Integrand size = 14, antiderivative size = 72 \[ \int \frac {e^{-\frac {1}{2} \text {arctanh}(a x)}}{x^2} \, dx=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{x}-a \arctan \left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )+a \text {arctanh}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right ) \] Output:
-(-a*x+1)^(1/4)*(a*x+1)^(3/4)/x-a*arctan((a*x+1)^(1/4)/(-a*x+1)^(1/4))+a*a rctanh((a*x+1)^(1/4)/(-a*x+1)^(1/4))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.76 \[ \int \frac {e^{-\frac {1}{2} \text {arctanh}(a x)}}{x^2} \, dx=\frac {\sqrt [4]{1-a x} \left (-1-a x+2 a x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},1,\frac {5}{4},\frac {1-a x}{1+a x}\right )\right )}{x \sqrt [4]{1+a x}} \] Input:
Integrate[1/(E^(ArcTanh[a*x]/2)*x^2),x]
Output:
((1 - a*x)^(1/4)*(-1 - a*x + 2*a*x*Hypergeometric2F1[1/4, 1, 5/4, (1 - a*x )/(1 + a*x)]))/(x*(1 + a*x)^(1/4))
Time = 0.41 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6676, 105, 104, 25, 827, 216, 219}
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.
\(\displaystyle \int \frac {e^{-\frac {1}{2} \text {arctanh}(a x)}}{x^2} \, dx\) |
\(\Big \downarrow \) 6676 |
\(\displaystyle \int \frac {\sqrt [4]{1-a x}}{x^2 \sqrt [4]{a x+1}}dx\) |
\(\Big \downarrow \) 105 |
\(\displaystyle -\frac {1}{2} a \int \frac {1}{x (1-a x)^{3/4} \sqrt [4]{a x+1}}dx-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{x}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle -2 a \int -\frac {\sqrt {a x+1}}{\sqrt {1-a x} \left (1-\frac {a x+1}{1-a x}\right )}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 2 a \int \frac {\sqrt {a x+1}}{\sqrt {1-a x} \left (1-\frac {a x+1}{1-a x}\right )}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{x}\) |
\(\Big \downarrow \) 827 |
\(\displaystyle -2 a \left (\frac {1}{2} \int \frac {1}{\frac {\sqrt {a x+1}}{\sqrt {1-a x}}+1}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a x+1}}{\sqrt {1-a x}}}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{x}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -2 a \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a x+1}}{\sqrt {1-a x}}}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{x}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -2 a \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )\right )-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{x}\) |
Input:
Int[1/(E^(ArcTanh[a*x]/2)*x^2),x]
Output:
-(((1 - a*x)^(1/4)*(1 + a*x)^(3/4))/x) - 2*a*(ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]/2 - ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]/2)
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x) ^m*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c, m, n}, x] && !Int egerQ[(n - 1)/2]
\[\int \frac {1}{\sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}\, x^{2}}d x\]
Input:
int(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^2,x)
Output:
int(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^2,x)
Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (60) = 120\).
Time = 0.08 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.81 \[ \int \frac {e^{-\frac {1}{2} \text {arctanh}(a x)}}{x^2} \, dx=-\frac {2 \, a x \arctan \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) - a x \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) + a x \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{2 \, x} \] Input:
integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^2,x, algorithm="fricas")
Output:
-1/2*(2*a*x*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - a*x*log(sqrt(-sq rt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + a*x*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) + 2*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x
\[ \int \frac {e^{-\frac {1}{2} \text {arctanh}(a x)}}{x^2} \, dx=\int \frac {1}{x^{2} \sqrt {\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}}}\, dx \] Input:
integrate(1/((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/x**2,x)
Output:
Integral(1/(x**2*sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1))), x)
\[ \int \frac {e^{-\frac {1}{2} \text {arctanh}(a x)}}{x^2} \, dx=\int { \frac {1}{x^{2} \sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}} \,d x } \] Input:
integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^2,x, algorithm="maxima")
Output:
integrate(1/(x^2*sqrt((a*x + 1)/sqrt(-a^2*x^2 + 1))), x)
\[ \int \frac {e^{-\frac {1}{2} \text {arctanh}(a x)}}{x^2} \, dx=\int { \frac {1}{x^{2} \sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}} \,d x } \] Input:
integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^2,x, algorithm="giac")
Output:
integrate(1/(x^2*sqrt((a*x + 1)/sqrt(-a^2*x^2 + 1))), x)
Timed out. \[ \int \frac {e^{-\frac {1}{2} \text {arctanh}(a x)}}{x^2} \, dx=\int \frac {1}{x^2\,\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}} \,d x \] Input:
int(1/(x^2*((a*x + 1)/(1 - a^2*x^2)^(1/2))^(1/2)),x)
Output:
int(1/(x^2*((a*x + 1)/(1 - a^2*x^2)^(1/2))^(1/2)), x)
\[ \int \frac {e^{-\frac {1}{2} \text {arctanh}(a x)}}{x^2} \, dx=\frac {-2 \sqrt {a x +1}\, \left (-a^{2} x^{2}+1\right )^{\frac {1}{4}}+\left (\int \frac {\sqrt {a x +1}\, \left (-a^{2} x^{2}+1\right )^{\frac {1}{4}}}{a^{2} x^{4}-x^{2}}d x \right ) x}{x} \] Input:
int(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^2,x)
Output:
( - 2*sqrt(a*x + 1)*( - a**2*x**2 + 1)**(1/4) + int((sqrt(a*x + 1)*( - a** 2*x**2 + 1)**(1/4))/(a**2*x**4 - x**2),x)*x)/x