Integrand size = 12, antiderivative size = 46 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x} \, dx=-\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}+\csc ^{-1}(a x)+\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right ) \] Output:
-4*a*(1-1/a^2/x^2)^(1/2)/(a-1/x)+arccsc(a*x)+arctanh((1-1/a^2/x^2)^(1/2))
Time = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.15 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x} \, dx=-\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}} x}{-1+a x}+\arcsin \left (\frac {1}{a x}\right )+\log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right ) \] Input:
Integrate[E^(3*ArcCoth[a*x])/x,x]
Output:
(-4*a*Sqrt[1 - 1/(a^2*x^2)]*x)/(-1 + a*x) + ArcSin[1/(a*x)] + Log[(1 + Sqr t[1 - 1/(a^2*x^2)])*x]
Time = 0.98 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.80, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {6719, 2351, 27, 564, 25, 27, 243, 73, 221, 671, 223}
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.
\(\displaystyle \int \frac {e^{3 \coth ^{-1}(a x)}}{x} \, dx\) |
\(\Big \downarrow \) 6719 |
\(\displaystyle -\int \frac {\left (1+\frac {1}{a x}\right )^2 x}{\sqrt {1-\frac {1}{a^2 x^2}} \left (1-\frac {1}{a x}\right )}d\frac {1}{x}\) |
\(\Big \downarrow \) 2351 |
\(\displaystyle -\int \frac {\frac {2}{a}+\frac {1}{x a^2}}{\sqrt {1-\frac {1}{a^2 x^2}} \left (1-\frac {1}{a x}\right )}d\frac {1}{x}-\int \frac {a x}{\sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )}d\frac {1}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\int \frac {\frac {2}{a}+\frac {1}{x a^2}}{\sqrt {1-\frac {1}{a^2 x^2}} \left (1-\frac {1}{a x}\right )}d\frac {1}{x}-a \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )}d\frac {1}{x}\) |
\(\Big \downarrow \) 564 |
\(\displaystyle -\int \frac {\frac {2}{a}+\frac {1}{x a^2}}{\sqrt {1-\frac {1}{a^2 x^2}} \left (1-\frac {1}{a x}\right )}d\frac {1}{x}-a \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}-\int -\frac {x}{a \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\frac {2}{a}+\frac {1}{x a^2}}{\sqrt {1-\frac {1}{a^2 x^2}} \left (1-\frac {1}{a x}\right )}d\frac {1}{x}-a \left (\int \frac {x}{a \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\int \frac {\frac {2}{a}+\frac {1}{x a^2}}{\sqrt {1-\frac {1}{a^2 x^2}} \left (1-\frac {1}{a x}\right )}d\frac {1}{x}-a \left (\frac {\int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{a}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}\right )\) |
\(\Big \downarrow \) 243 |
\(\displaystyle -\int \frac {\frac {2}{a}+\frac {1}{x a^2}}{\sqrt {1-\frac {1}{a^2 x^2}} \left (1-\frac {1}{a x}\right )}d\frac {1}{x}-a \left (\frac {\int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}}{2 a}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -a \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}-a \int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}\right )-\int \frac {\frac {2}{a}+\frac {1}{x a^2}}{\sqrt {1-\frac {1}{a^2 x^2}} \left (1-\frac {1}{a x}\right )}d\frac {1}{x}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\int \frac {\frac {2}{a}+\frac {1}{x a^2}}{\sqrt {1-\frac {1}{a^2 x^2}} \left (1-\frac {1}{a x}\right )}d\frac {1}{x}-a \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}\right )\) |
\(\Big \downarrow \) 671 |
\(\displaystyle \frac {\int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{a}-\left (a \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}\right )\right )-\frac {3 a \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle -\left (a \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}\right )\right )-\frac {3 a \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}+\arcsin \left (\frac {1}{a x}\right )\) |
Input:
Int[E^(3*ArcCoth[a*x])/x,x]
Output:
(-3*a*Sqrt[1 - 1/(a^2*x^2)])/(a - x^(-1)) + ArcSin[1/(a*x)] - a*(Sqrt[1 - 1/(a^2*x^2)]/(a - x^(-1)) - ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]/a)
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[(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[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_ ), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + c*x^2)^(p + 1)/(2*c*d*(m + p + 1))), x] + Simp[(m*(g*c*d + c*e*f) + 2*e*c*f*(p + 1))/(e*(2*c*d)*(m + p + 1)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && ((LtQ[m, -1] && !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]
Int[((Px_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.))/(x_), x_S ymbol] :> Int[PolynomialQuotient[Px, x, x]*(c + d*x)^n*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, x, x] Int[(c + d*x)^n*((a + b*x^2)^p/x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && PolynomialQ[Px, x]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(x_)^(m_.), x_Symbol] :> -Subst[Int[(1 + x/a)^((n + 1)/2)/(x^(m + 2)*(1 - x/a)^((n - 1)/2)*Sqrt[1 - x^2/a^2]), x], x , 1/x] /; FreeQ[a, x] && IntegerQ[(n - 1)/2] && IntegerQ[m]
Leaf count of result is larger than twice the leaf count of optimal. \(371\) vs. \(2(42)=84\).
Time = 0.09 (sec) , antiderivative size = 372, normalized size of antiderivative = 8.09
method | result | size |
default | \(-\frac {-\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}-\sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{2} x^{2}-a^{2} \sqrt {a^{2}}\, x^{2} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-\ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+2 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+2 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a x +2 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x +2 a \sqrt {a^{2}}\, x \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+2 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x -\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}-\arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {a^{2}}-a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )}{\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(372\) |
Input:
int(1/((a*x-1)/(a*x+1))^(3/2)/x,x,method=_RETURNVERBOSE)
Output:
-(-((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*a^2*x^2-(a^2)^(1/2)*(a^2*x^2-1)^(1/ 2)*a^2*x^2-a^2*(a^2)^(1/2)*x^2*arctan(1/(a^2*x^2-1)^(1/2))-ln((a^2*x+((a*x -1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*a^3*x^2+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*(a^2)^(1/2)*( a^2*x^2-1)^(1/2)*a*x+2*a*(a^2)^(1/2)*x*arctan(1/(a^2*x^2-1)^(1/2))+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^2*x^2-1)^(1/2)*(a^2)^(1/2)-arctan(1/(a^2*x^2-1)^ (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)*(a*x+1))^(1/2)/(a*x+1)/((a*x-1)/(a*x+1))^(3/2 )
Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (42) = 84\).
Time = 0.07 (sec) , antiderivative size = 104, normalized size of antiderivative = 2.26 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x} \, dx=-\frac {2 \, {\left (a x - 1\right )} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + {\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + 4 \, {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x - 1} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/x,x, algorithm="fricas")
Output:
-(2*(a*x - 1)*arctan(sqrt((a*x - 1)/(a*x + 1))) - (a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) + (a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + 4*(a *x + 1)*sqrt((a*x - 1)/(a*x + 1)))/(a*x - 1)
\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x} \, dx=\int \frac {1}{x \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/((a*x-1)/(a*x+1))**(3/2)/x,x)
Output:
Integral(1/(x*((a*x - 1)/(a*x + 1))**(3/2)), x)
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (42) = 84\).
Time = 0.11 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.96 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x} \, dx=-a {\left (\frac {2 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a} - \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a} + \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a} + \frac {4}{a \sqrt {\frac {a x - 1}{a x + 1}}}\right )} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/x,x, algorithm="maxima")
Output:
-a*(2*arctan(sqrt((a*x - 1)/(a*x + 1)))/a - log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a + log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a + 4/(a*sqrt((a*x - 1)/(a*x + 1))))
\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x} \, dx=\int { \frac {1}{x \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/x,x, algorithm="giac")
Output:
undef
Time = 23.36 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.17 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x} \, dx=2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )-2\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )-\frac {4}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \] Input:
int(1/(x*((a*x - 1)/(a*x + 1))^(3/2)),x)
Output:
2*atanh(((a*x - 1)/(a*x + 1))^(1/2)) - 2*atan(((a*x - 1)/(a*x + 1))^(1/2)) - 4/((a*x - 1)/(a*x + 1))^(1/2)
Time = 0.16 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.13 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x} \, dx=\frac {-2 \sqrt {a x -1}\, \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}-1\right )+2 \sqrt {a x -1}\, \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}+1\right )+2 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right )-4 \sqrt {a x -1}-4 \sqrt {a x +1}}{\sqrt {a x -1}} \] Input:
int(1/((a*x-1)/(a*x+1))^(3/2)/x,x)
Output:
(2*( - sqrt(a*x - 1)*atan(sqrt(a*x - 1) + sqrt(a*x + 1) - 1) + sqrt(a*x - 1)*atan(sqrt(a*x - 1) + sqrt(a*x + 1) + 1) + sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2)) - 2*sqrt(a*x - 1) - 2*sqrt(a*x + 1)))/sqrt(a *x - 1)