Integrand size = 34, antiderivative size = 150 \[ \int \frac {e^{\text {arctanh}(a+b x)}}{x^2 \left (1-a^2-2 a b x-b^2 x^2\right )} \, dx=\frac {(2+a) b \sqrt {1+a+b x}}{(1-a)^2 (1+a) \sqrt {1-a-b x}}-\frac {\sqrt {1+a+b x}}{\left (1-a^2\right ) x \sqrt {1-a-b x}}-\frac {2 (1+2 a) b \text {arctanh}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {1-a-b x}}\right )}{(1-a)^2 (1+a) \sqrt {1-a^2}} \] Output:
(2+a)*b*(b*x+a+1)^(1/2)/(1-a)^2/(1+a)/(-b*x-a+1)^(1/2)-(b*x+a+1)^(1/2)/(-a ^2+1)/x/(-b*x-a+1)^(1/2)-2*(1+2*a)*b*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 = 0.17 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.99 \[ \int \frac {e^{\text {arctanh}(a+b x)}}{x^2 \left (1-a^2-2 a b x-b^2 x^2\right )} \, dx=-\frac {\sqrt {1-a^2-2 a b x-b^2 x^2} \left (\frac {1}{x+a x}+\frac {b}{-1+a+b x}\right )-\frac {(1+2 a) b \log (x)}{(1+a) \sqrt {1-a^2}}+\frac {(1+2 a) b \log \left (1-a^2-a b x+\sqrt {1-a^2} \sqrt {1-a^2-2 a b x-b^2 x^2}\right )}{(1+a) \sqrt {1-a^2}}}{(-1+a)^2} \] Input:
Integrate[E^ArcTanh[a + b*x]/(x^2*(1 - a^2 - 2*a*b*x - b^2*x^2)),x]
Output:
-((Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*((x + a*x)^(-1) + b/(-1 + a + b*x)) - ((1 + 2*a)*b*Log[x])/((1 + a)*Sqrt[1 - a^2]) + ((1 + 2*a)*b*Log[1 - a^2 - a*b*x + Sqrt[1 - a^2]*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]])/((1 + a)*Sqrt[1 - a^2]))/(-1 + a)^2)
Time = 0.41 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.265, Rules used = {6714, 114, 25, 27, 169, 25, 27, 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.
| \(\displaystyle \int \frac {e^{\text {arctanh}(a+b x)}}{x^2 \left (-a^2-2 a b x-b^2 x^2+1\right )} \, dx\) |
| \(\Big \downarrow \) 6714 |
| \(\displaystyle \int \frac {1}{x^2 (-a-b x+1)^{3/2} \sqrt {a+b x+1}}dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {\int -\frac {b (2 a+b x+1)}{x (-a-b x+1)^{3/2} \sqrt {a+b x+1}}dx}{1-a^2}-\frac {\sqrt {a+b x+1}}{\left (1-a^2\right ) x \sqrt {-a-b x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {b (2 a+b x+1)}{x (-a-b x+1)^{3/2} \sqrt {a+b x+1}}dx}{1-a^2}-\frac {\sqrt {a+b x+1}}{\left (1-a^2\right ) x \sqrt {-a-b x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \int \frac {2 a+b x+1}{x (-a-b x+1)^{3/2} \sqrt {a+b x+1}}dx}{1-a^2}-\frac {\sqrt {a+b x+1}}{\left (1-a^2\right ) x \sqrt {-a-b x+1}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {b \left (\frac {(a+2) \sqrt {a+b x+1}}{(1-a) \sqrt {-a-b x+1}}-\frac {\int -\frac {(2 a+1) b}{x \sqrt {-a-b x+1} \sqrt {a+b x+1}}dx}{(1-a) b}\right )}{1-a^2}-\frac {\sqrt {a+b x+1}}{\left (1-a^2\right ) x \sqrt {-a-b x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {\int \frac {(2 a+1) b}{x \sqrt {-a-b x+1} \sqrt {a+b x+1}}dx}{(1-a) b}+\frac {(a+2) \sqrt {a+b x+1}}{(1-a) \sqrt {-a-b x+1}}\right )}{1-a^2}-\frac {\sqrt {a+b x+1}}{\left (1-a^2\right ) x \sqrt {-a-b x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {(2 a+1) \int \frac {1}{x \sqrt {-a-b x+1} \sqrt {a+b x+1}}dx}{1-a}+\frac {(a+2) \sqrt {a+b x+1}}{(1-a) \sqrt {-a-b x+1}}\right )}{1-a^2}-\frac {\sqrt {a+b x+1}}{\left (1-a^2\right ) x \sqrt {-a-b x+1}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {b \left (\frac {2 (2 a+1) \int \frac {1}{-a+\frac {(1-a) (a+b x+1)}{-a-b x+1}-1}d\frac {\sqrt {a+b x+1}}{\sqrt {-a-b x+1}}}{1-a}+\frac {(a+2) \sqrt {a+b x+1}}{(1-a) \sqrt {-a-b x+1}}\right )}{1-a^2}-\frac {\sqrt {a+b x+1}}{\left (1-a^2\right ) x \sqrt {-a-b x+1}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {(a+2) \sqrt {a+b x+1}}{(1-a) \sqrt {-a-b x+1}}-\frac {2 (2 a+1) \text {arctanh}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {-a-b x+1}}\right )}{(1-a) \sqrt {1-a^2}}\right )}{1-a^2}-\frac {\sqrt {a+b x+1}}{\left (1-a^2\right ) x \sqrt {-a-b x+1}}\) |
Input:
Int[E^ArcTanh[a + b*x]/(x^2*(1 - a^2 - 2*a*b*x - b^2*x^2)),x]
Output:
-(Sqrt[1 + a + b*x]/((1 - a^2)*x*Sqrt[1 - a - b*x])) + (b*(((2 + a)*Sqrt[1 + a + b*x])/((1 - a)*Sqrt[1 - a - b*x]) - (2*(1 + 2*a)*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/((1 - a)*Sqrt[1 - a^2])))/(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[(((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*(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[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*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(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] + S imp[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*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
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[(a_) + (b_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_) + (e_.)*( x_)^2)^(p_.), x_Symbol] :> Simp[(c/(1 - a^2))^p Int[u*(1 - a - b*x)^(p - n/2)*(1 + a + b*x)^(p + n/2), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d - 2*a*e, 0] && EqQ[b^2*c + e*(1 - a^2), 0] && (IntegerQ[p] || GtQ[c /(1 - a^2), 0])
Time = 0.78 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.25
| method | result | size |
| risch | \(\frac {b^{2} x^{2}+2 a b x +a^{2}-1}{\left (a^{2}-1\right ) \left (-1+a \right ) x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {b \left (-\frac {\left (1+2 a \right ) \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{\sqrt {-a^{2}+1}}-\frac {\left (a +1\right ) \sqrt {-\left (x +\frac {-1+a}{b}\right )^{2} b^{2}-2 \left (x +\frac {-1+a}{b}\right ) b}}{b \left (x +\frac {-1+a}{b}\right )}\right )}{\left (-1+a \right ) \left (a^{2}-1\right )}\) | \(188\) |
| default | \(b \left (\frac {1}{\left (-a^{2}+1\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {2 a b \left (-2 b^{2} x -2 a b \right )}{\left (-a^{2}+1\right ) \left (-4 b^{2} \left (-a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {\ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}\right )+\left (a +1\right ) \left (-\frac {1}{\left (-a^{2}+1\right ) x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {3 a b \left (\frac {1}{\left (-a^{2}+1\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {2 a b \left (-2 b^{2} x -2 a b \right )}{\left (-a^{2}+1\right ) \left (-4 b^{2} \left (-a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {\ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}\right )}{-a^{2}+1}+\frac {4 b^{2} \left (-2 b^{2} x -2 a b \right )}{\left (-a^{2}+1\right ) \left (-4 b^{2} \left (-a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )\) | \(454\) |
Input:
int((b*x+a+1)/(1-(b*x+a)^2)^(1/2)/x^2/(-b^2*x^2-2*a*b*x-a^2+1),x,method=_R ETURNVERBOSE)
Output:
1/(a^2-1)/(-1+a)*(b^2*x^2+2*a*b*x+a^2-1)/x/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+ 1/(-1+a)/(a^2-1)*b*(-(1+2*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)-1/b*(a+1)/(x+(-1+a)/b)*(-(x+(-1 +a)/b)^2*b^2-2*(x+(-1+a)/b)*b)^(1/2))
Time = 0.12 (sec) , antiderivative size = 459, normalized size of antiderivative = 3.06 \[ \int \frac {e^{\text {arctanh}(a+b x)}}{x^2 \left (1-a^2-2 a b x-b^2 x^2\right )} \, dx=\left [-\frac {{\left ({\left (2 \, a + 1\right )} b^{2} x^{2} + {\left (2 \, a^{2} - a - 1\right )} b x\right )} \sqrt {-a^{2} + 1} \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x + 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right ) + 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{3} + {\left (a^{3} + 2 \, a^{2} - a - 2\right )} b x - a^{2} - a + 1\right )}}{2 \, {\left ({\left (a^{5} - a^{4} - 2 \, a^{3} + 2 \, a^{2} + a - 1\right )} b x^{2} + {\left (a^{6} - 2 \, a^{5} - a^{4} + 4 \, a^{3} - a^{2} - 2 \, a + 1\right )} x\right )}}, \frac {{\left ({\left (2 \, a + 1\right )} b^{2} x^{2} + {\left (2 \, a^{2} - a - 1\right )} b x\right )} \sqrt {a^{2} - 1} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) - \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{3} + {\left (a^{3} + 2 \, a^{2} - a - 2\right )} b x - a^{2} - a + 1\right )}}{{\left (a^{5} - a^{4} - 2 \, a^{3} + 2 \, a^{2} + a - 1\right )} b x^{2} + {\left (a^{6} - 2 \, a^{5} - a^{4} + 4 \, a^{3} - a^{2} - 2 \, a + 1\right )} x}\right ] \] Input:
integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2)/x^2/(-b^2*x^2-2*a*b*x-a^2+1),x, al gorithm="fricas")
Output:
[-1/2*(((2*a + 1)*b^2*x^2 + (2*a^2 - a - 1)*b*x)*sqrt(-a^2 + 1)*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*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^3 + (a^3 + 2*a^2 - a - 2)*b*x - a^2 - a + 1))/((a ^5 - a^4 - 2*a^3 + 2*a^2 + a - 1)*b*x^2 + (a^6 - 2*a^5 - a^4 + 4*a^3 - a^2 - 2*a + 1)*x), (((2*a + 1)*b^2*x^2 + (2*a^2 - a - 1)*b*x)*sqrt(a^2 - 1)*a rctan(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)) - sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^3 + (a^3 + 2*a^2 - a - 2)*b*x - a^2 - a + 1))/((a^5 - a^4 - 2*a^3 + 2*a^2 + a - 1)*b*x^2 + (a^6 - 2*a^5 - a^4 + 4*a^3 - a^2 - 2*a + 1)*x)]
\[ \int \frac {e^{\text {arctanh}(a+b x)}}{x^2 \left (1-a^2-2 a b x-b^2 x^2\right )} \, dx=- \int \frac {1}{a x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} + b x^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} - x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}\, dx \] Input:
integrate((b*x+a+1)/(1-(b*x+a)**2)**(1/2)/x**2/(-b**2*x**2-2*a*b*x-a**2+1) ,x)
Output:
-Integral(1/(a*x**2*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1) + b*x**3*sqrt(-a **2 - 2*a*b*x - b**2*x**2 + 1) - x**2*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1 )), x)
\[ \int \frac {e^{\text {arctanh}(a+b x)}}{x^2 \left (1-a^2-2 a b x-b^2 x^2\right )} \, dx=\int { -\frac {b x + a + 1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )} \sqrt {-{\left (b x + a\right )}^{2} + 1} x^{2}} \,d x } \] Input:
integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2)/x^2/(-b^2*x^2-2*a*b*x-a^2+1),x, al gorithm="maxima")
Output:
-integrate((b*x + a + 1)/((b^2*x^2 + 2*a*b*x + a^2 - 1)*sqrt(-(b*x + a)^2 + 1)*x^2), x)
Leaf count of result is larger than twice the leaf count of optimal. 627 vs. \(2 (125) = 250\).
Time = 0.15 (sec) , antiderivative size = 627, normalized size of antiderivative = 4.18 \[ \int \frac {e^{\text {arctanh}(a+b x)}}{x^2 \left (1-a^2-2 a b x-b^2 x^2\right )} \, dx=\frac {2 \, {\left (2 \, a b^{2} + b^{2}\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 )}{{\left (a^{3} {\left | b \right |} - a^{2} {\left | b \right |} - a {\left | b \right |} + {\left | b \right |}\right )} \sqrt {a^{2} - 1}} + \frac {2 \, {\left (\frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a^{3} b^{2}}{{\left (b^{2} x + a b\right )}^{2}} + a^{3} b^{2} - \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a^{2} b^{2}}{b^{2} x + a b} + \frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a^{2} b^{2}}{{\left (b^{2} x + a b\right )}^{2}} + a^{2} b^{2} - \frac {3 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a b^{2}}{b^{2} x + a b} + a b^{2} - \frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} b^{2}}{b^{2} x + a b} + \frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} b^{2}}{{\left (b^{2} x + a b\right )}^{2}}\right )}}{{\left (a^{4} {\left | b \right |} - a^{3} {\left | b \right |} - a^{2} {\left | b \right |} + a {\left | b \right |}\right )} {\left (\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} - \frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a}{{\left (b^{2} x + a b\right )}^{2}} + \frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{3} a}{{\left (b^{2} x + a b\right )}^{3}} - a + \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}}{b^{2} x + a b} - \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2}}{{\left (b^{2} x + a b\right )}^{2}}\right )}} \] Input:
integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2)/x^2/(-b^2*x^2-2*a*b*x-a^2+1),x, al gorithm="giac")
Output:
2*(2*a*b^2 + b^2)*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) + abs(b))*sqrt(a^2 - 1)) + 2*((sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a^3*b^2/(b^2*x + a*b)^2 + a^3*b^2 - 2*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*a^2*b^2/(b^2*x + a*b) + (sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a^2*b^2/(b^2*x + a*b)^2 + a^2*b^2 - 3*(sqrt(-b^2*x^2 - 2* a*b*x - a^2 + 1)*abs(b) + b)*a*b^2/(b^2*x + a*b) + a*b^2 - (sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*b^2/(b^2*x + a*b) + (sqrt(-b^2*x^2 - 2*a* b*x - a^2 + 1)*abs(b) + b)^2*b^2/(b^2*x + a*b)^2)/((a^4*abs(b) - a^3*abs(b ) - a^2*abs(b) + a*abs(b))*((sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b )*a/(b^2*x + a*b) - (sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a/(b ^2*x + a*b)^2 + (sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^3*a/(b^2*x + a*b)^3 - a + 2*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)/(b^2*x + a*b) - 2*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2/(b^2*x + a*b)^ 2))
Timed out. \[ \int \frac {e^{\text {arctanh}(a+b x)}}{x^2 \left (1-a^2-2 a b x-b^2 x^2\right )} \, dx=\int -\frac {a+b\,x+1}{x^2\,\sqrt {1-{\left (a+b\,x\right )}^2}\,\left (a^2+2\,a\,b\,x+b^2\,x^2-1\right )} \,d x \] Input:
int(-(a + b*x + 1)/(x^2*(1 - (a + b*x)^2)^(1/2)*(a^2 + b^2*x^2 + 2*a*b*x - 1)),x)
Output:
int(-(a + b*x + 1)/(x^2*(1 - (a + b*x)^2)^(1/2)*(a^2 + b^2*x^2 + 2*a*b*x - 1)), x)
Time = 0.16 (sec) , antiderivative size = 1078, normalized size of antiderivative = 7.19 \[ \int \frac {e^{\text {arctanh}(a+b x)}}{x^2 \left (1-a^2-2 a b x-b^2 x^2\right )} \, dx =\text {Too large to display} \] Input:
int((b*x+a+1)/(1-(b*x+a)^2)^(1/2)/x^2/(-b^2*x^2-2*a*b*x-a^2+1),x)
Output:
(2*b*( - 2*sqrt(a**2 - 1)*atan((tan(asin(a + b*x)/2)*a - 1)/sqrt(a**2 - 1) )*tan(asin(a + b*x)/2)**3*a**3 - 5*sqrt(a**2 - 1)*atan((tan(asin(a + b*x)/ 2)*a - 1)/sqrt(a**2 - 1))*tan(asin(a + b*x)/2)**3*a**2 - 2*sqrt(a**2 - 1)* atan((tan(asin(a + b*x)/2)*a - 1)/sqrt(a**2 - 1))*tan(asin(a + b*x)/2)**3* a + 2*sqrt(a**2 - 1)*atan((tan(asin(a + b*x)/2)*a - 1)/sqrt(a**2 - 1))*tan (asin(a + b*x)/2)**2*a**3 + 9*sqrt(a**2 - 1)*atan((tan(asin(a + b*x)/2)*a - 1)/sqrt(a**2 - 1))*tan(asin(a + b*x)/2)**2*a**2 + 12*sqrt(a**2 - 1)*atan ((tan(asin(a + b*x)/2)*a - 1)/sqrt(a**2 - 1))*tan(asin(a + b*x)/2)**2*a + 4*sqrt(a**2 - 1)*atan((tan(asin(a + b*x)/2)*a - 1)/sqrt(a**2 - 1))*tan(asi n(a + b*x)/2)**2 - 2*sqrt(a**2 - 1)*atan((tan(asin(a + b*x)/2)*a - 1)/sqrt (a**2 - 1))*tan(asin(a + b*x)/2)*a**3 - 9*sqrt(a**2 - 1)*atan((tan(asin(a + b*x)/2)*a - 1)/sqrt(a**2 - 1))*tan(asin(a + b*x)/2)*a**2 - 12*sqrt(a**2 - 1)*atan((tan(asin(a + b*x)/2)*a - 1)/sqrt(a**2 - 1))*tan(asin(a + b*x)/2 )*a - 4*sqrt(a**2 - 1)*atan((tan(asin(a + b*x)/2)*a - 1)/sqrt(a**2 - 1))*t an(asin(a + b*x)/2) + 2*sqrt(a**2 - 1)*atan((tan(asin(a + b*x)/2)*a - 1)/s qrt(a**2 - 1))*a**3 + 5*sqrt(a**2 - 1)*atan((tan(asin(a + b*x)/2)*a - 1)/s qrt(a**2 - 1))*a**2 + 2*sqrt(a**2 - 1)*atan((tan(asin(a + b*x)/2)*a - 1)/s qrt(a**2 - 1))*a - tan(asin(a + b*x)/2)**3*a**5 - tan(asin(a + b*x)/2)**3* a**4 + tan(asin(a + b*x)/2)**3*a**3 + tan(asin(a + b*x)/2)**3 - tan(asin(a + b*x)/2)*a**5 - tan(asin(a + b*x)/2)*a**4 + 6*tan(asin(a + b*x)/2)*a*...