Integrand size = 34, antiderivative size = 134 \[ \int \frac {e^{\text {arctanh}(a+b x)} x^3}{1-a^2-2 a b x-b^2 x^2} \, dx=\frac {(1-a) x^2 \sqrt {1+a+b x}}{b^2 \sqrt {1-a-b x}}+\frac {\left (7-14 a+4 a^2\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^4}-\frac {(3-2 a) (1-a-b x)^{3/2} \sqrt {1+a+b x}}{2 b^4}-\frac {3 \left (1-2 a+2 a^2\right ) \arcsin (a+b x)}{2 b^4} \] Output:
(1-a)*x^2*(b*x+a+1)^(1/2)/b^2/(-b*x-a+1)^(1/2)+1/2*(4*a^2-14*a+7)*(-b*x-a+ 1)^(1/2)*(b*x+a+1)^(1/2)/b^4-1/2*(3-2*a)*(-b*x-a+1)^(3/2)*(b*x+a+1)^(1/2)/ b^4-3/2*(2*a^2-2*a+1)*arcsin(b*x+a)/b^4
Time = 0.16 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.67 \[ \int \frac {e^{\text {arctanh}(a+b x)} x^3}{1-a^2-2 a b x-b^2 x^2} \, dx=-\frac {-\frac {\sqrt {1-a^2-2 a b x-b^2 x^2} \left (-4-11 a^2+2 a^3+b x+b^2 x^2+a (13-4 b x)\right )}{-1+a+b x}+3 \left (1-2 a+2 a^2\right ) \arcsin (a+b x)}{2 b^4} \] Input:
Integrate[(E^ArcTanh[a + b*x]*x^3)/(1 - a^2 - 2*a*b*x - b^2*x^2),x]
Output:
-1/2*(-((Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*(-4 - 11*a^2 + 2*a^3 + b*x + b^ 2*x^2 + a*(13 - 4*b*x)))/(-1 + a + b*x)) + 3*(1 - 2*a + 2*a^2)*ArcSin[a + b*x])/b^4
Time = 0.67 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {6714, 109, 164, 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.
| \(\displaystyle \int \frac {x^3 e^{\text {arctanh}(a+b x)}}{-a^2-2 a b x-b^2 x^2+1} \, dx\) |
| \(\Big \downarrow \) 6714 |
| \(\displaystyle \int \frac {x^3}{(-a-b x+1)^{3/2} \sqrt {a+b x+1}}dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {(1-a) x^2 \sqrt {a+b x+1}}{b^2 \sqrt {-a-b x+1}}-\frac {\int \frac {x \left (2 \left (1-a^2\right )+(3-2 a) b x\right )}{\sqrt {-a-b x+1} \sqrt {a+b x+1}}dx}{b^2}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {(1-a) x^2 \sqrt {a+b x+1}}{b^2 \sqrt {-a-b x+1}}-\frac {\frac {3 \left (2 a^2-2 a+1\right ) \int \frac {1}{\sqrt {-a-b x+1} \sqrt {a+b x+1}}dx}{2 b}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1} ((3-2 a) b x+(1-2 a) (4-a))}{2 b^2}}{b^2}\) |
| \(\Big \downarrow \) 62 |
| \(\displaystyle \frac {(1-a) x^2 \sqrt {a+b x+1}}{b^2 \sqrt {-a-b x+1}}-\frac {\frac {3 \left (2 a^2-2 a+1\right ) \int \frac {1}{\sqrt {-b^2 x^2-2 a b x+(1-a) (a+1)}}dx}{2 b}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1} ((3-2 a) b x+(1-2 a) (4-a))}{2 b^2}}{b^2}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {(1-a) x^2 \sqrt {a+b x+1}}{b^2 \sqrt {-a-b x+1}}-\frac {-\frac {3 \left (2 a^2-2 a+1\right ) \int \frac {1}{\sqrt {1-\frac {\left (-2 x b^2-2 a b\right )^2}{4 b^2}}}d\left (-2 x b^2-2 a b\right )}{4 b^3}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1} ((3-2 a) b x+(1-2 a) (4-a))}{2 b^2}}{b^2}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {(1-a) x^2 \sqrt {a+b x+1}}{b^2 \sqrt {-a-b x+1}}-\frac {-\frac {3 \left (2 a^2-2 a+1\right ) \arcsin \left (\frac {-2 a b-2 b^2 x}{2 b}\right )}{2 b^2}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1} ((3-2 a) b x+(1-2 a) (4-a))}{2 b^2}}{b^2}\) |
Input:
Int[(E^ArcTanh[a + b*x]*x^3)/(1 - a^2 - 2*a*b*x - b^2*x^2),x]
Output:
((1 - a)*x^2*Sqrt[1 + a + b*x])/(b^2*Sqrt[1 - a - b*x]) - (-1/2*(Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x]*((1 - 2*a)*(4 - a) + (3 - 2*a)*b*x))/b^2 - (3*( 1 - 2*a + 2*a^2)*ArcSin[(-2*a*b - 2*b^2*x)/(2*b)])/(2*b^2))/b^2
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_) )^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ ))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h *(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)) Int[( a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 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[(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])
Leaf count of result is larger than twice the leaf count of optimal. \(254\) vs. \(2(116)=232\).
Time = 0.52 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.90
| method | result | size |
| risch | \(\frac {\left (-b x +5 a -2\right ) \left (b^{2} x^{2}+2 a b x +a^{2}-1\right )}{2 b^{4} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {\frac {\left (-2 a^{3}+6 a^{2}-6 a +2\right ) \sqrt {-\left (x +\frac {-1+a}{b}\right )^{2} b^{2}-2 \left (x +\frac {-1+a}{b}\right ) b}}{b^{2} \left (x +\frac {-1+a}{b}\right )}+\frac {3 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{\sqrt {b^{2}}}-\frac {6 a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{\sqrt {b^{2}}}+\frac {6 a^{2} \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{\sqrt {b^{2}}}}{2 b^{3}}\) | \(255\) |
| default | \(b \left (-\frac {x^{3}}{2 b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {5 a \left (-\frac {x^{2}}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {3 a \left (\frac {x}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {a \left (\frac {1}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {2 a \left (-2 b^{2} x -2 a b \right )}{b \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 )}{b}-\frac {\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 )}{b^{2} \sqrt {b^{2}}}\right )}{b}+\frac {2 \left (-a^{2}+1\right ) \left (\frac {1}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {2 a \left (-2 b^{2} x -2 a b \right )}{b \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 )}{b^{2}}\right )}{2 b}+\frac {3 \left (-a^{2}+1\right ) \left (\frac {x}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {a \left (\frac {1}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {2 a \left (-2 b^{2} x -2 a b \right )}{b \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 )}{b}-\frac {\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 )}{b^{2} \sqrt {b^{2}}}\right )}{2 b^{2}}\right )+\left (a +1\right ) \left (-\frac {x^{2}}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {3 a \left (\frac {x}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {a \left (\frac {1}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {2 a \left (-2 b^{2} x -2 a b \right )}{b \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 )}{b}-\frac {\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 )}{b^{2} \sqrt {b^{2}}}\right )}{b}+\frac {2 \left (-a^{2}+1\right ) \left (\frac {1}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {2 a \left (-2 b^{2} x -2 a b \right )}{b \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 )}{b^{2}}\right )\) | \(840\) |
Input:
int((b*x+a+1)/(1-(b*x+a)^2)^(1/2)*x^3/(-b^2*x^2-2*a*b*x-a^2+1),x,method=_R ETURNVERBOSE)
Output:
1/2*(-b*x+5*a-2)*(b^2*x^2+2*a*b*x+a^2-1)/b^4/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2 )-1/2/b^3*((-2*a^3+6*a^2-6*a+2)/b^2/(x+(-1+a)/b)*(-(x+(-1+a)/b)^2*b^2-2*(x +(-1+a)/b)*b)^(1/2)+3/(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))-6*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))+6*a^2/(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 = 0.09 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.12 \[ \int \frac {e^{\text {arctanh}(a+b x)} x^3}{1-a^2-2 a b x-b^2 x^2} \, dx=\frac {3 \, {\left (2 \, a^{3} + {\left (2 \, a^{2} - 2 \, a + 1\right )} b x - 4 \, a^{2} + 3 \, a - 1\right )} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + {\left (b^{2} x^{2} + 2 \, a^{3} - {\left (4 \, a - 1\right )} b x - 11 \, a^{2} + 13 \, a - 4\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \, {\left (b^{5} x + {\left (a - 1\right )} b^{4}\right )}} \] Input:
integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2)*x^3/(-b^2*x^2-2*a*b*x-a^2+1),x, al gorithm="fricas")
Output:
1/2*(3*(2*a^3 + (2*a^2 - 2*a + 1)*b*x - 4*a^2 + 3*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)) + (b^2*x ^2 + 2*a^3 - (4*a - 1)*b*x - 11*a^2 + 13*a - 4)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/(b^5*x + (a - 1)*b^4)
\[ \int \frac {e^{\text {arctanh}(a+b x)} x^3}{1-a^2-2 a b x-b^2 x^2} \, dx=- \int \frac {x^{3}}{a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} + b x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} - \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}\, dx \] Input:
integrate((b*x+a+1)/(1-(b*x+a)**2)**(1/2)*x**3/(-b**2*x**2-2*a*b*x-a**2+1) ,x)
Output:
-Integral(x**3/(a*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1) + b*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1) - sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)), x)
Leaf count of result is larger than twice the leaf count of optimal. 576 vs. \(2 (115) = 230\).
Time = 0.28 (sec) , antiderivative size = 576, normalized size of antiderivative = 4.30 \[ \int \frac {e^{\text {arctanh}(a+b x)} x^3}{1-a^2-2 a b x-b^2 x^2} \, dx=\frac {{\left (\frac {2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{3}}{b^{6} x + a b^{5} - b^{5}} - \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2} b}{b^{7} x + a b^{6} + b^{6}} - \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2} b}{b^{7} x + a b^{6} - b^{6}} + \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2}}{b^{6} x + a b^{5} + b^{5}} - \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2}}{b^{6} x + a b^{5} - b^{5}} + \frac {6 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{b^{6} x + a b^{5} - b^{5}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b}{b^{7} x + a b^{6} + b^{6}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b}{b^{7} x + a b^{6} - b^{6}} + \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{6} x + a b^{5} + b^{5}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{6} x + a b^{5} - b^{5}} - \frac {6 \, a^{2} \arcsin \left (b x + a\right )}{b^{5}} + \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} x}{b^{4}} + \frac {6 \, a \arcsin \left (b x + a\right )}{b^{5}} - \frac {5 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{b^{5}} - \frac {3 \, \arcsin \left (b x + a\right )}{b^{5}} + \frac {2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{5}}\right )} b^{2}}{2 \, \sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}} \] Input:
integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2)*x^3/(-b^2*x^2-2*a*b*x-a^2+1),x, al gorithm="maxima")
Output:
1/2*(2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a^3/(b^6*x + a*b^5 - b^5) - 3*sq rt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a^2*b/(b^7*x + a*b^6 + b^6) - 3*sqrt(-b^2 *x^2 - 2*a*b*x - a^2 + 1)*a^2*b/(b^7*x + a*b^6 - b^6) + 3*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a^2/(b^6*x + a*b^5 + b^5) - 3*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a^2/(b^6*x + a*b^5 - b^5) + 6*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1) *a/(b^6*x + a*b^5 - b^5) - sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b/(b^7*x + a *b^6 + b^6) - sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b/(b^7*x + a*b^6 - b^6) + sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)/(b^6*x + a*b^5 + b^5) - sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)/(b^6*x + a*b^5 - b^5) - 6*a^2*arcsin(b*x + a)/b^5 + s qrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*x/b^4 + 6*a*arcsin(b*x + a)/b^5 - 5*sqrt (-b^2*x^2 - 2*a*b*x - a^2 + 1)*a/b^5 - 3*arcsin(b*x + a)/b^5 + 2*sqrt(-b^2 *x^2 - 2*a*b*x - a^2 + 1)/b^5)*b^2/sqrt(a^2*b^2 - (a^2 - 1)*b^2)
Time = 0.17 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.07 \[ \int \frac {e^{\text {arctanh}(a+b x)} x^3}{1-a^2-2 a b x-b^2 x^2} \, dx=\frac {1}{2} \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (\frac {x}{b^{3}} - \frac {5 \, a b^{6} - 2 \, b^{6}}{b^{10}}\right )} + \frac {3 \, {\left (2 \, a^{2} - 2 \, a + 1\right )} \arcsin \left (-b x - a\right ) \mathrm {sgn}\left (b\right )}{2 \, b^{3} {\left | b \right |}} - \frac {2 \, {\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )}}{b^{3} {\left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b}{b^{2} x + a b} - 1\right )} {\left | b \right |}} \] Input:
integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2)*x^3/(-b^2*x^2-2*a*b*x-a^2+1),x, al gorithm="giac")
Output:
1/2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(x/b^3 - (5*a*b^6 - 2*b^6)/b^10) + 3/2*(2*a^2 - 2*a + 1)*arcsin(-b*x - a)*sgn(b)/(b^3*abs(b)) - 2*(a^3 - 3*a^ 2 + 3*a - 1)/(b^3*((sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)/(b^2*x + a*b) - 1)*abs(b))
Timed out. \[ \int \frac {e^{\text {arctanh}(a+b x)} x^3}{1-a^2-2 a b x-b^2 x^2} \, dx=-\int \frac {x^3\,\left (a+b\,x+1\right )}{\sqrt {1-{\left (a+b\,x\right )}^2}\,\left (a^2+2\,a\,b\,x+b^2\,x^2-1\right )} \,d x \] Input:
int(-(x^3*(a + b*x + 1))/((1 - (a + b*x)^2)^(1/2)*(a^2 + b^2*x^2 + 2*a*b*x - 1)),x)
Output:
-int((x^3*(a + b*x + 1))/((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 = 428, normalized size of antiderivative = 3.19 \[ \int \frac {e^{\text {arctanh}(a+b x)} x^3}{1-a^2-2 a b x-b^2 x^2} \, dx=\frac {6-11 a -6 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \mathit {asin} \left (b x +a \right ) a^{2}+6 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \mathit {asin} \left (b x +a \right ) a -3 \mathit {asin} \left (b x +a \right ) b x +3 a \,b^{2} x^{2}-3 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \mathit {asin} \left (b x +a \right )+9 a^{2} b x -4 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a b x -2 b^{2} x^{2}-6 \mathit {asin} \left (b x +a \right ) a^{3}+12 \mathit {asin} \left (b x +a \right ) a^{2}-9 \mathit {asin} \left (b x +a \right ) a +4 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{3}-17 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{2}+13 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a -b^{3} x^{3}+4 a^{2}+a^{3}+\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b^{2} x^{2}+\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b x -6 \mathit {asin} \left (b x +a \right ) a^{2} b x +6 \mathit {asin} \left (b x +a \right ) a b x +3 \mathit {asin} \left (b x +a \right )-6 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}-10 a b x +b x}{2 b^{4} \left (\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}+a +b x -1\right )} \] Input:
int((b*x+a+1)/(1-(b*x+a)^2)^(1/2)*x^3/(-b^2*x^2-2*a*b*x-a^2+1),x)
Output:
( - 6*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)*a**2 + 6*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)*a - 3*sqrt( - a**2 - 2*a* b*x - b**2*x**2 + 1)*asin(a + b*x) - 6*asin(a + b*x)*a**3 - 6*asin(a + b*x )*a**2*b*x + 12*asin(a + b*x)*a**2 + 6*asin(a + b*x)*a*b*x - 9*asin(a + b* x)*a - 3*asin(a + b*x)*b*x + 3*asin(a + b*x) + 4*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a**3 - 17*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a**2 - 4* sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a*b*x + 13*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**2*x**2 + s qrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*b*x - 6*sqrt( - a**2 - 2*a*b*x - b* *2*x**2 + 1) + a**3 + 9*a**2*b*x + 4*a**2 + 3*a*b**2*x**2 - 10*a*b*x - 11* a - b**3*x**3 - 2*b**2*x**2 + b*x + 6)/(2*b**4*(sqrt( - a**2 - 2*a*b*x - b **2*x**2 + 1) + a + b*x - 1))