Integrand size = 18, antiderivative size = 129 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {16 c^2 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {35}{3} c^2 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {5}{2} a c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {1}{3} a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {35 c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a} \]
-35/2*c^2*arctanh((1-1/a^2/x^2)^(1/2))/a+16*c^2*(a-1/x)/a^2/(1-1/a^2/x^2)^ (1/2)+35/3*c^2*x*(1-1/a^2/x^2)^(1/2)-5/2*a*c^2*x^2*(1-1/a^2/x^2)^(1/2)+1/3 *a^2*c^2*x^3*(1-1/a^2/x^2)^(1/2)
Time = 0.22 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.60 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {1}{6} c^2 \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (166+55 a x-13 a^2 x^2+2 a^3 x^3\right )}{1+a x}-\frac {105 \log \left (a \left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a}\right ) \]
(c^2*((Sqrt[1 - 1/(a^2*x^2)]*x*(166 + 55*a*x - 13*a^2*x^2 + 2*a^3*x^3))/(1 + a*x) - (105*Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/a))/6
Time = 0.56 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.02, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6724, 25, 27, 528, 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.
| \(\displaystyle \int (c-a c x)^2 e^{-3 \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6724 |
| \(\displaystyle \frac {\int -\frac {c^5 \left (a-\frac {1}{x}\right )^5 x^4}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{a^3 c^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {c^5 \left (a-\frac {1}{x}\right )^5 x^4}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{a^3 c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^2 \int \frac {\left (a-\frac {1}{x}\right )^5 x^4}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{a^3}\) |
| \(\Big \downarrow \) 528 |
| \(\displaystyle -\frac {c^2 \left (a^2 \int \frac {\left (a^3-\frac {5 a^2}{x}+\frac {11 a}{x^2}-\frac {15}{x^3}\right ) x^4}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {16 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^3}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle -\frac {c^2 \left (a^2 \left (-\frac {1}{3} \int \frac {5 \left (3 a^2-\frac {7 a}{x}+\frac {9}{x^2}\right ) x^3}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {1}{3} a^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {16 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^2 \left (a^2 \left (-\frac {5}{3} \int \frac {\left (3 a^2-\frac {7 a}{x}+\frac {9}{x^2}\right ) x^3}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {1}{3} a^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {16 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^3}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle -\frac {c^2 \left (a^2 \left (-\frac {5}{3} \left (-\frac {1}{2} \int \frac {7 \left (2 a-\frac {3}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {3}{2} a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {1}{3} a^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {16 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^2 \left (a^2 \left (-\frac {5}{3} \left (-\frac {7}{2} \int \frac {\left (2 a-\frac {3}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {3}{2} a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {1}{3} a^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {16 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^3}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle -\frac {c^2 \left (a^2 \left (-\frac {5}{3} \left (-\frac {7}{2} \left (-3 \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-2 a x \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {3}{2} a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {1}{3} a^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {16 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^3}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {c^2 \left (a^2 \left (-\frac {5}{3} \left (-\frac {7}{2} \left (-\frac {3}{2} \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}-2 a x \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {3}{2} a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {1}{3} a^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {16 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {c^2 \left (a^2 \left (-\frac {5}{3} \left (-\frac {7}{2} \left (3 a^2 \int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}-2 a x \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {3}{2} a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {1}{3} a^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {16 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {c^2 \left (a^2 \left (-\frac {5}{3} \left (-\frac {7}{2} \left (3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-2 a x \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {3}{2} a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {1}{3} a^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {16 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^3}\) |
-((c^2*((-16*a*(a - x^(-1)))/Sqrt[1 - 1/(a^2*x^2)] + a^2*(-1/3*(a^3*Sqrt[1 - 1/(a^2*x^2)]*x^3) - (5*((-3*a^2*Sqrt[1 - 1/(a^2*x^2)]*x^2)/2 - (7*(-2*a *Sqrt[1 - 1/(a^2*x^2)]*x + 3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]))/2))/3)))/a^3 )
3.3.18.3.1 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 = 0.41 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.15
| method | result | size |
| risch | \(\frac {\left (2 a^{2} x^{2}-15 a x +70\right ) \left (a x +1\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}}{6 a}+\frac {\left (-\frac {35 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{2 \sqrt {a^{2}}}+\frac {16 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{a^{2} \left (x +\frac {1}{a}\right )}\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a x -1}\) | \(148\) |
| default | \(-\frac {\left (15 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}-2 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}+30 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}-15 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-4 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -120 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}+120 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+15 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x -30 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x +46 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-240 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x +240 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -15 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a -120 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}+120 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )\right ) c^{2} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{6 a \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )}\) | \(474\) |
1/6*(2*a^2*x^2-15*a*x+70)*(a*x+1)/a*c^2*((a*x-1)/(a*x+1))^(1/2)+(-35/2*ln( a^2*x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2))/(a^2)^(1/2)+16/a^2/(x+1/a)*(a^2*(x+1/ a)^2-2*a*(x+1/a))^(1/2))*c^2/(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 = 104, normalized size of antiderivative = 0.81 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=-\frac {105 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 105 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (2 \, a^{3} c^{2} x^{3} - 13 \, a^{2} c^{2} x^{2} + 55 \, a c^{2} x + 166 \, c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, a} \]
-1/6*(105*c^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 105*c^2*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (2*a^3*c^2*x^3 - 13*a^2*c^2*x^2 + 55*a*c^2*x + 166*c ^2)*sqrt((a*x - 1)/(a*x + 1)))/a
\[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=c^{2} \left (\int \left (- \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \frac {3 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx + \int \left (- \frac {3 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \frac {a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx\right ) \]
c**2*(Integral(-sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x) + Integral (3*a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x) + Integral(-3*a**2* x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x) + Integral(a**3*x**3* sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x))
Time = 0.20 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.58 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=-\frac {1}{6} \, a {\left (\frac {105 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {105 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {96 \, c^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2}} + \frac {2 \, {\left (87 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 136 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 57 \, c^{2} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {3 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {3 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - a^{2}}\right )} \]
-1/6*a*(105*c^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 105*c^2*log(sqrt( (a*x - 1)/(a*x + 1)) - 1)/a^2 - 96*c^2*sqrt((a*x - 1)/(a*x + 1))/a^2 + 2*( 87*c^2*((a*x - 1)/(a*x + 1))^(5/2) - 136*c^2*((a*x - 1)/(a*x + 1))^(3/2) + 57*c^2*sqrt((a*x - 1)/(a*x + 1)))/(3*(a*x - 1)*a^2/(a*x + 1) - 3*(a*x - 1 )^2*a^2/(a*x + 1)^2 + (a*x - 1)^3*a^2/(a*x + 1)^3 - a^2))
\[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=\int { {\left (a c x - c\right )}^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]
Time = 4.54 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.26 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {19\,c^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}-\frac {136\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}+29\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{a-\frac {3\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {3\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}}+\frac {16\,c^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a}-\frac {35\,c^2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
(19*c^2*((a*x - 1)/(a*x + 1))^(1/2) - (136*c^2*((a*x - 1)/(a*x + 1))^(3/2) )/3 + 29*c^2*((a*x - 1)/(a*x + 1))^(5/2))/(a - (3*a*(a*x - 1))/(a*x + 1) + (3*a*(a*x - 1)^2)/(a*x + 1)^2 - (a*(a*x - 1)^3)/(a*x + 1)^3) + (16*c^2*(( a*x - 1)/(a*x + 1))^(1/2))/a - (35*c^2*atanh(((a*x - 1)/(a*x + 1))^(1/2))) /a