Integrand size = 18, antiderivative size = 53 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {2 \left (a-\frac {1}{x}\right )}{a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c} \] Output:
2*(a-1/x)/a^2/c/(1-1/a^2/x^2)^(1/2)-arctanh((1-1/a^2/x^2)^(1/2))/a/c
Time = 0.19 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.02 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{1+a x}-\frac {\log \left (a \left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a}}{c} \] Input:
Integrate[1/(E^(3*ArcCoth[a*x])*(c - a*c*x)),x]
Output:
((2*Sqrt[1 - 1/(a^2*x^2)]*x)/(1 + a*x) - Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)]) *x]/a)/c
Time = 0.46 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6724, 27, 528, 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.
\(\displaystyle \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-a c x} \, dx\) |
\(\Big \downarrow \) 6724 |
\(\displaystyle \frac {\int \frac {c^2 \left (a-\frac {1}{x}\right )^2 x}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{a^3 c^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\left (a-\frac {1}{x}\right )^2 x}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{a^3 c}\) |
\(\Big \downarrow \) 528 |
\(\displaystyle \frac {a^2 \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {2 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}}{a^3 c}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\frac {1}{2} a^2 \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}+\frac {2 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}}{a^3 c}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {2 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}-a^4 \int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}}{a^3 c}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {2 a \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}-a^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a^3 c}\) |
Input:
Int[1/(E^(3*ArcCoth[a*x])*(c - a*c*x)),x]
Output:
((2*a*(a - x^(-1)))/Sqrt[1 - 1/(a^2*x^2)] - a^2*ArcTanh[Sqrt[1 - 1/(a^2*x^ 2)]])/(a^3*c)
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[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]
Leaf count of result is larger than twice the leaf count of optimal. \(247\) vs. \(2(49)=98\).
Time = 0.12 (sec) , antiderivative size = 248, normalized size of antiderivative = 4.68
method | result | size |
default | \(-\frac {\left (-\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}+\ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+\left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-2 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a x +2 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x -\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}+a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{a \sqrt {a^{2}}\, c \left (a x -1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}}\) | \(248\) |
Input:
int(((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c),x,method=_RETURNVERBOSE)
Output:
-(-((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*a^2*x^2+ln((a^2*x+((a*x-1)*(a*x+1)) ^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*a^3*x^2+((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/ 2)-2*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*a*x+2*ln((a^2*x+((a*x-1)*(a*x+1)) ^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*a^2*x-((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2) +a*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2)))/a*((a*x-1) /(a*x+1))^(3/2)/(a^2)^(1/2)/c/(a*x-1)/((a*x-1)*(a*x+1))^(1/2)
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.19 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {2 \, \sqrt {\frac {a x - 1}{a x + 1}} - \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a c} \] Input:
integrate(((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c),x, algorithm="fricas")
Output:
(2*sqrt((a*x - 1)/(a*x + 1)) - log(sqrt((a*x - 1)/(a*x + 1)) + 1) + log(sq rt((a*x - 1)/(a*x + 1)) - 1))/(a*c)
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-a c x} \, dx=- \frac {\int \left (- \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{2} x^{2} - 1}\right )\, dx + \int \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{2} x^{2} - 1}\, dx}{c} \] Input:
integrate(((a*x-1)/(a*x+1))**(3/2)/(-a*c*x+c),x)
Output:
-(Integral(-sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**2*x**2 - 1), x) + Integr al(a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**2*x**2 - 1), x))/c
Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.47 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-a c x} \, dx=-a {\left (\frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c} - \frac {2 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c}\right )} \] Input:
integrate(((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c),x, algorithm="maxima")
Output:
-a*(log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c) - log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c) - 2*sqrt((a*x - 1)/(a*x + 1))/(a^2*c))
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-a c x} \, dx=\int { -\frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{a c x - c} \,d x } \] Input:
integrate(((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c),x, algorithm="giac")
Output:
undef
Time = 13.32 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.91 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c}-\frac {2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c} \] Input:
int(((a*x - 1)/(a*x + 1))^(3/2)/(c - a*c*x),x)
Output:
(2*((a*x - 1)/(a*x + 1))^(1/2))/(a*c) - (2*atanh(((a*x - 1)/(a*x + 1))^(1/ 2)))/(a*c)
Time = 0.14 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.45 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {2 \sqrt {a x +1}\, \sqrt {a x -1}-2 \,\mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a x -2 \,\mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right )+2 a x +2}{a c \left (a x +1\right )} \] Input:
int(((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c),x)
Output:
(2*(sqrt(a*x + 1)*sqrt(a*x - 1) - log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt (2))*a*x - log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2)) + a*x + 1))/(a*c*( a*x + 1))