Integrand size = 22, antiderivative size = 182 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=-\frac {11 a^3 \sqrt {1-\frac {1}{a^2 x^2}}}{7 c^3 \left (a-\frac {1}{x}\right )^4}-\frac {15 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{7 c^3 \left (a-\frac {1}{x}\right )^3}-\frac {24 a \sqrt {1-\frac {1}{a^2 x^2}}}{7 c^3 \left (a-\frac {1}{x}\right )^2}-\frac {66 \sqrt {1-\frac {1}{a^2 x^2}}}{7 c^3 \left (a-\frac {1}{x}\right )}+\frac {a^4 \sqrt {1-\frac {1}{a^2 x^2}} x}{c^3 \left (a-\frac {1}{x}\right )^4}+\frac {6 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^3} \] Output:
-11/7*a^3*(1-1/a^2/x^2)^(1/2)/c^3/(a-1/x)^4-15/7*a^2*(1-1/a^2/x^2)^(1/2)/c ^3/(a-1/x)^3-24/7*a*(1-1/a^2/x^2)^(1/2)/c^3/(a-1/x)^2-66/7*(1-1/a^2/x^2)^( 1/2)/c^3/(a-1/x)+a^4*(1-1/a^2/x^2)^(1/2)*x/c^3/(a-1/x)^4+6*arctanh((1-1/a^ 2/x^2)^(1/2))/a/c^3
Time = 0.10 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.62 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {66-156 a x+39 a^2 x^2+145 a^3 x^3-109 a^4 x^4+7 a^5 x^5+42 a \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{7 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^3} \] Input:
Integrate[E^(3*ArcCoth[a*x])/(c - c/(a*x))^3,x]
Output:
(66 - 156*a*x + 39*a^2*x^2 + 145*a^3*x^3 - 109*a^4*x^4 + 7*a^5*x^5 + 42*a* Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(7*a^ 2*c^3*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^3)
Time = 1.32 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.87, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {6731, 27, 570, 532, 25, 2336, 27, 2336, 27, 2336, 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.
| \(\displaystyle \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx\) |
| \(\Big \downarrow \) 6731 |
| \(\displaystyle -c^3 \int \frac {a^6 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^2}{c^6 \left (a-\frac {1}{x}\right )^6}d\frac {1}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^6 \int \frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^2}{\left (a-\frac {1}{x}\right )^6}d\frac {1}{x}}{c^3}\) |
| \(\Big \downarrow \) 570 |
| \(\displaystyle -\frac {\int \frac {\left (a+\frac {1}{x}\right )^6 x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{9/2}}d\frac {1}{x}}{a^6 c^3}\) |
| \(\Big \downarrow \) 532 |
| \(\displaystyle -\frac {\frac {32 a^4 \left (a+\frac {1}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {1}{7} \int -\frac {\left (7 a^6+\frac {42 a^5}{x}+\frac {80 a^4}{x^2}-\frac {42 a^3}{x^3}-\frac {7 a^2}{x^4}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{7/2}}d\frac {1}{x}}{a^6 c^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{7} \int \frac {\left (7 a^6+\frac {42 a^5}{x}+\frac {80 a^4}{x^2}-\frac {42 a^3}{x^3}-\frac {7 a^2}{x^4}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{7/2}}d\frac {1}{x}+\frac {32 a^4 \left (a+\frac {1}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}}{a^6 c^3}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle -\frac {\frac {1}{7} \left (\frac {16 a^4}{x \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {1}{5} \int -\frac {5 \left (7 a^6+\frac {42 a^5}{x}+\frac {71 a^4}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2}}d\frac {1}{x}\right )+\frac {32 a^4 \left (a+\frac {1}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}}{a^6 c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{7} \left (\int \frac {\left (7 a^6+\frac {42 a^5}{x}+\frac {71 a^4}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2}}d\frac {1}{x}+\frac {16 a^4}{x \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}\right )+\frac {32 a^4 \left (a+\frac {1}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}}{a^6 c^3}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle -\frac {\frac {1}{7} \left (-\frac {1}{3} \int -\frac {3 \left (7 a^6+\frac {42 a^5}{x}+\frac {52 a^4}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}+\frac {2 a^4 \left (7 a+\frac {13}{x}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {16 a^4}{x \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}\right )+\frac {32 a^4 \left (a+\frac {1}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}}{a^6 c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{7} \left (\int \frac {\left (7 a^6+\frac {42 a^5}{x}+\frac {52 a^4}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}+\frac {2 a^4 \left (7 a+\frac {13}{x}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {16 a^4}{x \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}\right )+\frac {32 a^4 \left (a+\frac {1}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}}{a^6 c^3}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle -\frac {\frac {1}{7} \left (-\int -\frac {7 a^5 \left (a+\frac {6}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {2 a^4 \left (7 a+\frac {13}{x}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {a^4 \left (42 a+\frac {59}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {16 a^4}{x \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}\right )+\frac {32 a^4 \left (a+\frac {1}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}}{a^6 c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{7} \left (7 a^5 \int \frac {\left (a+\frac {6}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {2 a^4 \left (7 a+\frac {13}{x}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {a^4 \left (42 a+\frac {59}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {16 a^4}{x \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}\right )+\frac {32 a^4 \left (a+\frac {1}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}}{a^6 c^3}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle -\frac {\frac {1}{7} \left (7 a^5 \left (6 \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {2 a^4 \left (7 a+\frac {13}{x}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {a^4 \left (42 a+\frac {59}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {16 a^4}{x \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}\right )+\frac {32 a^4 \left (a+\frac {1}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}}{a^6 c^3}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {\frac {1}{7} \left (7 a^5 \left (3 \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {2 a^4 \left (7 a+\frac {13}{x}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {a^4 \left (42 a+\frac {59}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {16 a^4}{x \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}\right )+\frac {32 a^4 \left (a+\frac {1}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}}{a^6 c^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\frac {1}{7} \left (7 a^5 \left (-6 a^2 \int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {2 a^4 \left (7 a+\frac {13}{x}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {a^4 \left (42 a+\frac {59}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {16 a^4}{x \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}\right )+\frac {32 a^4 \left (a+\frac {1}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}}{a^6 c^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {32 a^4 \left (a+\frac {1}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}+\frac {1}{7} \left (7 a^5 \left (-6 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {2 a^4 \left (7 a+\frac {13}{x}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {a^4 \left (42 a+\frac {59}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {16 a^4}{x \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}\right )}{a^6 c^3}\) |
Input:
Int[E^(3*ArcCoth[a*x])/(c - c/(a*x))^3,x]
Output:
-(((32*a^4*(a + x^(-1)))/(7*(1 - 1/(a^2*x^2))^(7/2)) + ((2*a^4*(7*a + 13/x ))/(1 - 1/(a^2*x^2))^(3/2) + (a^4*(42*a + 59/x))/Sqrt[1 - 1/(a^2*x^2)] + ( 16*a^4)/((1 - 1/(a^2*x^2))^(5/2)*x) + 7*a^5*(-(a*Sqrt[1 - 1/(a^2*x^2)]*x) - 6*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]))/7)/(a^6*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[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> S imp[-c^n Subst[Int[(c + d*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^2), x], x, 1/ x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]
Time = 0.14 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.46
| method | result | size |
| risch | \(\frac {a x -1}{a \,c^{3} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {6 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{3} \sqrt {a^{2}}}-\frac {88 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{7 a^{5} \left (x -\frac {1}{a}\right )}-\frac {4 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{7 a^{8} \left (x -\frac {1}{a}\right )^{4}}-\frac {20 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{7 a^{7} \left (x -\frac {1}{a}\right )^{3}}-\frac {45 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{7 a^{6} \left (x -\frac {1}{a}\right )^{2}}\right ) a^{3} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{3} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(265\) |
| default | \(-\frac {-42 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}-42 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}+35 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{3} x^{3}+210 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{4} x^{4}+210 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}-87 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}-420 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}-420 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+78 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x +420 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}+420 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-24 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-210 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a x -210 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x +42 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}+42 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )}{7 a \left (a x -1\right )^{3} \sqrt {a^{2}}\, c^{3} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(530\) |
Input:
int(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^3,x,method=_RETURNVERBOSE)
Output:
1/a*(a*x-1)/c^3/((a*x-1)/(a*x+1))^(1/2)+(6/a^3*ln(a^2*x/(a^2)^(1/2)+(a^2*x ^2-1)^(1/2))/(a^2)^(1/2)-88/7/a^5/(x-1/a)*((x-1/a)^2*a^2+2*a*(x-1/a))^(1/2 )-4/7/a^8/(x-1/a)^4*((x-1/a)^2*a^2+2*a*(x-1/a))^(1/2)-20/7/a^7/(x-1/a)^3*( (x-1/a)^2*a^2+2*a*(x-1/a))^(1/2)-45/7/a^6/(x-1/a)^2*((x-1/a)^2*a^2+2*a*(x- 1/a))^(1/2))*a^3/c^3/((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^(1/2)/(a*x+ 1)
Time = 0.12 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.12 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {42 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 42 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (7 \, a^{5} x^{5} - 109 \, a^{4} x^{4} + 145 \, a^{3} x^{3} + 39 \, a^{2} x^{2} - 156 \, a x + 66\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{7 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^3,x, algorithm="fricas")
Output:
1/7*(42*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*log(sqrt((a*x - 1)/( a*x + 1)) + 1) - 42*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*log(sqrt ((a*x - 1)/(a*x + 1)) - 1) + (7*a^5*x^5 - 109*a^4*x^4 + 145*a^3*x^3 + 39*a ^2*x^2 - 156*a*x + 66)*sqrt((a*x - 1)/(a*x + 1)))/(a^5*c^3*x^4 - 4*a^4*c^3 *x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^3)
\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {a^{3} \int \frac {x^{3}}{\frac {a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {4 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {6 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {4 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx}{c^{3}} \] Input:
integrate(1/((a*x-1)/(a*x+1))**(3/2)/(c-c/a/x)**3,x)
Output:
a**3*Integral(x**3/(a**4*x**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - 4*a**3*x**3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) + 6*a**2*x**2*sq rt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - 4*a*x*sqrt(a*x/(a*x + 1) - 1/( a*x + 1))/(a*x + 1) + sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1)), x)/c** 3
Time = 0.04 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.93 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {1}{14} \, a {\left (\frac {\frac {6 \, {\left (a x - 1\right )}}{a x + 1} + \frac {21 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {112 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac {168 \, {\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 1}{a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} - a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}}} + \frac {84 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} - \frac {84 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{3}}\right )} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^3,x, algorithm="maxima")
Output:
1/14*a*((6*(a*x - 1)/(a*x + 1) + 21*(a*x - 1)^2/(a*x + 1)^2 + 112*(a*x - 1 )^3/(a*x + 1)^3 - 168*(a*x - 1)^4/(a*x + 1)^4 + 1)/(a^2*c^3*((a*x - 1)/(a* x + 1))^(9/2) - a^2*c^3*((a*x - 1)/(a*x + 1))^(7/2)) + 84*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^3) - 84*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2* c^3))
Exception generated. \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^3,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Time = 0.06 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.75 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {12\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^3}-\frac {\frac {3\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {16\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {24\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}+\frac {6\,\left (a\,x-1\right )}{7\,\left (a\,x+1\right )}+\frac {1}{7}}{2\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}-2\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}} \] Input:
int(1/((c - c/(a*x))^3*((a*x - 1)/(a*x + 1))^(3/2)),x)
Output:
(12*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(a*c^3) - ((3*(a*x - 1)^2)/(a*x + 1)^2 + (16*(a*x - 1)^3)/(a*x + 1)^3 - (24*(a*x - 1)^4)/(a*x + 1)^4 + (6*(a *x - 1))/(7*(a*x + 1)) + 1/7)/(2*a*c^3*((a*x - 1)/(a*x + 1))^(7/2) - 2*a*c ^3*((a*x - 1)/(a*x + 1))^(9/2))
Time = 0.18 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.47 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {84 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{3} x^{3}-252 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{2} x^{2}+252 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a x -84 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right )+60 \sqrt {a x -1}\, a^{3} x^{3}-180 \sqrt {a x -1}\, a^{2} x^{2}+180 \sqrt {a x -1}\, a x -60 \sqrt {a x -1}+7 \sqrt {a x +1}\, a^{4} x^{4}-116 \sqrt {a x +1}\, a^{3} x^{3}+261 \sqrt {a x +1}\, a^{2} x^{2}-222 \sqrt {a x +1}\, a x +66 \sqrt {a x +1}}{7 \sqrt {a x -1}\, a \,c^{3} \left (a^{3} x^{3}-3 a^{2} x^{2}+3 a x -1\right )} \] Input:
int(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^3,x)
Output:
(84*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a**3*x**3 - 252*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a**2*x**2 + 252*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a*x - 84* sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2)) + 60*sqrt(a*x - 1)*a**3*x**3 - 180*sqrt(a*x - 1)*a**2*x**2 + 180*sqrt(a*x - 1)*a*x - 60*s qrt(a*x - 1) + 7*sqrt(a*x + 1)*a**4*x**4 - 116*sqrt(a*x + 1)*a**3*x**3 + 2 61*sqrt(a*x + 1)*a**2*x**2 - 222*sqrt(a*x + 1)*a*x + 66*sqrt(a*x + 1))/(7* sqrt(a*x - 1)*a*c**3*(a**3*x**3 - 3*a**2*x**2 + 3*a*x - 1))