### 3.421 $$\int e^{-2 \coth ^{-1}(a x)} (c-\frac{c}{a x})^4 \, dx$$

Optimal. Leaf size=65 $-\frac{3 c^4}{a^3 x^2}+\frac{c^4}{3 a^4 x^3}+\frac{16 c^4}{a^2 x}+\frac{26 c^4 \log (x)}{a}-\frac{32 c^4 \log (a x+1)}{a}+c^4 x$

[Out]

c^4/(3*a^4*x^3) - (3*c^4)/(a^3*x^2) + (16*c^4)/(a^2*x) + c^4*x + (26*c^4*Log[x])/a - (32*c^4*Log[1 + a*x])/a

________________________________________________________________________________________

Rubi [A]  time = 0.145399, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.182, Rules used = {6167, 6131, 6129, 88} $-\frac{3 c^4}{a^3 x^2}+\frac{c^4}{3 a^4 x^3}+\frac{16 c^4}{a^2 x}+\frac{26 c^4 \log (x)}{a}-\frac{32 c^4 \log (a x+1)}{a}+c^4 x$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

c^4/(3*a^4*x^3) - (3*c^4)/(a^3*x^2) + (16*c^4)/(a^2*x) + c^4*x + (26*c^4*Log[x])/a - (32*c^4*Log[1 + a*x])/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 6131

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> Dist[d^p, Int[(u*(1 + (c*x)/d)
^p*E^(n*ArcTanh[a*x]))/x^p, x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[c^2 - a^2*d^2, 0] && IntegerQ[p]

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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^4 \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^4 \, dx\\ &=-\frac{c^4 \int \frac{e^{-2 \tanh ^{-1}(a x)} (1-a x)^4}{x^4} \, dx}{a^4}\\ &=-\frac{c^4 \int \frac{(1-a x)^5}{x^4 (1+a x)} \, dx}{a^4}\\ &=-\frac{c^4 \int \left (-a^4+\frac{1}{x^4}-\frac{6 a}{x^3}+\frac{16 a^2}{x^2}-\frac{26 a^3}{x}+\frac{32 a^4}{1+a x}\right ) \, dx}{a^4}\\ &=\frac{c^4}{3 a^4 x^3}-\frac{3 c^4}{a^3 x^2}+\frac{16 c^4}{a^2 x}+c^4 x+\frac{26 c^4 \log (x)}{a}-\frac{32 c^4 \log (1+a x)}{a}\\ \end{align*}

Mathematica [A]  time = 0.17904, size = 67, normalized size = 1.03 $-\frac{3 c^4}{a^3 x^2}+\frac{c^4}{3 a^4 x^3}+\frac{16 c^4}{a^2 x}+\frac{26 c^4 \log (a x)}{a}-\frac{32 c^4 \log (a x+1)}{a}+c^4 x$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

c^4/(3*a^4*x^3) - (3*c^4)/(a^3*x^2) + (16*c^4)/(a^2*x) + c^4*x + (26*c^4*Log[a*x])/a - (32*c^4*Log[1 + a*x])/a

________________________________________________________________________________________

Maple [A]  time = 0.048, size = 64, normalized size = 1. \begin{align*}{\frac{{c}^{4}}{3\,{a}^{4}{x}^{3}}}-3\,{\frac{{c}^{4}}{{x}^{2}{a}^{3}}}+16\,{\frac{{c}^{4}}{{a}^{2}x}}+{c}^{4}x+26\,{\frac{{c}^{4}\ln \left ( x \right ) }{a}}-32\,{\frac{{c}^{4}\ln \left ( ax+1 \right ) }{a}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*c^4/a^4/x^3-3*c^4/x^2/a^3+16*c^4/a^2/x+c^4*x+26*c^4*ln(x)/a-32*c^4*ln(a*x+1)/a

________________________________________________________________________________________

Maxima [A]  time = 1.06117, size = 81, normalized size = 1.25 \begin{align*} c^{4} x - \frac{32 \, c^{4} \log \left (a x + 1\right )}{a} + \frac{26 \, c^{4} \log \left (x\right )}{a} + \frac{48 \, a^{2} c^{4} x^{2} - 9 \, a c^{4} x + c^{4}}{3 \, a^{4} x^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^4*x - 32*c^4*log(a*x + 1)/a + 26*c^4*log(x)/a + 1/3*(48*a^2*c^4*x^2 - 9*a*c^4*x + c^4)/(a^4*x^3)

________________________________________________________________________________________

Fricas [A]  time = 1.85412, size = 162, normalized size = 2.49 \begin{align*} \frac{3 \, a^{4} c^{4} x^{4} - 96 \, a^{3} c^{4} x^{3} \log \left (a x + 1\right ) + 78 \, a^{3} c^{4} x^{3} \log \left (x\right ) + 48 \, a^{2} c^{4} x^{2} - 9 \, a c^{4} x + c^{4}}{3 \, a^{4} x^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*a^4*c^4*x^4 - 96*a^3*c^4*x^3*log(a*x + 1) + 78*a^3*c^4*x^3*log(x) + 48*a^2*c^4*x^2 - 9*a*c^4*x + c^4)/(
a^4*x^3)

________________________________________________________________________________________

Sympy [A]  time = 0.661916, size = 56, normalized size = 0.86 \begin{align*} c^{4} x + \frac{2 c^{4} \left (13 \log{\left (x \right )} - 16 \log{\left (x + \frac{1}{a} \right )}\right )}{a} + \frac{48 a^{2} c^{4} x^{2} - 9 a c^{4} x + c^{4}}{3 a^{4} x^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**4*x + 2*c**4*(13*log(x) - 16*log(x + 1/a))/a + (48*a**2*c**4*x**2 - 9*a*c**4*x + c**4)/(3*a**4*x**3)

________________________________________________________________________________________

Giac [A]  time = 1.13839, size = 84, normalized size = 1.29 \begin{align*} c^{4} x - \frac{32 \, c^{4} \log \left ({\left | a x + 1 \right |}\right )}{a} + \frac{26 \, c^{4} \log \left ({\left | x \right |}\right )}{a} + \frac{48 \, a^{2} c^{4} x^{2} - 9 \, a c^{4} x + c^{4}}{3 \, a^{4} x^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^4*x - 32*c^4*log(abs(a*x + 1))/a + 26*c^4*log(abs(x))/a + 1/3*(48*a^2*c^4*x^2 - 9*a*c^4*x + c^4)/(a^4*x^3)