Integrand size = 20, antiderivative size = 49 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=c \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {c \csc ^{-1}(a x)}{a}+\frac {2 c \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \]
Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.49 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {c \left (a \sqrt {1-\frac {1}{a^2 x^2}} x-2 \arcsin \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )-2 \arcsin \left (\frac {1}{a x}\right )+2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a} \]
(c*(a*Sqrt[1 - 1/(a^2*x^2)]*x - 2*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] - 2*Ar cSin[1/(a*x)] + 2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]))/a
Time = 0.33 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6731, 27, 570, 540, 25, 538, 223, 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.
\(\displaystyle \int \left (c-\frac {c}{a x}\right ) e^{3 \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6731 |
\(\displaystyle -c^3 \int \frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^2}{c^2 \left (a-\frac {1}{x}\right )^2}d\frac {1}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -a^2 c \int \frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^2}{\left (a-\frac {1}{x}\right )^2}d\frac {1}{x}\) |
\(\Big \downarrow \) 570 |
\(\displaystyle -\frac {c \int \frac {\left (a+\frac {1}{x}\right )^2 x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{a^2}\) |
\(\Big \downarrow \) 540 |
\(\displaystyle -\frac {c \left (a^2 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )-\int -\frac {\left (2 a+\frac {1}{x}\right ) x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )}{a^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {c \left (\int \frac {\left (2 a+\frac {1}{x}\right ) x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-a^2 x \sqrt {1-\frac {1}{a^2 x^2}}\right )}{a^2}\) |
\(\Big \downarrow \) 538 |
\(\displaystyle -\frac {c \left (2 a \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+a^2 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a^2}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle -\frac {c \left (2 a \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+a^2 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )+a \arcsin \left (\frac {1}{a x}\right )\right )}{a^2}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle -\frac {c \left (a \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}+a^2 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )+a \arcsin \left (\frac {1}{a x}\right )\right )}{a^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {c \left (-2 a^3 \int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}-a^2 x \sqrt {1-\frac {1}{a^2 x^2}}+a \arcsin \left (\frac {1}{a x}\right )\right )}{a^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {c \left (-2 a \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )+a^2 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )+a \arcsin \left (\frac {1}{a x}\right )\right )}{a^2}\) |
-((c*(-(a^2*Sqrt[1 - 1/(a^2*x^2)]*x) + a*ArcSin[1/(a*x)] - 2*a*ArcTanh[Sqr t[1 - 1/(a^2*x^2)]]))/a^2)
3.4.99.3.1 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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[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[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp [c Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d Int[1/Sqrt[a + b*x^2], x] , x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol ] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain der[(c + d*x)^n, x, x]}, Simp[R*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))) , x] + Simp[1/(a*(m + 1)) Int[x^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*(m + 1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c^(2*n)/a^n Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I LtQ[n, -1] && !(IGtQ[m, 0] && ILtQ[m + n, 0] && !GtQ[p, 1])
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> S imp[-c^n Subst[Int[(c + d*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^2), x], x, 1/ x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(144\) vs. \(2(45)=90\).
Time = 0.12 (sec) , antiderivative size = 145, normalized size of antiderivative = 2.96
method | result | size |
default | \(-\frac {\left (a x -1\right )^{2} c \left (\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}+\arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {a^{2}}-2 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )-2 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\right )}{\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a \sqrt {a^{2}}}\) | \(145\) |
-(a*x-1)^2*c*((a^2*x^2-1)^(1/2)*(a^2)^(1/2)+arctan(1/(a^2*x^2-1)^(1/2))*(a ^2)^(1/2)-2*a*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))- 2*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/((a*x-1)/(a*x+1))^(3/2)/(a*x+1)/((a *x-1)*(a*x+1))^(1/2)/a/(a^2)^(1/2)
Time = 0.26 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.80 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {2 \, c \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + 2 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 2 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (a c x + c\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a} \]
(2*c*arctan(sqrt((a*x - 1)/(a*x + 1))) + 2*c*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 2*c*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (a*c*x + c)*sqrt((a*x - 1 )/(a*x + 1)))/a
\[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {c \left (\int \frac {a}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \left (- \frac {1}{\frac {a x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx\right )}{a} \]
c*(Integral(a/(a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - sqrt(a*x/ (a*x + 1) - 1/(a*x + 1))/(a*x + 1)), x) + Integral(-1/(a*x**2*sqrt(a*x/(a* x + 1) - 1/(a*x + 1))/(a*x + 1) - x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1)), x))/a
Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (45) = 90\).
Time = 0.29 (sec) , antiderivative size = 114, normalized size of antiderivative = 2.33 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=-2 \, a {\left (\frac {c \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )} a^{2}}{a x + 1} - a^{2}} - \frac {c \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} \]
-2*a*(c*sqrt((a*x - 1)/(a*x + 1))/((a*x - 1)*a^2/(a*x + 1) - a^2) - c*arct an(sqrt((a*x - 1)/(a*x + 1)))/a^2 - c*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a ^2 + c*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2)
Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (45) = 90\).
Time = 0.28 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.86 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {2 \, c \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right )}{a \mathrm {sgn}\left (a x + 1\right )} - \frac {2 \, c \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{{\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1} c}{a \mathrm {sgn}\left (a x + 1\right )} \]
2*c*arctan(-x*abs(a) + sqrt(a^2*x^2 - 1))/(a*sgn(a*x + 1)) - 2*c*log(abs(- x*abs(a) + sqrt(a^2*x^2 - 1)))/(abs(a)*sgn(a*x + 1)) + sqrt(a^2*x^2 - 1)*c /(a*sgn(a*x + 1))
Time = 0.10 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.67 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {2\,c\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}+\frac {4\,c\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}+\frac {2\,c\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a-\frac {a\,\left (a\,x-1\right )}{a\,x+1}} \]