Integrand size = 16, antiderivative size = 52 \[ \int \frac {e^{\coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {2 \sqrt {1-\frac {1}{a^2 x^2}}}{c \left (a-\frac {1}{x}\right )}-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c} \] Output:
2*(1-1/a^2/x^2)^(1/2)/c/(a-1/x)-arctanh((1-1/a^2/x^2)^(1/2))/a/c
Time = 0.13 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.15 \[ \int \frac {e^{\coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {2 a \sqrt {1-\frac {1}{a^2 x^2}} x+(1-a x) \log \left (a \left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a c (-1+a x)} \] Input:
Integrate[E^ArcCoth[a*x]/(c - a*c*x),x]
Output:
(2*a*Sqrt[1 - 1/(a^2*x^2)]*x + (1 - a*x)*Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)]) *x])/(a*c*(-1 + a*x))
Time = 0.47 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6724, 27, 564, 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.
\(\displaystyle \int \frac {e^{\coth ^{-1}(a x)}}{c-a c x} \, dx\) |
\(\Big \downarrow \) 6724 |
\(\displaystyle a c \int \frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^2 \left (a-\frac {1}{x}\right )^2}d\frac {1}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a \int \frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{\left (a-\frac {1}{x}\right )^2}d\frac {1}{x}}{c}\) |
\(\Big \downarrow \) 564 |
\(\displaystyle \frac {a \left (\frac {\int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{a^2}+\frac {2 \sqrt {1-\frac {1}{a^2 x^2}}}{a \left (a-\frac {1}{x}\right )}\right )}{c}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {a \left (\frac {\int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}}{2 a^2}+\frac {2 \sqrt {1-\frac {1}{a^2 x^2}}}{a \left (a-\frac {1}{x}\right )}\right )}{c}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {a \left (\frac {2 \sqrt {1-\frac {1}{a^2 x^2}}}{a \left (a-\frac {1}{x}\right )}-\int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}\right )}{c}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {a \left (\frac {2 \sqrt {1-\frac {1}{a^2 x^2}}}{a \left (a-\frac {1}{x}\right )}-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a^2}\right )}{c}\) |
Input:
Int[E^ArcCoth[a*x]/(c - a*c*x),x]
Output:
(a*((2*Sqrt[1 - 1/(a^2*x^2)])/(a*(a - x^(-1))) - ArcTanh[Sqrt[1 - 1/(a^2*x ^2)]]/a^2))/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)^(p_), x_Symbol ] :> Simp[(-(-c)^(m - n - 2))*d^(2*n - m + 3)*(Sqrt[a + b*x^2]/(2^(n + 1)*b ^(n + 2)*(c + d*x))), x] - Simp[d^(2*n + 2)/b^(n + 1) Int[(x^m/Sqrt[a + b *x^2])*ExpandToSum[((2^(-n - 1)*(-c)^(m - n - 1))/(d^m*x^m) - (-c + d*x)^(- n - 1))/(c + d*x), x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^ 2, 0] && ILtQ[m, 0] && ILtQ[n, 0] && EqQ[n + p, -3/2]
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. \(248\) vs. \(2(48)=96\).
Time = 0.13 (sec) , antiderivative size = 249, normalized size of antiderivative = 4.79
method | result | size |
default | \(\frac {-\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}-\ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+\left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+2 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a x +2 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x -\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}-a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )}{a \sqrt {a^{2}}\, \left (a x -1\right ) c \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {\frac {a x -1}{a x +1}}}\) | \(249\) |
Input:
int(1/((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c),x,method=_RETURNVERBOSE)
Output:
1/a*(-((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^2)^( 1/2)/(a*x-1)/c/((a*x-1)*(a*x+1))^(1/2)/((a*x-1)/(a*x+1))^(1/2)
Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.67 \[ \int \frac {e^{\coth ^{-1}(a x)}}{c-a c x} \, dx=-\frac {{\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - {\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - 2 \, {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c x - a c} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c),x, algorithm="fricas")
Output:
-((a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - (a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - 2*(a*x + 1)*sqrt((a*x - 1)/(a*x + 1)))/(a^2*c*x - a* c)
\[ \int \frac {e^{\coth ^{-1}(a x)}}{c-a c x} \, dx=- \frac {\int \frac {1}{a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx}{c} \] Input:
integrate(1/((a*x-1)/(a*x+1))**(1/2)/(-a*c*x+c),x)
Output:
-Integral(1/(a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) - sqrt(a*x/(a*x + 1) - 1/(a*x + 1))), x)/c
Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\coth ^{-1}(a x)}}{c-a c x} \, dx=-a {\left (\frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c} - \frac {2}{a^{2} c \sqrt {\frac {a x - 1}{a x + 1}}}\right )} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)/(-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/(a^2*c*sqrt((a*x - 1)/(a*x + 1))))
\[ \int \frac {e^{\coth ^{-1}(a x)}}{c-a c x} \, dx=\int { -\frac {1}{{\left (a c x - c\right )} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c),x, algorithm="giac")
Output:
undef
Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.92 \[ \int \frac {e^{\coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {2}{a\,c\,\sqrt {\frac {a\,x-1}{a\,x+1}}}-\frac {2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c} \] Input:
int(1/((c - a*c*x)*((a*x - 1)/(a*x + 1))^(1/2)),x)
Output:
2/(a*c*((a*x - 1)/(a*x + 1))^(1/2)) - (2*atanh(((a*x - 1)/(a*x + 1))^(1/2) ))/(a*c)
Time = 0.15 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.08 \[ \int \frac {e^{\coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {-2 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right )+2 \sqrt {a x -1}+2 \sqrt {a x +1}}{\sqrt {a x -1}\, a c} \] Input:
int(1/((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c),x)
Output:
(2*( - sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2)) + sqrt(a *x - 1) + sqrt(a*x + 1)))/(sqrt(a*x - 1)*a*c)