Integrand size = 12, antiderivative size = 162 \[ \int \frac {e^{\text {arctanh}(a+b x)}}{x^3} \, dx=-\frac {(1+2 a) b \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 (1-a)^2 (1+a) x}-\frac {\sqrt {1-a-b x} (1+a+b x)^{3/2}}{2 \left (1-a^2\right ) x^2}-\frac {(1+2 a) b^2 \text {arctanh}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {1-a-b x}}\right )}{(1-a)^2 (1+a) \sqrt {1-a^2}} \] Output:
-1/2*(1+2*a)*b*(-b*x-a+1)^(1/2)*(b*x+a+1)^(1/2)/(1-a)^2/(1+a)/x-1/2*(-b*x- a+1)^(1/2)*(b*x+a+1)^(3/2)/(-a^2+1)/x^2-(1+2*a)*b^2*arctanh((1-a)^(1/2)*(b *x+a+1)^(1/2)/(1+a)^(1/2)/(-b*x-a+1)^(1/2))/(1-a)^2/(1+a)/(-a^2+1)^(1/2)
Time = 0.16 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.76 \[ \int \frac {e^{\text {arctanh}(a+b x)}}{x^3} \, dx=\frac {\left (-1+a^2-2 b x-a b x\right ) \sqrt {1-a^2-2 a b x-b^2 x^2}}{2 (-1+a)^2 (1+a) x^2}-\frac {(1+2 a) b^2 \text {arctanh}\left (\frac {\sqrt {-1-a} \sqrt {1-a-b x}}{\sqrt {-1+a} \sqrt {1+a+b x}}\right )}{(-1-a)^{3/2} (-1+a)^{5/2}} \] Input:
Integrate[E^ArcTanh[a + b*x]/x^3,x]
Output:
((-1 + a^2 - 2*b*x - a*b*x)*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2])/(2*(-1 + a) ^2*(1 + a)*x^2) - ((1 + 2*a)*b^2*ArcTanh[(Sqrt[-1 - a]*Sqrt[1 - a - b*x])/ (Sqrt[-1 + a]*Sqrt[1 + a + b*x])])/((-1 - a)^(3/2)*(-1 + a)^(5/2))
Time = 0.33 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6713, 107, 105, 104, 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.
| \(\displaystyle \int \frac {e^{\text {arctanh}(a+b x)}}{x^3} \, dx\) |
| \(\Big \downarrow \) 6713 |
| \(\displaystyle \int \frac {\sqrt {a+b x+1}}{x^3 \sqrt {-a-b x+1}}dx\) |
| \(\Big \downarrow \) 107 |
| \(\displaystyle \frac {(2 a+1) b \int \frac {\sqrt {a+b x+1}}{x^2 \sqrt {-a-b x+1}}dx}{2 \left (1-a^2\right )}-\frac {\sqrt {-a-b x+1} (a+b x+1)^{3/2}}{2 \left (1-a^2\right ) x^2}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {(2 a+1) b \left (\frac {b \int \frac {1}{x \sqrt {-a-b x+1} \sqrt {a+b x+1}}dx}{1-a}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{(1-a) x}\right )}{2 \left (1-a^2\right )}-\frac {\sqrt {-a-b x+1} (a+b x+1)^{3/2}}{2 \left (1-a^2\right ) x^2}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {(2 a+1) b \left (\frac {2 b \int \frac {1}{-a+\frac {(1-a) (a+b x+1)}{-a-b x+1}-1}d\frac {\sqrt {a+b x+1}}{\sqrt {-a-b x+1}}}{1-a}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{(1-a) x}\right )}{2 \left (1-a^2\right )}-\frac {\sqrt {-a-b x+1} (a+b x+1)^{3/2}}{2 \left (1-a^2\right ) x^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(2 a+1) b \left (-\frac {2 b \text {arctanh}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {-a-b x+1}}\right )}{(1-a) \sqrt {1-a^2}}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{(1-a) x}\right )}{2 \left (1-a^2\right )}-\frac {\sqrt {-a-b x+1} (a+b x+1)^{3/2}}{2 \left (1-a^2\right ) x^2}\) |
Input:
Int[E^ArcTanh[a + b*x]/x^3,x]
Output:
-1/2*(Sqrt[1 - a - b*x]*(1 + a + b*x)^(3/2))/((1 - a^2)*x^2) + ((1 + 2*a)* b*(-((Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/((1 - a)*x)) - (2*b*ArcTanh[(Sq rt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/((1 - a)*Sq rt[1 - a^2])))/(2*(1 - a^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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 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[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]
Time = 0.44 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.94
| method | result | size |
| risch | \(-\frac {\left (b^{2} x^{2}+2 a b x +a^{2}-1\right ) \left (-a b x +a^{2}-2 b x -1\right )}{2 \left (-1+a \right ) x^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \left (a^{2}-1\right )}-\frac {b^{2} \left (1+2 a \right ) \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{2 \left (a^{2}-1\right ) \left (-1+a \right ) \sqrt {-a^{2}+1}}\) | \(153\) |
| default | \(b \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{\left (-a^{2}+1\right ) x}-\frac {a b \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}\right )+\left (a +1\right ) \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 \left (-a^{2}+1\right ) x^{2}}+\frac {3 a b \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{\left (-a^{2}+1\right ) x}-\frac {a b \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (-a^{2}+1\right )}-\frac {b^{2} \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{2 \left (-a^{2}+1\right )^{\frac {3}{2}}}\right )\) | \(322\) |
Input:
int((b*x+a+1)/(1-(b*x+a)^2)^(1/2)/x^3,x,method=_RETURNVERBOSE)
Output:
-1/2*(b^2*x^2+2*a*b*x+a^2-1)*(-a*b*x+a^2-2*b*x-1)/(-1+a)/x^2/(-b^2*x^2-2*a *b*x-a^2+1)^(1/2)/(a^2-1)-1/2*b^2*(1+2*a)/(a^2-1)/(-1+a)/(-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)
Time = 0.11 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.22 \[ \int \frac {e^{\text {arctanh}(a+b x)}}{x^3} \, dx=\left [-\frac {\sqrt {-a^{2} + 1} {\left (2 \, a + 1\right )} b^{2} x^{2} \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x + 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right ) - 2 \, {\left (a^{4} - {\left (a^{3} + 2 \, a^{2} - a - 2\right )} b x - 2 \, a^{2} + 1\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{4 \, {\left (a^{5} - a^{4} - 2 \, a^{3} + 2 \, a^{2} + a - 1\right )} x^{2}}, \frac {\sqrt {a^{2} - 1} {\left (2 \, a + 1\right )} b^{2} x^{2} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) + {\left (a^{4} - {\left (a^{3} + 2 \, a^{2} - a - 2\right )} b x - 2 \, a^{2} + 1\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \, {\left (a^{5} - a^{4} - 2 \, a^{3} + 2 \, a^{2} + a - 1\right )} x^{2}}\right ] \] Input:
integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2)/x^3,x, algorithm="fricas")
Output:
[-1/4*(sqrt(-a^2 + 1)*(2*a + 1)*b^2*x^2*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x + 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)* sqrt(-a^2 + 1) - 4*a^2 + 2)/x^2) - 2*(a^4 - (a^3 + 2*a^2 - a - 2)*b*x - 2* a^2 + 1)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^5 - a^4 - 2*a^3 + 2*a^2 + a - 1)*x^2), 1/2*(sqrt(a^2 - 1)*(2*a + 1)*b^2*x^2*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(a^2 - 1)/((a^2 - 1)*b^2*x^2 + a^ 4 + 2*(a^3 - a)*b*x - 2*a^2 + 1)) + (a^4 - (a^3 + 2*a^2 - a - 2)*b*x - 2*a ^2 + 1)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^5 - a^4 - 2*a^3 + 2*a^2 + a - 1)*x^2)]
\[ \int \frac {e^{\text {arctanh}(a+b x)}}{x^3} \, dx=\int \frac {a + b x + 1}{x^{3} \sqrt {- \left (a + b x - 1\right ) \left (a + b x + 1\right )}}\, dx \] Input:
integrate((b*x+a+1)/(1-(b*x+a)**2)**(1/2)/x**3,x)
Output:
Integral((a + b*x + 1)/(x**3*sqrt(-(a + b*x - 1)*(a + b*x + 1))), x)
Exception generated. \[ \int \frac {e^{\text {arctanh}(a+b x)}}{x^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2)/x^3,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
Leaf count of result is larger than twice the leaf count of optimal. 748 vs. \(2 (134) = 268\).
Time = 0.13 (sec) , antiderivative size = 748, normalized size of antiderivative = 4.62 \[ \int \frac {e^{\text {arctanh}(a+b x)}}{x^3} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2)/x^3,x, algorithm="giac")
Output:
(2*a*b^3 + b^3)*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))/((a^3*abs(b) - a^2*abs(b) - a*abs(b) + a bs(b))*sqrt(a^2 - 1)) - (2*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b) ^2*a^4*b^3/(b^2*x + a*b)^2 + 2*a^4*b^3 - 5*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*a^3*b^3/(b^2*x + a*b) + 2*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a^3*b^3/(b^2*x + a*b)^2 - 3*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^3*a^3*b^3/(b^2*x + a*b)^3 + 2*a^3*b^3 - 6*(sqrt(-b^2* x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*a^2*b^3/(b^2*x + a*b) + 3*(sqrt(-b^2* x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a^2*b^3/(b^2*x + a*b)^2 - 2*(sqrt(- b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^3*a^2*b^3/(b^2*x + a*b)^3 - a^2*b ^3 + 2*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*a*b^3/(b^2*x + a*b) + 4*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a*b^3/(b^2*x + a*b) ^2 + 2*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^3*a*b^3/(b^2*x + a* b)^3 - 2*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*b^3/(b^2*x + a* b)^2)/((a^5*abs(b) - a^4*abs(b) - a^3*abs(b) + a^2*abs(b))*((sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a/(b^2*x + a*b)^2 + a - 2*(sqrt(-b^2*x ^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)/(b^2*x + a*b))^2)
Timed out. \[ \int \frac {e^{\text {arctanh}(a+b x)}}{x^3} \, dx=\int \frac {a+b\,x+1}{x^3\,\sqrt {1-{\left (a+b\,x\right )}^2}} \,d x \] Input:
int((a + b*x + 1)/(x^3*(1 - (a + b*x)^2)^(1/2)),x)
Output:
int((a + b*x + 1)/(x^3*(1 - (a + b*x)^2)^(1/2)), x)
Time = 0.15 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.06 \[ \int \frac {e^{\text {arctanh}(a+b x)}}{x^3} \, dx=\frac {-8 \sqrt {a^{2}-1}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (b x +a \right )}{2}\right ) a -1}{\sqrt {a^{2}-1}}\right ) a^{2} b^{2} x^{2}-4 \sqrt {a^{2}-1}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (b x +a \right )}{2}\right ) a -1}{\sqrt {a^{2}-1}}\right ) a \,b^{2} x^{2}+2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{5}-2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{4} b x -4 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{3} b x -4 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{3}+2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{2} b x +4 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a b x +2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a +a^{3} b^{2} x^{2}+2 a^{2} b^{2} x^{2}-a \,b^{2} x^{2}-2 b^{2} x^{2}}{4 a \,x^{2} \left (a^{5}-a^{4}-2 a^{3}+2 a^{2}+a -1\right )} \] Input:
int((b*x+a+1)/(1-(b*x+a)^2)^(1/2)/x^3,x)
Output:
( - 8*sqrt(a**2 - 1)*atan((tan(asin(a + b*x)/2)*a - 1)/sqrt(a**2 - 1))*a** 2*b**2*x**2 - 4*sqrt(a**2 - 1)*atan((tan(asin(a + b*x)/2)*a - 1)/sqrt(a**2 - 1))*a*b**2*x**2 + 2*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a**5 - 2*sq rt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a**4*b*x - 4*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a**3*b*x - 4*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a**3 + 2*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a**2*b*x + 4*sqrt( - a**2 - 2 *a*b*x - b**2*x**2 + 1)*a*b*x + 2*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)* a + a**3*b**2*x**2 + 2*a**2*b**2*x**2 - a*b**2*x**2 - 2*b**2*x**2)/(4*a*x* *2*(a**5 - a**4 - 2*a**3 + 2*a**2 + a - 1))