorithm="giac")

[Out]

-32*c^4*log(abs(a*x + 1))/a + 1/30*(6*a^9*c^4*x^5 - 45*a^8*c^4*x^4 + 160*a^7*c^4*x^3 - 390*a^6*c^4*x^2 + 930*a
^5*c^4*x)/a^5

________________________________________________________________________________________

maple [A]  time = 0.04, size = 64, normalized size = 0.70 \[ \frac {a^{4} c^{4} x^{5}}{5}-\frac {3 c^{4} x^{4} a^{3}}{2}+\frac {16 a^{2} c^{4} x^{3}}{3}-13 c^{4} x^{2} a +31 c^{4} x -\frac {32 c^{4} \ln \left (a x +1\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a*c*x+c)^4/(a*x+1)*(a*x-1),x)

[Out]

1/5*a^4*c^4*x^5-3/2*c^4*x^4*a^3+16/3*a^2*c^4*x^3-13*c^4*x^2*a+31*c^4*x-32*c^4*ln(a*x+1)/a

________________________________________________________________________________________

maxima [A]  time = 0.31, size = 63, normalized size = 0.69 \[ \frac {1}{5} \, a^{4} c^{4} x^{5} - \frac {3}{2} \, a^{3} c^{4} x^{4} + \frac {16}{3} \, a^{2} c^{4} x^{3} - 13 \, a c^{4} x^{2} + 31 \, c^{4} x - \frac {32 \, c^{4} \log \left (a x + 1\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^4*(a*x-1)/(a*x+1),x, algorithm="maxima")

[Out]

1/5*a^4*c^4*x^5 - 3/2*a^3*c^4*x^4 + 16/3*a^2*c^4*x^3 - 13*a*c^4*x^2 + 31*c^4*x - 32*c^4*log(a*x + 1)/a

________________________________________________________________________________________

mupad [B]  time = 1.17, size = 63, normalized size = 0.69 \[ 31\,c^4\,x-13\,a\,c^4\,x^2+\frac {16\,a^2\,c^4\,x^3}{3}-\frac {3\,a^3\,c^4\,x^4}{2}+\frac {a^4\,c^4\,x^5}{5}-\frac {32\,c^4\,\ln \left (a\,x+1\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c - a*c*x)^4*(a*x - 1))/(a*x + 1),x)

[Out]

31*c^4*x - 13*a*c^4*x^2 + (16*a^2*c^4*x^3)/3 - (3*a^3*c^4*x^4)/2 + (a^4*c^4*x^5)/5 - (32*c^4*log(a*x + 1))/a

________________________________________________________________________________________

sympy [A]  time = 0.16, size = 68, normalized size = 0.75 \[ \frac {a^{4} c^{4} x^{5}}{5} - \frac {3 a^{3} c^{4} x^{4}}{2} + \frac {16 a^{2} c^{4} x^{3}}{3} - 13 a c^{4} x^{2} + 31 c^{4} x - \frac {32 c^{4} \log {\left (a x + 1 \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)**4*(a*x-1)/(a*x+1),x)

[Out]

a**4*c**4*x**5/5 - 3*a**3*c**4*x**4/2 + 16*a**2*c**4*x**3/3 - 13*a*c**4*x**2 + 31*c**4*x - 32*c**4*log(a*x + 1
)/a

________________________________________________________________________________________