Integrand size = 16, antiderivative size = 92 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x) \, dx=\frac {8 c \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+4 c \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {1}{2} a c \sqrt {1-\frac {1}{a^2 x^2}} x^2-\frac {15 c \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a} \]
-15/2*c*arctanh((1-1/a^2/x^2)^(1/2))/a+8*c*(a-1/x)/a^2/(1-1/a^2/x^2)^(1/2) +4*c*x*(1-1/a^2/x^2)^(1/2)-1/2*a*c*x^2*(1-1/a^2/x^2)^(1/2)
Time = 0.18 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.74 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x) \, dx=\frac {1}{2} c \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (24+7 a x-a^2 x^2\right )}{1+a x}-\frac {15 \log \left (a \left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a}\right ) \]
(c*((Sqrt[1 - 1/(a^2*x^2)]*x*(24 + 7*a*x - a^2*x^2))/(1 + a*x) - (15*Log[a *(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/a))/2
Time = 0.41 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6724, 27, 528, 2338, 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.
| \(\displaystyle \int (c-a c x) e^{-3 \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6724 |
| \(\displaystyle \frac {\int \frac {c^4 \left (a-\frac {1}{x}\right )^4 x^3}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{a^3 c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \int \frac {\left (a-\frac {1}{x}\right )^4 x^3}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{a^3}\) |
| \(\Big \downarrow \) 528 |
| \(\displaystyle \frac {c \left (a^2 \int \frac {\left (a^2-\frac {4 a}{x}+\frac {7}{x^2}\right ) x^3}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {8 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^3}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {c \left (a^2 \left (-\frac {1}{2} \int \frac {\left (8 a-\frac {15}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {1}{2} a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {8 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^3}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {c \left (a^2 \left (\frac {1}{2} \left (15 \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+8 a x \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {1}{2} a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {8 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^3}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {c \left (a^2 \left (\frac {1}{2} \left (\frac {15}{2} \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}+8 a x \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {1}{2} a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {8 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {c \left (a^2 \left (\frac {1}{2} \left (8 a x \sqrt {1-\frac {1}{a^2 x^2}}-15 a^2 \int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {1}{2} a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {8 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {c \left (a^2 \left (\frac {1}{2} \left (8 a x \sqrt {1-\frac {1}{a^2 x^2}}-15 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )-\frac {1}{2} a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {8 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^3}\) |
(c*((8*a*(a - x^(-1)))/Sqrt[1 - 1/(a^2*x^2)] + a^2*(-1/2*(a^2*Sqrt[1 - 1/( a^2*x^2)]*x^2) + (8*a*Sqrt[1 - 1/(a^2*x^2)]*x - 15*ArcTanh[Sqrt[1 - 1/(a^2 *x^2)]])/2)))/a^3
3.3.19.3.1 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 = 0.13 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.48
| method | result | size |
| risch | \(-\frac {\left (a x -8\right ) \left (a x +1\right ) c \sqrt {\frac {a x -1}{a x +1}}}{2 a}-\frac {\left (\frac {15 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{2 \sqrt {a^{2}}}-\frac {8 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{a^{2} \left (x +\frac {1}{a}\right )}\right ) c \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a x -1}\) | \(136\) |
| default | \(-\frac {\left (\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}+2 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}-\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-16 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}+16 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x -2 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x +8 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-32 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x +32 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a -16 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}+16 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )\right ) c \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{2 a \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )}\) | \(422\) |
-1/2*(a*x-8)*(a*x+1)/a*c*((a*x-1)/(a*x+1))^(1/2)-(15/2*ln(a^2*x/(a^2)^(1/2 )+(a^2*x^2-1)^(1/2))/(a^2)^(1/2)-8/a^2/(x+1/a)*(a^2*(x+1/a)^2-2*a*(x+1/a)) ^(1/2))*c*((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^(1/2)/(a*x-1)
Time = 0.25 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.88 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x) \, dx=-\frac {15 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 15 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (a^{2} c x^{2} - 7 \, a c x - 24 \, c\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, a} \]
-1/2*(15*c*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 15*c*log(sqrt((a*x - 1)/(a *x + 1)) - 1) + (a^2*c*x^2 - 7*a*c*x - 24*c)*sqrt((a*x - 1)/(a*x + 1)))/a
\[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x) \, dx=- c \left (\int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx + \int \left (- \frac {2 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \frac {a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx\right ) \]
-c*(Integral(sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x) + Integral(-2 *a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x) + Integral(a**2*x**2* sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x))
Time = 0.21 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.70 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x) \, dx=\frac {1}{2} \, a {\left (\frac {2 \, {\left (9 \, c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 7 \, c \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {2 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} - \frac {15 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {15 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} + \frac {16 \, c \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2}}\right )} \]
1/2*a*(2*(9*c*((a*x - 1)/(a*x + 1))^(3/2) - 7*c*sqrt((a*x - 1)/(a*x + 1))) /(2*(a*x - 1)*a^2/(a*x + 1) - (a*x - 1)^2*a^2/(a*x + 1)^2 - a^2) - 15*c*lo g(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 + 15*c*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2 + 16*c*sqrt((a*x - 1)/(a*x + 1))/a^2)
\[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x) \, dx=\int { -{\left (a c x - c\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]
Time = 0.08 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.27 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x) \, dx=\frac {7\,c\,\sqrt {\frac {a\,x-1}{a\,x+1}}-9\,c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{a-\frac {2\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}}-\frac {15\,c\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}+\frac {8\,c\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a} \]