Integrand size = 18, antiderivative size = 152 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=\frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+30 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {67}{8} a c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2+2 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {1}{4} a^3 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^4-\frac {315 c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a} \]
-315/8*c^3*arctanh((1-1/a^2/x^2)^(1/2))/a+32*c^3*(a-1/x)/a^2/(1-1/a^2/x^2) ^(1/2)+30*c^3*x*(1-1/a^2/x^2)^(1/2)-67/8*a*c^3*x^2*(1-1/a^2/x^2)^(1/2)+2*a ^2*c^3*x^3*(1-1/a^2/x^2)^(1/2)-1/4*a^3*c^3*x^4*(1-1/a^2/x^2)^(1/2)
Time = 0.29 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.57 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=\frac {1}{8} c^3 \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (496+173 a x-51 a^2 x^2+14 a^3 x^3-2 a^4 x^4\right )}{1+a x}-\frac {315 \log \left (a \left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a}\right ) \]
(c^3*((Sqrt[1 - 1/(a^2*x^2)]*x*(496 + 173*a*x - 51*a^2*x^2 + 14*a^3*x^3 - 2*a^4*x^4))/(1 + a*x) - (315*Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/a))/8
Time = 0.69 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6724, 27, 528, 2338, 2338, 27, 2338, 27, 534, 243, 73, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (c-a c x)^3 e^{-3 \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6724 |
| \(\displaystyle \frac {\int \frac {c^6 \left (a-\frac {1}{x}\right )^6 x^5}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{a^3 c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c^3 \int \frac {\left (a-\frac {1}{x}\right )^6 x^5}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{a^3}\) |
| \(\Big \downarrow \) 528 |
| \(\displaystyle \frac {c^3 \left (a^2 \int \frac {\left (a^4-\frac {6 a^3}{x}+\frac {16 a^2}{x^2}-\frac {26 a}{x^3}+\frac {31}{x^4}\right ) x^5}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {32 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^3}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {c^3 \left (a^2 \left (-\frac {1}{4} \int \frac {\left (24 a^3-\frac {67 a^2}{x}+\frac {104 a}{x^2}-\frac {124}{x^3}\right ) x^4}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {1}{4} a^4 x^4 \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {32 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^3}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {c^3 \left (a^2 \left (\frac {1}{4} \left (\frac {1}{3} \int \frac {3 \left (67 a^2-\frac {120 a}{x}+\frac {124}{x^2}\right ) x^3}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+8 a^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {1}{4} a^4 x^4 \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {32 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c^3 \left (a^2 \left (\frac {1}{4} \left (\int \frac {\left (67 a^2-\frac {120 a}{x}+\frac {124}{x^2}\right ) x^3}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+8 a^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {1}{4} a^4 x^4 \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {32 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^3}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {c^3 \left (a^2 \left (\frac {1}{4} \left (-\frac {1}{2} \int \frac {15 \left (16 a-\frac {21}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {67}{2} a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}+8 a^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {1}{4} a^4 x^4 \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {32 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c^3 \left (a^2 \left (\frac {1}{4} \left (-\frac {15}{2} \int \frac {\left (16 a-\frac {21}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {67}{2} a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}+8 a^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {1}{4} a^4 x^4 \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {32 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^3}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {c^3 \left (a^2 \left (\frac {1}{4} \left (-\frac {15}{2} \left (-21 \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-16 a x \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {67}{2} a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}+8 a^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {1}{4} a^4 x^4 \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {32 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^3}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {c^3 \left (a^2 \left (\frac {1}{4} \left (-\frac {15}{2} \left (-\frac {21}{2} \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}-16 a x \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {67}{2} a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}+8 a^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {1}{4} a^4 x^4 \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {32 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {c^3 \left (a^2 \left (\frac {1}{4} \left (-\frac {15}{2} \left (21 a^2 \int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}-16 a x \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {67}{2} a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}+8 a^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {1}{4} a^4 x^4 \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {32 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {c^3 \left (\frac {32 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}+a^2 \left (\frac {1}{4} \left (-\frac {15}{2} \left (21 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-16 a x \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {67}{2} a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}+8 a^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {1}{4} a^4 x^4 \sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a^3}\) |
(c^3*((32*a*(a - x^(-1)))/Sqrt[1 - 1/(a^2*x^2)] + a^2*(-1/4*(a^4*Sqrt[1 - 1/(a^2*x^2)]*x^4) + ((-67*a^2*Sqrt[1 - 1/(a^2*x^2)]*x^2)/2 + 8*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^3 - (15*(-16*a*Sqrt[1 - 1/(a^2*x^2)]*x + 21*ArcTanh[Sqrt[ 1 - 1/(a^2*x^2)]]))/2)/4)))/a^3
3.3.17.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((x_)^(m_)*((c_) + (d_.)*(x_))^(n_.))/((a_) + (b_.)*(x_)^2)^(3/2), x_Sy mbol] :> Simp[(-2^(n - 1))*c^(m + n - 2)*((c + d*x)/(b*d^(m - 1)*Sqrt[a + b *x^2])), x] + Simp[c^2/a Int[(x^m/Sqrt[a + b*x^2])*ExpandToSum[((c + d*x) ^(n - 1) - (2^(n - 1)*c^(m + n - 1))/(d^m*x^m))/(c - d*x), x], x], x] /; Fr eeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && EqQ[b*c^2 + a*d^2, 0]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( m + 1)) Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> S imp[-d^n Subst[Int[(d + c*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^(p + 2)), x], x, 1/x], x] /; FreeQ[{a, c, d}, x] && EqQ[a*c + d, 0] && IntegerQ[p] && In tegerQ[n]
Time = 0.42 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.03
| method | result | size |
| risch | \(-\frac {\left (2 a^{3} x^{3}-16 a^{2} x^{2}+67 a x -240\right ) \left (a x +1\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}}{8 a}-\frac {\left (\frac {315 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{8 \sqrt {a^{2}}}-\frac {32 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{a^{2} \left (x +\frac {1}{a}\right )}\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a x -1}\) | \(157\) |
| default | \(-\frac {\left (2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}+4 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}+69 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}-16 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}+2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x +138 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}-69 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-32 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -384 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}+384 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+69 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x -138 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x +112 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-768 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x +768 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -69 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a -384 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}+384 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )\right ) c^{3} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{8 a \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )}\) | \(542\) |
-1/8*(2*a^3*x^3-16*a^2*x^2+67*a*x-240)*(a*x+1)/a*c^3*((a*x-1)/(a*x+1))^(1/ 2)-(315/8*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2))/(a^2)^(1/2)-32/a^2/(x+1/ a)*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2))*c^3/(a*x-1)*((a*x-1)/(a*x+1))^(1/2)* ((a*x-1)*(a*x+1))^(1/2)
Time = 0.25 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.75 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=-\frac {315 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 315 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (2 \, a^{4} c^{3} x^{4} - 14 \, a^{3} c^{3} x^{3} + 51 \, a^{2} c^{3} x^{2} - 173 \, a c^{3} x - 496 \, c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{8 \, a} \]
-1/8*(315*c^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 315*c^3*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (2*a^4*c^3*x^4 - 14*a^3*c^3*x^3 + 51*a^2*c^3*x^2 - 1 73*a*c^3*x - 496*c^3)*sqrt((a*x - 1)/(a*x + 1)))/a
\[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=- c^{3} \left (\int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx + \int \left (- \frac {4 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \frac {6 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx + \int \left (- \frac {4 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \frac {a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx\right ) \]
-c**3*(Integral(sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x) + Integral (-4*a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x) + Integral(6*a**2* x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x) + Integral(-4*a**3*x* *3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x) + Integral(a**4*x**4*sq rt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x))
Time = 0.20 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.61 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=-\frac {1}{8} \, {\left (\frac {315 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {315 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {256 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2}} - \frac {2 \, {\left (325 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 765 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 643 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 187 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {4 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {6 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {4 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} - a^{2}}\right )} a \]
-1/8*(315*c^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 315*c^3*log(sqrt((a *x - 1)/(a*x + 1)) - 1)/a^2 - 256*c^3*sqrt((a*x - 1)/(a*x + 1))/a^2 - 2*(3 25*c^3*((a*x - 1)/(a*x + 1))^(7/2) - 765*c^3*((a*x - 1)/(a*x + 1))^(5/2) + 643*c^3*((a*x - 1)/(a*x + 1))^(3/2) - 187*c^3*sqrt((a*x - 1)/(a*x + 1)))/ (4*(a*x - 1)*a^2/(a*x + 1) - 6*(a*x - 1)^2*a^2/(a*x + 1)^2 + 4*(a*x - 1)^3 *a^2/(a*x + 1)^3 - (a*x - 1)^4*a^2/(a*x + 1)^4 - a^2))*a
\[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=\int { -{\left (a c x - c\right )}^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]
Time = 0.09 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.31 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=\frac {\frac {187\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{4}-\frac {643\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{4}+\frac {765\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{4}-\frac {325\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{4}}{a-\frac {4\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {6\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {4\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}}+\frac {32\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a}-\frac {315\,c^3\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4\,a} \]
((187*c^3*((a*x - 1)/(a*x + 1))^(1/2))/4 - (643*c^3*((a*x - 1)/(a*x + 1))^ (3/2))/4 + (765*c^3*((a*x - 1)/(a*x + 1))^(5/2))/4 - (325*c^3*((a*x - 1)/( a*x + 1))^(7/2))/4)/(a - (4*a*(a*x - 1))/(a*x + 1) + (6*a*(a*x - 1)^2)/(a* x + 1)^2 - (4*a*(a*x - 1)^3)/(a*x + 1)^3 + (a*(a*x - 1)^4)/(a*x + 1)^4) + (32*c^3*((a*x - 1)/(a*x + 1))^(1/2))/a - (315*c^3*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(4*a)