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

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*(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

________________________________________________________________________________________

Rubi [A]  time = 0.0487565, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {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*ArcTanh[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 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 \tanh ^{-1}(a x)} (c-a c x)^4 \, dx &=c^4 \int \frac{(1-a x)^5}{1+a x} \, dx\\ &=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\\ &=-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 = 0.0191269, 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*ArcTanh[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)

________________________________________________________________________________________

Maple [A]  time = 0.028, size = 64, normalized size = 0.7 \begin{align*} -{\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+32\,{\frac{{c}^{4}\ln \left ( ax+1 \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a*c*x+c)^4/(a*x+1)^2*(-a^2*x^2+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 = 0.941401, size = 85, normalized size = 0.93 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^4/(a*x+1)^2*(-a^2*x^2+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

________________________________________________________________________________________

Fricas [A]  time = 1.54748, size = 155, normalized size = 1.7 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^4/(a*x+1)^2*(-a^2*x^2+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

________________________________________________________________________________________

Sympy [A]  time = 0.346247, size = 68, normalized size = 0.75 \begin{align*} - \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)**4/(a*x+1)**2*(-a**2*x**2+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

________________________________________________________________________________________

Giac [A]  time = 1.19463, size = 127, normalized size = 1.4 \begin{align*} -\frac{{\left (6 \, c^{4} - \frac{75 \, c^{4}}{a x + 1} + \frac{400 \, c^{4}}{{\left (a x + 1\right )}^{2}} - \frac{1200 \, c^{4}}{{\left (a x + 1\right )}^{3}} + \frac{2400 \, c^{4}}{{\left (a x + 1\right )}^{4}}\right )}{\left (a x + 1\right )}^{5}}{30 \, a} - \frac{32 \, c^{4} \log \left (\frac{{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2}{\left | a \right |}}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^4/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="giac")

[Out]

-1/30*(6*c^4 - 75*c^4/(a*x + 1) + 400*c^4/(a*x + 1)^2 - 1200*c^4/(a*x + 1)^3 + 2400*c^4/(a*x + 1)^4)*(a*x + 1)
^5/a - 32*c^4*log(abs(a*x + 1)/((a*x + 1)^2*abs(a)))/a