Integrand size = 14, antiderivative size = 68 \[ \int \frac {e^{-\text {arctanh}(a+b x)}}{x} \, dx=-\arcsin (a+b x)-\frac {2 (1-a) \text {arctanh}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {1-a-b x}}\right )}{\sqrt {1-a^2}} \] Output:
-arcsin(b*x+a)-2*(1-a)*arctanh((1-a)^(1/2)*(b*x+a+1)^(1/2)/(1+a)^(1/2)/(-b *x-a+1)^(1/2))/(-a^2+1)^(1/2)
Time = 0.04 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.56 \[ \int \frac {e^{-\text {arctanh}(a+b x)}}{x} \, dx=\frac {2 \sqrt {-b} \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {1-a-b x}}{\sqrt {2} \sqrt {-b}}\right )}{\sqrt {b}}-\frac {2 \sqrt {-1+a} \text {arctanh}\left (\frac {\sqrt {-1-a} \sqrt {1-a-b x}}{\sqrt {-1+a} \sqrt {1+a+b x}}\right )}{\sqrt {-1-a}} \] Input:
Integrate[1/(E^ArcTanh[a + b*x]*x),x]
Output:
(2*Sqrt[-b]*ArcSinh[(Sqrt[b]*Sqrt[1 - a - b*x])/(Sqrt[2]*Sqrt[-b])])/Sqrt[ b] - (2*Sqrt[-1 + a]*ArcTanh[(Sqrt[-1 - a]*Sqrt[1 - a - b*x])/(Sqrt[-1 + a ]*Sqrt[1 + a + b*x])])/Sqrt[-1 - a]
Time = 0.30 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6713, 140, 27, 62, 104, 221, 1090, 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.
\(\displaystyle \int \frac {e^{-\text {arctanh}(a+b x)}}{x} \, dx\) |
\(\Big \downarrow \) 6713 |
\(\displaystyle \int \frac {\sqrt {-a-b x+1}}{x \sqrt {a+b x+1}}dx\) |
\(\Big \downarrow \) 140 |
\(\displaystyle \int \frac {1-a}{x \sqrt {-a-b x+1} \sqrt {a+b x+1}}dx-b \int \frac {1}{\sqrt {-a-b x+1} \sqrt {a+b x+1}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle (1-a) \int \frac {1}{x \sqrt {-a-b x+1} \sqrt {a+b x+1}}dx-b \int \frac {1}{\sqrt {-a-b x+1} \sqrt {a+b x+1}}dx\) |
\(\Big \downarrow \) 62 |
\(\displaystyle (1-a) \int \frac {1}{x \sqrt {-a-b x+1} \sqrt {a+b x+1}}dx-b \int \frac {1}{\sqrt {-b^2 x^2-2 a b x+(1-a) (a+1)}}dx\) |
\(\Big \downarrow \) 104 |
\(\displaystyle 2 (1-a) \int \frac {1}{-a+\frac {(1-a) (a+b x+1)}{-a-b x+1}-1}d\frac {\sqrt {a+b x+1}}{\sqrt {-a-b x+1}}-b \int \frac {1}{\sqrt {-b^2 x^2-2 a b x+(1-a) (a+1)}}dx\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -b \int \frac {1}{\sqrt {-b^2 x^2-2 a b x+(1-a) (a+1)}}dx-\frac {2 (1-a) \text {arctanh}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {-a-b x+1}}\right )}{\sqrt {1-a^2}}\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle \frac {\int \frac {1}{\sqrt {1-\frac {\left (-2 x b^2-2 a b\right )^2}{4 b^2}}}d\left (-2 x b^2-2 a b\right )}{2 b}-\frac {2 (1-a) \text {arctanh}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {-a-b x+1}}\right )}{\sqrt {1-a^2}}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \arcsin \left (\frac {-2 a b-2 b^2 x}{2 b}\right )-\frac {2 (1-a) \text {arctanh}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {-a-b x+1}}\right )}{\sqrt {1-a^2}}\) |
Input:
Int[1/(E^ArcTanh[a + b*x]*x),x]
Output:
ArcSin[(-2*a*b - 2*b^2*x)/(2*b)] - (2*(1 - a)*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/Sqrt[1 - a^2]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Int[ 1/Sqrt[a*c - b*(a - c)*x - b^2*x^2], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b + d, 0] && GtQ[a + c, 0]
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[b*d^(m + n)*f^p Int[(a + b*x)^(m - 1)/(c + d*x)^m, x] , x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandToSum[(a + b*x )*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 0] || SumSimplerQ[m, -1] || !(GtQ[n, 0] || SumSimplerQ[n, -1]))
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[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* (c/(b^2 - 4*a*c)))^p) Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]
Int[E^(ArcTanh[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.) , x_Symbol] :> Int[(d + e*x)^m*((1 + a*c + b*c*x)^(n/2)/(1 - a*c - b*c*x)^( n/2)), x] /; FreeQ[{a, b, c, d, e, m, n}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(234\) vs. \(2(58)=116\).
Time = 0.42 (sec) , antiderivative size = 235, normalized size of antiderivative = 3.46
method | result | size |
default | \(\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}-\frac {a b \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{\sqrt {b^{2}}}-\sqrt {-a^{2}+1}\, \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{a +1}-\frac {\sqrt {-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b}+\frac {b \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a +1}{b}-\frac {1}{b}\right )}{\sqrt {-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b}}\right )}{\sqrt {b^{2}}}}{a +1}\) | \(235\) |
Input:
int(1/(b*x+a+1)*(1-(b*x+a)^2)^(1/2)/x,x,method=_RETURNVERBOSE)
Output:
1/(a+1)*((-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-a*b/(b^2)^(1/2)*arctan((b^2)^(1/2) *(x+1/b*a)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))-(-a^2+1)^(1/2)*ln((-2*a^2+2-2*a *b*x+2*(-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))/x))-1/(a+1)*((-(x+(a +1)/b)^2*b^2+2*(x+(a+1)/b)*b)^(1/2)+b/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+(a +1)/b-1/b)/(-(x+(a+1)/b)^2*b^2+2*(x+(a+1)/b)*b)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (56) = 112\).
Time = 0.12 (sec) , antiderivative size = 303, normalized size of antiderivative = 4.46 \[ \int \frac {e^{-\text {arctanh}(a+b x)}}{x} \, dx=\left [\frac {1}{2} \, \sqrt {-\frac {a - 1}{a + 1}} \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x - 4 \, a^{2} + 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{3} + {\left (a^{2} + a\right )} b x + a^{2} - a - 1\right )} \sqrt {-\frac {a - 1}{a + 1}} + 2}{x^{2}}\right ) + \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ), -\sqrt {\frac {a - 1}{a + 1}} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {\frac {a - 1}{a + 1}}}{{\left (a - 1\right )} b^{2} x^{2} + a^{3} + 2 \, {\left (a^{2} - a\right )} b x - a^{2} - a + 1}\right ) + \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )\right ] \] Input:
integrate(1/(b*x+a+1)*(1-(b*x+a)^2)^(1/2)/x,x, algorithm="fricas")
Output:
[1/2*sqrt(-(a - 1)/(a + 1))*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a) *b*x - 4*a^2 + 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^3 + (a^2 + a)*b*x + a^2 - a - 1)*sqrt(-(a - 1)/(a + 1)) + 2)/x^2) + arctan(sqrt(-b^2*x^2 - 2* a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)), -sqrt((a - 1)/( a + 1))*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(( a - 1)/(a + 1))/((a - 1)*b^2*x^2 + a^3 + 2*(a^2 - a)*b*x - a^2 - a + 1)) + arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1))]
\[ \int \frac {e^{-\text {arctanh}(a+b x)}}{x} \, dx=\int \frac {\sqrt {- \left (a + b x - 1\right ) \left (a + b x + 1\right )}}{x \left (a + b x + 1\right )}\, dx \] Input:
integrate(1/(b*x+a+1)*(1-(b*x+a)**2)**(1/2)/x,x)
Output:
Integral(sqrt(-(a + b*x - 1)*(a + b*x + 1))/(x*(a + b*x + 1)), x)
Exception generated. \[ \int \frac {e^{-\text {arctanh}(a+b x)}}{x} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(b*x+a+1)*(1-(b*x+a)^2)^(1/2)/x,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a-1>0)', see `assume?` for more details)Is
Time = 0.14 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.31 \[ \int \frac {e^{-\text {arctanh}(a+b x)}}{x} \, dx=\frac {b \arcsin \left (-b x - a\right ) \mathrm {sgn}\left (b\right )}{{\left | b \right |}} - \frac {2 \, {\left (a b - b\right )} \arctan \left (\frac {\frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a}{b^{2} x + a b} - 1}{\sqrt {a^{2} - 1}}\right )}{\sqrt {a^{2} - 1} {\left | b \right |}} \] Input:
integrate(1/(b*x+a+1)*(1-(b*x+a)^2)^(1/2)/x,x, algorithm="giac")
Output:
b*arcsin(-b*x - a)*sgn(b)/abs(b) - 2*(a*b - b)*arctan(((sqrt(-b^2*x^2 - 2* a*b*x - a^2 + 1)*abs(b) + b)*a/(b^2*x + a*b) - 1)/sqrt(a^2 - 1))/(sqrt(a^2 - 1)*abs(b))
Timed out. \[ \int \frac {e^{-\text {arctanh}(a+b x)}}{x} \, dx=\int \frac {\sqrt {1-{\left (a+b\,x\right )}^2}}{x\,\left (a+b\,x+1\right )} \,d x \] Input:
int((1 - (a + b*x)^2)^(1/2)/(x*(a + b*x + 1)),x)
Output:
int((1 - (a + b*x)^2)^(1/2)/(x*(a + b*x + 1)), x)
Time = 0.15 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.81 \[ \int \frac {e^{-\text {arctanh}(a+b x)}}{x} \, dx=\frac {-\mathit {asin} \left (b x +a \right ) a -\mathit {asin} \left (b x +a \right )+2 \sqrt {a^{2}-1}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (b x +a \right )}{2}\right ) a -1}{\sqrt {a^{2}-1}}\right )}{a +1} \] Input:
int(1/(b*x+a+1)*(1-(b*x+a)^2)^(1/2)/x,x)
Output:
( - asin(a + b*x)*a - asin(a + b*x) + 2*sqrt(a**2 - 1)*atan((tan(asin(a + b*x)/2)*a - 1)/sqrt(a**2 - 1)))/(a + 1)