Integrand size = 14, antiderivative size = 260 \[ \int \frac {e^{3 \text {arctanh}(a+b x)}}{x^4} \, dx=\frac {\left (52+51 a+2 a^2\right ) b^3 \sqrt {1+a+b x}}{6 (1-a)^4 (1+a) \sqrt {1-a-b x}}-\frac {(1+a) \sqrt {1+a+b x}}{3 (1-a) x^3 \sqrt {1-a-b x}}-\frac {7 b \sqrt {1+a+b x}}{6 (1-a)^2 x^2 \sqrt {1-a-b x}}-\frac {(19+16 a) b^2 \sqrt {1+a+b x}}{6 (1-a)^3 (1+a) x \sqrt {1-a-b x}}-\frac {\left (11+18 a+6 a^2\right ) b^3 \text {arctanh}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {1-a-b x}}\right )}{(1-a)^4 (1+a) \sqrt {1-a^2}} \]
-(6*a^2+18*a+11)*b^3*arctanh((1-a)^(1/2)*(b*x+a+1)^(1/2)/(1+a)^(1/2)/(-b*x -a+1)^(1/2))/(1-a)^4/(1+a)/(-a^2+1)^(1/2)+1/6*(2*a^2+51*a+52)*b^3*(b*x+a+1 )^(1/2)/(1-a)^4/(1+a)/(-b*x-a+1)^(1/2)-1/3*(1+a)*(b*x+a+1)^(1/2)/(1-a)/x^3 /(-b*x-a+1)^(1/2)-7/6*b*(b*x+a+1)^(1/2)/(1-a)^2/x^2/(-b*x-a+1)^(1/2)-1/6*( 19+16*a)*b^2*(b*x+a+1)^(1/2)/(1-a)^3/(1+a)/x/(-b*x-a+1)^(1/2)
Time = 0.22 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.77 \[ \int \frac {e^{3 \text {arctanh}(a+b x)}}{x^4} \, dx=-\frac {-2 (-1+a)^{7/2} (1+a) (1+a+b x)^{5/2}+(-1+a)^{5/2} (3+4 a) b x (1+a+b x)^{5/2}-\left (11+18 a+6 a^2\right ) b^2 x^2 \left (\sqrt {-1+a} \sqrt {1+a+b x} \left (-1+a^2+5 b x+a b x\right )-6 \sqrt {-1-a} b x \sqrt {1-a-b x} \text {arctanh}\left (\frac {\sqrt {-1-a} \sqrt {1-a-b x}}{\sqrt {-1+a} \sqrt {1+a+b x}}\right )\right )}{6 (-1+a)^{5/2} \left (-1+a^2\right )^2 x^3 \sqrt {1-a-b x}} \]
-1/6*(-2*(-1 + a)^(7/2)*(1 + a)*(1 + a + b*x)^(5/2) + (-1 + a)^(5/2)*(3 + 4*a)*b*x*(1 + a + b*x)^(5/2) - (11 + 18*a + 6*a^2)*b^2*x^2*(Sqrt[-1 + a]*S qrt[1 + a + b*x]*(-1 + a^2 + 5*b*x + a*b*x) - 6*Sqrt[-1 - a]*b*x*Sqrt[1 - a - b*x]*ArcTanh[(Sqrt[-1 - a]*Sqrt[1 - a - b*x])/(Sqrt[-1 + a]*Sqrt[1 + a + b*x])]))/((-1 + a)^(5/2)*(-1 + a^2)^2*x^3*Sqrt[1 - a - b*x])
Time = 0.45 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.06, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6713, 109, 25, 27, 168, 25, 27, 168, 25, 27, 169, 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.
| \(\displaystyle \int \frac {e^{3 \text {arctanh}(a+b x)}}{x^4} \, dx\) |
| \(\Big \downarrow \) 6713 |
| \(\displaystyle \int \frac {(a+b x+1)^{3/2}}{x^4 (-a-b x+1)^{3/2}}dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle -\frac {\int -\frac {b (7 (a+1)+6 b x)}{x^3 (-a-b x+1)^{3/2} \sqrt {a+b x+1}}dx}{3 (1-a)}-\frac {(a+1) \sqrt {a+b x+1}}{3 (1-a) x^3 \sqrt {-a-b x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {b (7 (a+1)+6 b x)}{x^3 (-a-b x+1)^{3/2} \sqrt {a+b x+1}}dx}{3 (1-a)}-\frac {(a+1) \sqrt {a+b x+1}}{3 (1-a) x^3 \sqrt {-a-b x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \int \frac {7 (a+1)+6 b x}{x^3 (-a-b x+1)^{3/2} \sqrt {a+b x+1}}dx}{3 (1-a)}-\frac {(a+1) \sqrt {a+b x+1}}{3 (1-a) x^3 \sqrt {-a-b x+1}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {b \left (-\frac {\int -\frac {(a+1) b (16 a+14 b x+19)}{x^2 (-a-b x+1)^{3/2} \sqrt {a+b x+1}}dx}{2 \left (1-a^2\right )}-\frac {7 \sqrt {a+b x+1}}{2 (1-a) x^2 \sqrt {-a-b x+1}}\right )}{3 (1-a)}-\frac {(a+1) \sqrt {a+b x+1}}{3 (1-a) x^3 \sqrt {-a-b x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {\int \frac {(a+1) b (16 a+14 b x+19)}{x^2 (-a-b x+1)^{3/2} \sqrt {a+b x+1}}dx}{2 \left (1-a^2\right )}-\frac {7 \sqrt {a+b x+1}}{2 (1-a) x^2 \sqrt {-a-b x+1}}\right )}{3 (1-a)}-\frac {(a+1) \sqrt {a+b x+1}}{3 (1-a) x^3 \sqrt {-a-b x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {(a+1) b \int \frac {16 a+14 b x+19}{x^2 (-a-b x+1)^{3/2} \sqrt {a+b x+1}}dx}{2 \left (1-a^2\right )}-\frac {7 \sqrt {a+b x+1}}{2 (1-a) x^2 \sqrt {-a-b x+1}}\right )}{3 (1-a)}-\frac {(a+1) \sqrt {a+b x+1}}{3 (1-a) x^3 \sqrt {-a-b x+1}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {b \left (\frac {(a+1) b \left (-\frac {\int -\frac {b \left (3 \left (6 a^2+18 a+11\right )+(16 a+19) b x\right )}{x (-a-b x+1)^{3/2} \sqrt {a+b x+1}}dx}{1-a^2}-\frac {(16 a+19) \sqrt {a+b x+1}}{\left (1-a^2\right ) x \sqrt {-a-b x+1}}\right )}{2 \left (1-a^2\right )}-\frac {7 \sqrt {a+b x+1}}{2 (1-a) x^2 \sqrt {-a-b x+1}}\right )}{3 (1-a)}-\frac {(a+1) \sqrt {a+b x+1}}{3 (1-a) x^3 \sqrt {-a-b x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {(a+1) b \left (\frac {\int \frac {b \left (3 \left (6 a^2+18 a+11\right )+(16 a+19) b x\right )}{x (-a-b x+1)^{3/2} \sqrt {a+b x+1}}dx}{1-a^2}-\frac {(16 a+19) \sqrt {a+b x+1}}{\left (1-a^2\right ) x \sqrt {-a-b x+1}}\right )}{2 \left (1-a^2\right )}-\frac {7 \sqrt {a+b x+1}}{2 (1-a) x^2 \sqrt {-a-b x+1}}\right )}{3 (1-a)}-\frac {(a+1) \sqrt {a+b x+1}}{3 (1-a) x^3 \sqrt {-a-b x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {(a+1) b \left (\frac {b \int \frac {3 \left (6 a^2+18 a+11\right )+(16 a+19) b x}{x (-a-b x+1)^{3/2} \sqrt {a+b x+1}}dx}{1-a^2}-\frac {(16 a+19) \sqrt {a+b x+1}}{\left (1-a^2\right ) x \sqrt {-a-b x+1}}\right )}{2 \left (1-a^2\right )}-\frac {7 \sqrt {a+b x+1}}{2 (1-a) x^2 \sqrt {-a-b x+1}}\right )}{3 (1-a)}-\frac {(a+1) \sqrt {a+b x+1}}{3 (1-a) x^3 \sqrt {-a-b x+1}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {b \left (\frac {(a+1) b \left (\frac {b \left (\frac {\left (2 a^2+51 a+52\right ) \sqrt {a+b x+1}}{(1-a) \sqrt {-a-b x+1}}-\frac {\int -\frac {3 \left (6 a^2+18 a+11\right ) b}{x \sqrt {-a-b x+1} \sqrt {a+b x+1}}dx}{(1-a) b}\right )}{1-a^2}-\frac {(16 a+19) \sqrt {a+b x+1}}{\left (1-a^2\right ) x \sqrt {-a-b x+1}}\right )}{2 \left (1-a^2\right )}-\frac {7 \sqrt {a+b x+1}}{2 (1-a) x^2 \sqrt {-a-b x+1}}\right )}{3 (1-a)}-\frac {(a+1) \sqrt {a+b x+1}}{3 (1-a) x^3 \sqrt {-a-b x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {(a+1) b \left (\frac {b \left (\frac {3 \left (6 a^2+18 a+11\right ) \int \frac {1}{x \sqrt {-a-b x+1} \sqrt {a+b x+1}}dx}{1-a}+\frac {\left (2 a^2+51 a+52\right ) \sqrt {a+b x+1}}{(1-a) \sqrt {-a-b x+1}}\right )}{1-a^2}-\frac {(16 a+19) \sqrt {a+b x+1}}{\left (1-a^2\right ) x \sqrt {-a-b x+1}}\right )}{2 \left (1-a^2\right )}-\frac {7 \sqrt {a+b x+1}}{2 (1-a) x^2 \sqrt {-a-b x+1}}\right )}{3 (1-a)}-\frac {(a+1) \sqrt {a+b x+1}}{3 (1-a) x^3 \sqrt {-a-b x+1}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {b \left (\frac {(a+1) b \left (\frac {b \left (\frac {6 \left (6 a^2+18 a+11\right ) \int \frac {1}{-a+\frac {(1-a) (a+b x+1)}{-a-b x+1}-1}d\frac {\sqrt {a+b x+1}}{\sqrt {-a-b x+1}}}{1-a}+\frac {\left (2 a^2+51 a+52\right ) \sqrt {a+b x+1}}{(1-a) \sqrt {-a-b x+1}}\right )}{1-a^2}-\frac {(16 a+19) \sqrt {a+b x+1}}{\left (1-a^2\right ) x \sqrt {-a-b x+1}}\right )}{2 \left (1-a^2\right )}-\frac {7 \sqrt {a+b x+1}}{2 (1-a) x^2 \sqrt {-a-b x+1}}\right )}{3 (1-a)}-\frac {(a+1) \sqrt {a+b x+1}}{3 (1-a) x^3 \sqrt {-a-b x+1}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {(a+1) b \left (\frac {b \left (\frac {\left (2 a^2+51 a+52\right ) \sqrt {a+b x+1}}{(1-a) \sqrt {-a-b x+1}}-\frac {6 \left (6 a^2+18 a+11\right ) \text {arctanh}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {-a-b x+1}}\right )}{(1-a) \sqrt {1-a^2}}\right )}{1-a^2}-\frac {(16 a+19) \sqrt {a+b x+1}}{\left (1-a^2\right ) x \sqrt {-a-b x+1}}\right )}{2 \left (1-a^2\right )}-\frac {7 \sqrt {a+b x+1}}{2 (1-a) x^2 \sqrt {-a-b x+1}}\right )}{3 (1-a)}-\frac {(a+1) \sqrt {a+b x+1}}{3 (1-a) x^3 \sqrt {-a-b x+1}}\) |
-1/3*((1 + a)*Sqrt[1 + a + b*x])/((1 - a)*x^3*Sqrt[1 - a - b*x]) + (b*((-7 *Sqrt[1 + a + b*x])/(2*(1 - a)*x^2*Sqrt[1 - a - b*x]) + ((1 + a)*b*(-(((19 + 16*a)*Sqrt[1 + a + b*x])/((1 - a^2)*x*Sqrt[1 - a - b*x])) + (b*(((52 + 51*a + 2*a^2)*Sqrt[1 + a + b*x])/((1 - a)*Sqrt[1 - a - b*x]) - (6*(11 + 18 *a + 6*a^2)*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)))/(2*(1 - a^2))))/(3*(1 - a))
3.9.42.3.1 Defintions of rubi rules used
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*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_) )^(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] && ILtQ[m, -1]
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[(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 = 0.58 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.00
| method | result | size |
| risch | \(\frac {\left (b^{2} x^{2}+2 a b x +a^{2}-1\right ) \left (2 a^{2} b^{2} x^{2}-2 a^{3} b x +27 a \,b^{2} x^{2}+2 a^{4}-9 a^{2} b x +28 b^{2} x^{2}+2 a b x -4 a^{2}+9 b x +2\right )}{6 \left (-1+a \right )^{3} x^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \left (a^{2}-1\right )}+\frac {b^{3} \left (-\frac {\left (6 a^{2}+18 a +11\right ) \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{\sqrt {-a^{2}+1}}-\frac {8 \left (1+a \right ) \sqrt {-\left (x +\frac {-1+a}{b}\right )^{2} b^{2}-2 \left (x +\frac {-1+a}{b}\right ) b}}{b \left (x +\frac {-1+a}{b}\right )}\right )}{2 \left (a^{2}-1\right ) \left (-1+a \right )^{3}}\) | \(260\) |
| default | \(\text {Expression too large to display}\) | \(1852\) |
1/6*(b^2*x^2+2*a*b*x+a^2-1)*(2*a^2*b^2*x^2-2*a^3*b*x+27*a*b^2*x^2+2*a^4-9* a^2*b*x+28*b^2*x^2+2*a*b*x-4*a^2+9*b*x+2)/(-1+a)^3/x^3/(-b^2*x^2-2*a*b*x-a ^2+1)^(1/2)/(a^2-1)+1/2*b^3/(a^2-1)/(-1+a)^3*(-(6*a^2+18*a+11)/(-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)-8*(1+a)/b/(x+(-1+a)/b)*(-(x+(-1+a)/b)^2*b^2-2*(x+(-1+a)/b)*b)^(1/2))
Time = 0.30 (sec) , antiderivative size = 704, normalized size of antiderivative = 2.71 \[ \int \frac {e^{3 \text {arctanh}(a+b x)}}{x^4} \, dx=\left [-\frac {3 \, {\left ({\left (6 \, a^{2} + 18 \, a + 11\right )} b^{4} x^{4} + {\left (6 \, a^{3} + 12 \, a^{2} - 7 \, a - 11\right )} b^{3} x^{3}\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 \, {\left (2 \, a^{7} + {\left (2 \, a^{4} + 51 \, a^{3} + 50 \, a^{2} - 51 \, a - 52\right )} b^{3} x^{3} - 2 \, a^{6} - 6 \, a^{5} + {\left (16 \, a^{4} + 3 \, a^{3} - 35 \, a^{2} - 3 \, a + 19\right )} b^{2} x^{2} + 6 \, a^{4} + 6 \, a^{3} - 7 \, {\left (a^{5} - a^{4} - 2 \, a^{3} + 2 \, a^{2} + a - 1\right )} b x - 6 \, a^{2} - 2 \, a + 2\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{12 \, {\left ({\left (a^{7} - 3 \, a^{6} + a^{5} + 5 \, a^{4} - 5 \, a^{3} - a^{2} + 3 \, a - 1\right )} b x^{4} + {\left (a^{8} - 4 \, a^{7} + 4 \, a^{6} + 4 \, a^{5} - 10 \, a^{4} + 4 \, a^{3} + 4 \, a^{2} - 4 \, a + 1\right )} x^{3}\right )}}, \frac {3 \, {\left ({\left (6 \, a^{2} + 18 \, a + 11\right )} b^{4} x^{4} + {\left (6 \, a^{3} + 12 \, a^{2} - 7 \, a - 11\right )} b^{3} x^{3}\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 ) - {\left (2 \, a^{7} + {\left (2 \, a^{4} + 51 \, a^{3} + 50 \, a^{2} - 51 \, a - 52\right )} b^{3} x^{3} - 2 \, a^{6} - 6 \, a^{5} + {\left (16 \, a^{4} + 3 \, a^{3} - 35 \, a^{2} - 3 \, a + 19\right )} b^{2} x^{2} + 6 \, a^{4} + 6 \, a^{3} - 7 \, {\left (a^{5} - a^{4} - 2 \, a^{3} + 2 \, a^{2} + a - 1\right )} b x - 6 \, a^{2} - 2 \, a + 2\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{6 \, {\left ({\left (a^{7} - 3 \, a^{6} + a^{5} + 5 \, a^{4} - 5 \, a^{3} - a^{2} + 3 \, a - 1\right )} b x^{4} + {\left (a^{8} - 4 \, a^{7} + 4 \, a^{6} + 4 \, a^{5} - 10 \, a^{4} + 4 \, a^{3} + 4 \, a^{2} - 4 \, a + 1\right )} x^{3}\right )}}\right ] \]
[-1/12*(3*((6*a^2 + 18*a + 11)*b^4*x^4 + (6*a^3 + 12*a^2 - 7*a - 11)*b^3*x ^3)*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*(2*a^7 + (2*a^4 + 51*a^3 + 50*a^2 - 51*a - 52)*b^3*x^3 - 2 *a^6 - 6*a^5 + (16*a^4 + 3*a^3 - 35*a^2 - 3*a + 19)*b^2*x^2 + 6*a^4 + 6*a^ 3 - 7*(a^5 - a^4 - 2*a^3 + 2*a^2 + a - 1)*b*x - 6*a^2 - 2*a + 2)*sqrt(-b^2 *x^2 - 2*a*b*x - a^2 + 1))/((a^7 - 3*a^6 + a^5 + 5*a^4 - 5*a^3 - a^2 + 3*a - 1)*b*x^4 + (a^8 - 4*a^7 + 4*a^6 + 4*a^5 - 10*a^4 + 4*a^3 + 4*a^2 - 4*a + 1)*x^3), 1/6*(3*((6*a^2 + 18*a + 11)*b^4*x^4 + (6*a^3 + 12*a^2 - 7*a - 1 1)*b^3*x^3)*sqrt(a^2 - 1)*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)) - (2*a^7 + (2*a^4 + 51*a^3 + 50*a^2 - 51*a - 52)*b^3*x^3 - 2*a^6 - 6*a^5 + (16*a^4 + 3*a^3 - 35*a^2 - 3*a + 19)*b^2*x^2 + 6*a^4 + 6*a^3 - 7 *(a^5 - a^4 - 2*a^3 + 2*a^2 + a - 1)*b*x - 6*a^2 - 2*a + 2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^7 - 3*a^6 + a^5 + 5*a^4 - 5*a^3 - a^2 + 3*a - 1) *b*x^4 + (a^8 - 4*a^7 + 4*a^6 + 4*a^5 - 10*a^4 + 4*a^3 + 4*a^2 - 4*a + 1)* x^3)]
\[ \int \frac {e^{3 \text {arctanh}(a+b x)}}{x^4} \, dx=\int \frac {\left (a + b x + 1\right )^{3}}{x^{4} \left (- \left (a + b x - 1\right ) \left (a + b x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {e^{3 \text {arctanh}(a+b x)}}{x^4} \, dx=\text {Exception raised: ValueError} \]
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. 1836 vs. \(2 (216) = 432\).
Time = 0.35 (sec) , antiderivative size = 1836, normalized size of antiderivative = 7.06 \[ \int \frac {e^{3 \text {arctanh}(a+b x)}}{x^4} \, dx=\text {Too large to display} \]
8*b^4/((a^4*abs(b) - 4*a^3*abs(b) + 6*a^2*abs(b) - 4*a*abs(b) + abs(b))*(( sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)/(b^2*x + a*b) - 1)) + (6*a^ 2*b^4 + 18*a*b^4 + 11*b^4)*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^5*abs(b) - 3*a^4*abs(b) + 2*a^3*abs(b) + 2*a^2*abs(b) - 3*a*abs(b) + abs(b))*sqrt(a^2 - 1)) - 1/3*( 12*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a^7*b^4/(b^2*x + a*b) ^2 + 6*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^4*a^7*b^4/(b^2*x + a*b)^4 + 6*a^7*b^4 - 24*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*a^ 6*b^4/(b^2*x + a*b) + 72*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2 *a^6*b^4/(b^2*x + a*b)^2 - 36*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^3*a^6*b^4/(b^2*x + a*b)^3 + 36*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs (b) + b)^4*a^6*b^4/(b^2*x + a*b)^4 - 12*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1 )*abs(b) + b)^5*a^6*b^4/(b^2*x + a*b)^5 + 36*a^6*b^4 - 171*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*a^5*b^4/(b^2*x + a*b) + 84*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a^5*b^4/(b^2*x + a*b)^2 - 216*(sqrt(-b ^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^3*a^5*b^4/(b^2*x + a*b)^3 + 54*(sq rt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^4*a^5*b^4/(b^2*x + a*b)^4 - 4 5*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^5*a^5*b^4/(b^2*x + a*b)^ 5 + 22*a^5*b^4 - 120*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*a^4*b ^4/(b^2*x + a*b) + 252*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^...
Timed out. \[ \int \frac {e^{3 \text {arctanh}(a+b x)}}{x^4} \, dx=\int \frac {{\left (a+b\,x+1\right )}^3}{x^4\,{\left (1-{\left (a+b\,x\right )}^2\right )}^{3/2}} \,d x \]