Integrand size = 12, antiderivative size = 84 \[ \int e^{-\text {arctanh}(a+b x)} x \, dx=-\frac {(1+2 a) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^2}-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x}}{2 b^2}-\frac {(1+2 a) \arcsin (a+b x)}{2 b^2} \] Output:
-1/2*(1+2*a)*(-b*x-a+1)^(1/2)*(b*x+a+1)^(1/2)/b^2-1/2*(-b*x-a+1)^(3/2)*(b* x+a+1)^(1/2)/b^2-1/2*(1+2*a)*arcsin(b*x+a)/b^2
Time = 0.11 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.18 \[ \int e^{-\text {arctanh}(a+b x)} x \, dx=\frac {\sqrt {1+a+b x} \left (-2+a+a^2+3 b x-b^2 x^2\right )}{2 b^2 \sqrt {1-a-b x}}+\frac {(1+2 a) \sqrt {b} \text {arcsinh}\left (\frac {\sqrt {-b} \sqrt {1-a-b x}}{\sqrt {2} \sqrt {b}}\right )}{(-b)^{5/2}} \] Input:
Integrate[x/E^ArcTanh[a + b*x],x]
Output:
(Sqrt[1 + a + b*x]*(-2 + a + a^2 + 3*b*x - b^2*x^2))/(2*b^2*Sqrt[1 - a - b *x]) + ((1 + 2*a)*Sqrt[b]*ArcSinh[(Sqrt[-b]*Sqrt[1 - a - b*x])/(Sqrt[2]*Sq rt[b])])/(-b)^(5/2)
Time = 0.29 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6713, 90, 60, 62, 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.
\(\displaystyle \int x e^{-\text {arctanh}(a+b x)} \, dx\) |
\(\Big \downarrow \) 6713 |
\(\displaystyle \int \frac {x \sqrt {-a-b x+1}}{\sqrt {a+b x+1}}dx\) |
\(\Big \downarrow \) 90 |
\(\displaystyle -\frac {(2 a+1) \int \frac {\sqrt {-a-b x+1}}{\sqrt {a+b x+1}}dx}{2 b}-\frac {\sqrt {a+b x+1} (-a-b x+1)^{3/2}}{2 b^2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle -\frac {(2 a+1) \left (\int \frac {1}{\sqrt {-a-b x+1} \sqrt {a+b x+1}}dx+\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{b}\right )}{2 b}-\frac {\sqrt {a+b x+1} (-a-b x+1)^{3/2}}{2 b^2}\) |
\(\Big \downarrow \) 62 |
\(\displaystyle -\frac {(2 a+1) \left (\int \frac {1}{\sqrt {-b^2 x^2-2 a b x+(1-a) (a+1)}}dx+\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{b}\right )}{2 b}-\frac {\sqrt {a+b x+1} (-a-b x+1)^{3/2}}{2 b^2}\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle -\frac {(2 a+1) \left (\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{b}-\frac {\int \frac {1}{\sqrt {1-\frac {\left (-2 x b^2-2 a b\right )^2}{4 b^2}}}d\left (-2 x b^2-2 a b\right )}{2 b^2}\right )}{2 b}-\frac {\sqrt {a+b x+1} (-a-b x+1)^{3/2}}{2 b^2}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle -\frac {(2 a+1) \left (\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{b}-\frac {\arcsin \left (\frac {-2 a b-2 b^2 x}{2 b}\right )}{b}\right )}{2 b}-\frac {\sqrt {a+b x+1} (-a-b x+1)^{3/2}}{2 b^2}\) |
Input:
Int[x/E^ArcTanh[a + b*x],x]
Output:
-1/2*((1 - a - b*x)^(3/2)*Sqrt[1 + a + b*x])/b^2 - ((1 + 2*a)*((Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/b - ArcSin[(-2*a*b - 2*b^2*x)/(2*b)]/b))/(2*b)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
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]
Time = 0.38 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.24
method | result | size |
risch | \(\frac {\left (-b x +a +2\right ) \left (b^{2} x^{2}+2 a b x +a^{2}-1\right )}{2 b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {\left (1+2 a \right ) \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{2 b \sqrt {b^{2}}}\) | \(104\) |
default | \(\frac {-\frac {\left (-2 b^{2} x -2 a b \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{4 b^{2}}-\frac {\left (-4 b^{2} \left (-a^{2}+1\right )-4 a^{2} b^{2}\right ) \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{8 b^{2} \sqrt {b^{2}}}}{b}-\frac {\left (a +1\right ) \left (\sqrt {-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b}+\frac {b \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a +1}{b}-\frac {1}{b}\right )}{\sqrt {-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b}}\right )}{\sqrt {b^{2}}}\right )}{b^{2}}\) | \(211\) |
Input:
int(x/(b*x+a+1)*(1-(b*x+a)^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2*(-b*x+a+2)*(b^2*x^2+2*a*b*x+a^2-1)/b^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)- 1/2/b*(1+2*a)/(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))
Time = 0.10 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.08 \[ \int e^{-\text {arctanh}(a+b x)} x \, dx=\frac {{\left (2 \, a + 1\right )} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x - a - 2\right )}}{2 \, b^{2}} \] Input:
integrate(x/(b*x+a+1)*(1-(b*x+a)^2)^(1/2),x, algorithm="fricas")
Output:
1/2*((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)) + sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x - a - 2) )/b^2
\[ \int e^{-\text {arctanh}(a+b x)} x \, dx=\int \frac {x \sqrt {- \left (a + b x - 1\right ) \left (a + b x + 1\right )}}{a + b x + 1}\, dx \] Input:
integrate(x/(b*x+a+1)*(1-(b*x+a)**2)**(1/2),x)
Output:
Integral(x*sqrt(-(a + b*x - 1)*(a + b*x + 1))/(a + b*x + 1), x)
Time = 0.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.27 \[ \int e^{-\text {arctanh}(a+b x)} x \, dx=\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} x}{2 \, b} - \frac {a \arcsin \left (b x + a\right )}{b^{2}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{2 \, b^{2}} - \frac {\arcsin \left (b x + a\right )}{2 \, b^{2}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{2}} \] Input:
integrate(x/(b*x+a+1)*(1-(b*x+a)^2)^(1/2),x, algorithm="maxima")
Output:
1/2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*x/b - a*arcsin(b*x + a)/b^2 - 1/2*s qrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a/b^2 - 1/2*arcsin(b*x + a)/b^2 - sqrt(- b^2*x^2 - 2*a*b*x - a^2 + 1)/b^2
Time = 0.13 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.81 \[ \int e^{-\text {arctanh}(a+b x)} x \, dx=\frac {1}{2} \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (\frac {x}{b} - \frac {a b + 2 \, b}{b^{3}}\right )} + \frac {{\left (2 \, a + 1\right )} \arcsin \left (-b x - a\right ) \mathrm {sgn}\left (b\right )}{2 \, b {\left | b \right |}} \] Input:
integrate(x/(b*x+a+1)*(1-(b*x+a)^2)^(1/2),x, algorithm="giac")
Output:
1/2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(x/b - (a*b + 2*b)/b^3) + 1/2*(2*a + 1)*arcsin(-b*x - a)*sgn(b)/(b*abs(b))
Timed out. \[ \int e^{-\text {arctanh}(a+b x)} x \, dx=\int \frac {x\,\sqrt {1-{\left (a+b\,x\right )}^2}}{a+b\,x+1} \,d x \] Input:
int((x*(1 - (a + b*x)^2)^(1/2))/(a + b*x + 1),x)
Output:
int((x*(1 - (a + b*x)^2)^(1/2))/(a + b*x + 1), x)
Time = 0.15 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.17 \[ \int e^{-\text {arctanh}(a+b x)} x \, dx=\frac {-2 \mathit {asin} \left (b x +a \right ) a -\mathit {asin} \left (b x +a \right )-\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a +\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b x -2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}+2 a +2}{2 b^{2}} \] Input:
int(x/(b*x+a+1)*(1-(b*x+a)^2)^(1/2),x)
Output:
( - 2*asin(a + b*x)*a - asin(a + b*x) - sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a + sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*b*x - 2*sqrt( - a**2 - 2 *a*b*x - b**2*x**2 + 1) + 2*a + 2)/(2*b**2)