Integrand size = 19, antiderivative size = 220 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 (c-a c x)^3} \, dx=\frac {2 \sqrt {1-a^2 x^2}}{5 c^3 x^3 (1-a x)^3}+\frac {7 \sqrt {1-a^2 x^2}}{5 c^3 x^3 (1-a x)^2}+\frac {424 a^3 \sqrt {1-a^2 x^2}}{15 c^3 (1-a x)}-\frac {32 \sqrt {1-a^2 x^2}}{15 c^3 x^3 (1-a x)}-\frac {58 a \sqrt {1-a^2 x^2}}{15 c^3 x^2 (1-a x)}-\frac {154 a^2 \sqrt {1-a^2 x^2}}{15 c^3 x (1-a x)}-\frac {18 a^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{c^3} \] Output:
2/5*(-a^2*x^2+1)^(1/2)/c^3/x^3/(-a*x+1)^3+7/5*(-a^2*x^2+1)^(1/2)/c^3/x^3/( -a*x+1)^2+424/15*a^3*(-a^2*x^2+1)^(1/2)/c^3/(-a*x+1)-32/15*(-a^2*x^2+1)^(1 /2)/c^3/x^3/(-a*x+1)-58/15*a*(-a^2*x^2+1)^(1/2)/c^3/x^2/(-a*x+1)-154/15*a^ 2*(-a^2*x^2+1)^(1/2)/c^3/x/(-a*x+1)-18*a^3*arctanh((-a^2*x^2+1)^(1/2))/c^3
Time = 0.05 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.55 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 (c-a c x)^3} \, dx=\frac {-5-20 a x-85 a^2 x^2+604 a^3 x^3-328 a^4 x^4-578 a^5 x^5+424 a^6 x^6-270 a^3 x^3 (-1+a x)^2 \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{15 c^3 x^3 (-1+a x)^2 \sqrt {1-a^2 x^2}} \] Input:
Integrate[E^ArcTanh[a*x]/(x^4*(c - a*c*x)^3),x]
Output:
(-5 - 20*a*x - 85*a^2*x^2 + 604*a^3*x^3 - 328*a^4*x^4 - 578*a^5*x^5 + 424* a^6*x^6 - 270*a^3*x^3*(-1 + a*x)^2*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2* x^2]])/(15*c^3*x^3*(-1 + a*x)^2*Sqrt[1 - a^2*x^2])
Time = 0.84 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.80, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.895, Rules used = {6678, 27, 570, 532, 25, 2336, 27, 2336, 27, 2338, 25, 2338, 27, 534, 243, 73, 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.
| \(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{x^4 (c-a c x)^3} \, dx\) |
| \(\Big \downarrow \) 6678 |
| \(\displaystyle c \int \frac {\sqrt {1-a^2 x^2}}{c^4 x^4 (1-a x)^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {1-a^2 x^2}}{x^4 (1-a x)^4}dx}{c^3}\) |
| \(\Big \downarrow \) 570 |
| \(\displaystyle \frac {\int \frac {(a x+1)^4}{x^4 \left (1-a^2 x^2\right )^{7/2}}dx}{c^3}\) |
| \(\Big \downarrow \) 532 |
| \(\displaystyle \frac {\frac {8 a^3 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {1}{5} \int -\frac {32 a^4 x^4+40 a^3 x^3+35 a^2 x^2+20 a x+5}{x^4 \left (1-a^2 x^2\right )^{5/2}}dx}{c^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{5} \int \frac {32 a^4 x^4+40 a^3 x^3+35 a^2 x^2+20 a x+5}{x^4 \left (1-a^2 x^2\right )^{5/2}}dx+\frac {8 a^3 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {4 a^3 (6 a x+5)}{\left (1-a^2 x^2\right )^{3/2}}-\frac {1}{3} \int -\frac {3 \left (48 a^4 x^4+60 a^3 x^3+40 a^2 x^2+20 a x+5\right )}{x^4 \left (1-a^2 x^2\right )^{3/2}}dx\right )+\frac {8 a^3 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{5} \left (\int \frac {48 a^4 x^4+60 a^3 x^3+40 a^2 x^2+20 a x+5}{x^4 \left (1-a^2 x^2\right )^{3/2}}dx+\frac {4 a^3 (6 a x+5)}{\left (1-a^2 x^2\right )^{3/2}}\right )+\frac {8 a^3 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {1}{5} \left (-\int -\frac {5 \left (16 a^3 x^3+9 a^2 x^2+4 a x+1\right )}{x^4 \sqrt {1-a^2 x^2}}dx+\frac {a^3 (93 a x+80)}{\sqrt {1-a^2 x^2}}+\frac {4 a^3 (6 a x+5)}{\left (1-a^2 x^2\right )^{3/2}}\right )+\frac {8 a^3 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{5} \left (5 \int \frac {16 a^3 x^3+9 a^2 x^2+4 a x+1}{x^4 \sqrt {1-a^2 x^2}}dx+\frac {a^3 (93 a x+80)}{\sqrt {1-a^2 x^2}}+\frac {4 a^3 (6 a x+5)}{\left (1-a^2 x^2\right )^{3/2}}\right )+\frac {8 a^3 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {\frac {1}{5} \left (5 \left (-\frac {1}{3} \int -\frac {48 x^2 a^3+29 x a^2+12 a}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )+\frac {a^3 (93 a x+80)}{\sqrt {1-a^2 x^2}}+\frac {4 a^3 (6 a x+5)}{\left (1-a^2 x^2\right )^{3/2}}\right )+\frac {8 a^3 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{5} \left (5 \left (\frac {1}{3} \int \frac {48 x^2 a^3+29 x a^2+12 a}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )+\frac {a^3 (93 a x+80)}{\sqrt {1-a^2 x^2}}+\frac {4 a^3 (6 a x+5)}{\left (1-a^2 x^2\right )^{3/2}}\right )+\frac {8 a^3 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {\frac {1}{5} \left (5 \left (\frac {1}{3} \left (-\frac {1}{2} \int -\frac {2 a^2 (54 a x+29)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {6 a \sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )+\frac {a^3 (93 a x+80)}{\sqrt {1-a^2 x^2}}+\frac {4 a^3 (6 a x+5)}{\left (1-a^2 x^2\right )^{3/2}}\right )+\frac {8 a^3 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{5} \left (5 \left (\frac {1}{3} \left (a^2 \int \frac {54 a x+29}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {6 a \sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )+\frac {a^3 (93 a x+80)}{\sqrt {1-a^2 x^2}}+\frac {4 a^3 (6 a x+5)}{\left (1-a^2 x^2\right )^{3/2}}\right )+\frac {8 a^3 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {\frac {1}{5} \left (5 \left (\frac {1}{3} \left (a^2 \left (54 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {29 \sqrt {1-a^2 x^2}}{x}\right )-\frac {6 a \sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )+\frac {a^3 (93 a x+80)}{\sqrt {1-a^2 x^2}}+\frac {4 a^3 (6 a x+5)}{\left (1-a^2 x^2\right )^{3/2}}\right )+\frac {8 a^3 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {1}{5} \left (5 \left (\frac {1}{3} \left (a^2 \left (27 a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {29 \sqrt {1-a^2 x^2}}{x}\right )-\frac {6 a \sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )+\frac {a^3 (93 a x+80)}{\sqrt {1-a^2 x^2}}+\frac {4 a^3 (6 a x+5)}{\left (1-a^2 x^2\right )^{3/2}}\right )+\frac {8 a^3 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {1}{5} \left (5 \left (\frac {1}{3} \left (a^2 \left (-\frac {54 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {29 \sqrt {1-a^2 x^2}}{x}\right )-\frac {6 a \sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )+\frac {a^3 (93 a x+80)}{\sqrt {1-a^2 x^2}}+\frac {4 a^3 (6 a x+5)}{\left (1-a^2 x^2\right )^{3/2}}\right )+\frac {8 a^3 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {1}{5} \left (5 \left (\frac {1}{3} \left (a^2 \left (-54 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {29 \sqrt {1-a^2 x^2}}{x}\right )-\frac {6 a \sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )+\frac {a^3 (93 a x+80)}{\sqrt {1-a^2 x^2}}+\frac {4 a^3 (6 a x+5)}{\left (1-a^2 x^2\right )^{3/2}}\right )+\frac {8 a^3 (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
Input:
Int[E^ArcTanh[a*x]/(x^4*(c - a*c*x)^3),x]
Output:
((8*a^3*(1 + a*x))/(5*(1 - a^2*x^2)^(5/2)) + ((4*a^3*(5 + 6*a*x))/(1 - a^2 *x^2)^(3/2) + (a^3*(80 + 93*a*x))/Sqrt[1 - a^2*x^2] + 5*(-1/3*Sqrt[1 - a^2 *x^2]/x^3 + ((-6*a*Sqrt[1 - a^2*x^2])/x^2 + a^2*((-29*Sqrt[1 - a^2*x^2])/x - 54*a*ArcTanh[Sqrt[1 - a^2*x^2]]))/3))/5)/c^3
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_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) *((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[x^m *(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c^(2*n)/a^n Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I LtQ[n, -1] && !(IGtQ[m, 0] && ILtQ[m + n, 0] && !GtQ[p, 1])
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRema inder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x) ^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a* b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( m + 1)) Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)* (x_))^(m_.), x_Symbol] :> Simp[c^n Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1 , 0]) && IntegerQ[2*p]
Time = 0.35 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.50
| method | result | size |
| risch | \(\frac {29 a^{4} x^{4}+6 a^{3} x^{3}-28 a^{2} x^{2}-6 a x -1}{3 x^{3} \sqrt {-a^{2} x^{2}+1}\, c^{3}}+\frac {a^{3} \left (-18 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {2 \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{a^{2}}+\frac {\frac {7 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {7 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{a}-\frac {16 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{a \left (x -\frac {1}{a}\right )}\right )}{c^{3}}\) | \(331\) |
| default | \(-\frac {\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}+\frac {29 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}-4 a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )+16 a^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )-7 a^{2} \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )+\frac {16 a^{2} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{x -\frac {1}{a}}}{c^{3}}\) | \(355\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/x^4/(-a*c*x+c)^3,x,method=_RETURNVERBOSE)
Output:
1/3*(29*a^4*x^4+6*a^3*x^3-28*a^2*x^2-6*a*x-1)/x^3/(-a^2*x^2+1)^(1/2)/c^3+a ^3*(-18*arctanh(1/(-a^2*x^2+1)^(1/2))-2/a^2*(1/5/a/(x-1/a)^3*(-(x-1/a)^2*a ^2-2*a*(x-1/a))^(1/2)-2/5*a*(1/3/a/(x-1/a)^2*(-(x-1/a)^2*a^2-2*a*(x-1/a))^ (1/2)-1/3/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)))+7/a*(1/3/a/(x-1/a)^ 2*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)-1/3/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/ a))^(1/2))-16/a/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2))/c^3
Time = 0.11 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.82 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 (c-a c x)^3} \, dx=\frac {324 \, a^{6} x^{6} - 972 \, a^{5} x^{5} + 972 \, a^{4} x^{4} - 324 \, a^{3} x^{3} + 270 \, {\left (a^{6} x^{6} - 3 \, a^{5} x^{5} + 3 \, a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (424 \, a^{5} x^{5} - 1002 \, a^{4} x^{4} + 674 \, a^{3} x^{3} - 70 \, a^{2} x^{2} - 15 \, a x - 5\right )} \sqrt {-a^{2} x^{2} + 1}}{15 \, {\left (a^{3} c^{3} x^{6} - 3 \, a^{2} c^{3} x^{5} + 3 \, a c^{3} x^{4} - c^{3} x^{3}\right )}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^4/(-a*c*x+c)^3,x, algorithm="fricas ")
Output:
1/15*(324*a^6*x^6 - 972*a^5*x^5 + 972*a^4*x^4 - 324*a^3*x^3 + 270*(a^6*x^6 - 3*a^5*x^5 + 3*a^4*x^4 - a^3*x^3)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (424 *a^5*x^5 - 1002*a^4*x^4 + 674*a^3*x^3 - 70*a^2*x^2 - 15*a*x - 5)*sqrt(-a^2 *x^2 + 1))/(a^3*c^3*x^6 - 3*a^2*c^3*x^5 + 3*a*c^3*x^4 - c^3*x^3)
\[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 (c-a c x)^3} \, dx=- \frac {\int \frac {a x}{a^{3} x^{7} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a x^{5} \sqrt {- a^{2} x^{2} + 1} - x^{4} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{3} x^{7} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a x^{5} \sqrt {- a^{2} x^{2} + 1} - x^{4} \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**4/(-a*c*x+c)**3,x)
Output:
-(Integral(a*x/(a**3*x**7*sqrt(-a**2*x**2 + 1) - 3*a**2*x**6*sqrt(-a**2*x* *2 + 1) + 3*a*x**5*sqrt(-a**2*x**2 + 1) - x**4*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**3*x**7*sqrt(-a**2*x**2 + 1) - 3*a**2*x**6*sqrt(-a**2*x**2 + 1) + 3*a*x**5*sqrt(-a**2*x**2 + 1) - x**4*sqrt(-a**2*x**2 + 1)), x))/c** 3
\[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 (c-a c x)^3} \, dx=\int { -\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (a c x - c\right )}^{3} x^{4}} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^4/(-a*c*x+c)^3,x, algorithm="maxima ")
Output:
-integrate((a*x + 1)/(sqrt(-a^2*x^2 + 1)*(a*c*x - c)^3*x^4), x)
Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (188) = 376\).
Time = 0.13 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.79 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 (c-a c x)^3} \, dx=-\frac {18 \, a^{4} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{c^{3} {\left | a \right |}} - \frac {{\left (5 \, a^{4} + \frac {35 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{2}}{x} + \frac {335 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{x^{2}} - \frac {7559 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{2} x^{3}} + \frac {25195 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{4} x^{4}} - \frac {36035 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5}}{a^{6} x^{5}} + \frac {24225 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6}}{a^{8} x^{6}} - \frac {6585 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{7}}{a^{10} x^{7}}\right )} a^{6} x^{3}}{120 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{3} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{5} {\left | a \right |}} - \frac {\frac {117 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4} c^{6}}{x} + \frac {12 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{2} c^{6}}{x^{2}} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{6}}{x^{3}}}{24 \, a^{2} c^{9} {\left | a \right |}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^4/(-a*c*x+c)^3,x, algorithm="giac")
Output:
-18*a^4*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/(c^3 *abs(a)) - 1/120*(5*a^4 + 35*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^2/x + 335*( sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/x^2 - 7559*(sqrt(-a^2*x^2 + 1)*abs(a) + a )^3/(a^2*x^3) + 25195*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4/(a^4*x^4) - 36035* (sqrt(-a^2*x^2 + 1)*abs(a) + a)^5/(a^6*x^5) + 24225*(sqrt(-a^2*x^2 + 1)*ab s(a) + a)^6/(a^8*x^6) - 6585*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^7/(a^10*x^7)) *a^6*x^3/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c^3*((sqrt(-a^2*x^2 + 1)*abs(a ) + a)/(a^2*x) - 1)^5*abs(a)) - 1/24*(117*(sqrt(-a^2*x^2 + 1)*abs(a) + a)* a^4*c^6/x + 12*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a^2*c^6/x^2 + (sqrt(-a^2* x^2 + 1)*abs(a) + a)^3*c^6/x^3)/(a^2*c^9*abs(a))
Time = 14.04 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.49 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 (c-a c x)^3} \, dx=\frac {7\,a^5\,\sqrt {1-a^2\,x^2}}{3\,\left (a^4\,c^3\,x^2-2\,a^3\,c^3\,x+a^2\,c^3\right )}+\frac {4\,a^7\,\sqrt {1-a^2\,x^2}}{15\,\left (a^6\,c^3\,x^2-2\,a^5\,c^3\,x+a^4\,c^3\right )}-\frac {\sqrt {1-a^2\,x^2}}{3\,c^3\,x^3}-\frac {2\,a\,\sqrt {1-a^2\,x^2}}{c^3\,x^2}-\frac {29\,a^2\,\sqrt {1-a^2\,x^2}}{3\,c^3\,x}+\frac {93\,a^4\,\sqrt {1-a^2\,x^2}}{5\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}\right )}+\frac {2\,a^4\,\sqrt {1-a^2\,x^2}}{5\,\sqrt {-a^2}\,\left (3\,c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}+a^2\,c^3\,x^3\,\sqrt {-a^2}-3\,a\,c^3\,x^2\,\sqrt {-a^2}\right )}+\frac {a^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,18{}\mathrm {i}}{c^3} \] Input:
int((a*x + 1)/(x^4*(1 - a^2*x^2)^(1/2)*(c - a*c*x)^3),x)
Output:
(a^3*atan((1 - a^2*x^2)^(1/2)*1i)*18i)/c^3 + (7*a^5*(1 - a^2*x^2)^(1/2))/( 3*(a^2*c^3 - 2*a^3*c^3*x + a^4*c^3*x^2)) + (4*a^7*(1 - a^2*x^2)^(1/2))/(15 *(a^4*c^3 - 2*a^5*c^3*x + a^6*c^3*x^2)) - (1 - a^2*x^2)^(1/2)/(3*c^3*x^3) - (2*a*(1 - a^2*x^2)^(1/2))/(c^3*x^2) - (29*a^2*(1 - a^2*x^2)^(1/2))/(3*c^ 3*x) + (93*a^4*(1 - a^2*x^2)^(1/2))/(5*(-a^2)^(1/2)*(c^3*x*(-a^2)^(1/2) - (c^3*(-a^2)^(1/2))/a)) + (2*a^4*(1 - a^2*x^2)^(1/2))/(5*(-a^2)^(1/2)*(3*c^ 3*x*(-a^2)^(1/2) - (c^3*(-a^2)^(1/2))/a + a^2*c^3*x^3*(-a^2)^(1/2) - 3*a*c ^3*x^2*(-a^2)^(1/2)))
Time = 0.16 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.68 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 (c-a c x)^3} \, dx=\frac {1080 \sqrt {-a^{2} x^{2}+1}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{5} x^{5}-2160 \sqrt {-a^{2} x^{2}+1}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{4} x^{4}+1080 \sqrt {-a^{2} x^{2}+1}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{3} x^{3}-531 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+1678 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-1531 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+280 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+60 \sqrt {-a^{2} x^{2}+1}\, a x +20 \sqrt {-a^{2} x^{2}+1}+1080 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{6} x^{6}-3240 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{5} x^{5}+3240 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{4} x^{4}-1080 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{3} x^{3}+2861 a^{6} x^{6}-5807 a^{5} x^{5}+2183 a^{4} x^{4}+1251 a^{3} x^{3}-340 a^{2} x^{2}-80 a x -20}{60 c^{3} x^{3} \left (\sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-2 \sqrt {-a^{2} x^{2}+1}\, a x +\sqrt {-a^{2} x^{2}+1}+a^{3} x^{3}-3 a^{2} x^{2}+3 a x -1\right )} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/x^4/(-a*c*x+c)^3,x)
Output:
(1080*sqrt( - a**2*x**2 + 1)*log(tan(asin(a*x)/2))*a**5*x**5 - 2160*sqrt( - a**2*x**2 + 1)*log(tan(asin(a*x)/2))*a**4*x**4 + 1080*sqrt( - a**2*x**2 + 1)*log(tan(asin(a*x)/2))*a**3*x**3 - 531*sqrt( - a**2*x**2 + 1)*a**5*x** 5 + 1678*sqrt( - a**2*x**2 + 1)*a**4*x**4 - 1531*sqrt( - a**2*x**2 + 1)*a* *3*x**3 + 280*sqrt( - a**2*x**2 + 1)*a**2*x**2 + 60*sqrt( - a**2*x**2 + 1) *a*x + 20*sqrt( - a**2*x**2 + 1) + 1080*log(tan(asin(a*x)/2))*a**6*x**6 - 3240*log(tan(asin(a*x)/2))*a**5*x**5 + 3240*log(tan(asin(a*x)/2))*a**4*x** 4 - 1080*log(tan(asin(a*x)/2))*a**3*x**3 + 2861*a**6*x**6 - 5807*a**5*x**5 + 2183*a**4*x**4 + 1251*a**3*x**3 - 340*a**2*x**2 - 80*a*x - 20)/(60*c**3 *x**3*(sqrt( - a**2*x**2 + 1)*a**2*x**2 - 2*sqrt( - a**2*x**2 + 1)*a*x + s qrt( - a**2*x**2 + 1) + a**3*x**3 - 3*a**2*x**2 + 3*a*x - 1))