Integrand size = 22, antiderivative size = 159 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {23 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{a}-\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 a}+\frac {64 c^4 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {7 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^4 \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {91 c^4 \csc ^{-1}(a x)}{2 a}-\frac {7 c^4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \] Output:
23*c^4*(1-1/a^2/x^2)^(1/2)/a-1/3*c^4*(1-1/a^2/x^2)^(3/2)/a+64*c^4*(a-1/x)/ a^2/(1-1/a^2/x^2)^(1/2)-7/2*c^4*(1-1/a^2/x^2)^(1/2)/a^2/x+c^4*(1-1/a^2/x^2 )^(1/2)*x+91/2*c^4*arccsc(a*x)/a-7*c^4*arctanh((1-1/a^2/x^2)^(1/2))/a
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.39 (sec) , antiderivative size = 567, normalized size of antiderivative = 3.57 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {c^4 \left (2772 \sqrt {2} a^3 x^3 (-1+a x)^3 (1+a x) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {5}{2},\frac {7}{2},\frac {1}{2} \left (1-\frac {1}{a x}\right )\right )+1980 \sqrt {2} a^2 x^2 (-1+a x)^4 (1+a x) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {7}{2},\frac {9}{2},\frac {1}{2} \left (1-\frac {1}{a x}\right )\right )+35 \left (-198 a^2 \sqrt {1+\frac {1}{a x}} x^2+1716 a^3 \sqrt {1+\frac {1}{a x}} x^3-7425 a^4 \sqrt {1+\frac {1}{a x}} x^4+26268 a^5 \sqrt {1+\frac {1}{a x}} x^5+29403 a^6 \sqrt {1+\frac {1}{a x}} x^6-50160 a^7 \sqrt {1+\frac {1}{a x}} x^7+396 a^8 \sqrt {1+\frac {1}{a x}} x^8+66726 a^6 \sqrt {1-\frac {1}{a x}} x^6 \arcsin \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )+66726 a^7 \sqrt {1-\frac {1}{a x}} x^7 \arcsin \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )-1980 a^6 \sqrt {1-\frac {1}{a x}} x^6 \arcsin \left (\frac {1}{a x}\right )-1980 a^7 \sqrt {1-\frac {1}{a x}} x^7 \arcsin \left (\frac {1}{a x}\right )-2772 a^7 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {1+\frac {1}{a x}} x^7 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )+44 \sqrt {2} a x (-1+a x)^5 (1+a x) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {9}{2},\frac {11}{2},\frac {1}{2} \left (1-\frac {1}{a x}\right )\right )+36 \sqrt {2} (-1+a x)^6 (1+a x) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {11}{2},\frac {13}{2},\frac {1}{2} \left (1-\frac {1}{a x}\right )\right )\right )\right )}{13860 a^7 \sqrt {1-\frac {1}{a x}} x^6 (1+a x)} \] Input:
Integrate[(c - c/(a*x))^4/E^(3*ArcCoth[a*x]),x]
Output:
(c^4*(2772*Sqrt[2]*a^3*x^3*(-1 + a*x)^3*(1 + a*x)*Hypergeometric2F1[3/2, 5 /2, 7/2, (1 - 1/(a*x))/2] + 1980*Sqrt[2]*a^2*x^2*(-1 + a*x)^4*(1 + a*x)*Hy pergeometric2F1[3/2, 7/2, 9/2, (1 - 1/(a*x))/2] + 35*(-198*a^2*Sqrt[1 + 1/ (a*x)]*x^2 + 1716*a^3*Sqrt[1 + 1/(a*x)]*x^3 - 7425*a^4*Sqrt[1 + 1/(a*x)]*x ^4 + 26268*a^5*Sqrt[1 + 1/(a*x)]*x^5 + 29403*a^6*Sqrt[1 + 1/(a*x)]*x^6 - 5 0160*a^7*Sqrt[1 + 1/(a*x)]*x^7 + 396*a^8*Sqrt[1 + 1/(a*x)]*x^8 + 66726*a^6 *Sqrt[1 - 1/(a*x)]*x^6*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] + 66726*a^7*Sqrt[ 1 - 1/(a*x)]*x^7*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] - 1980*a^6*Sqrt[1 - 1/( a*x)]*x^6*ArcSin[1/(a*x)] - 1980*a^7*Sqrt[1 - 1/(a*x)]*x^7*ArcSin[1/(a*x)] - 2772*a^7*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[1 + 1/(a*x)]*x^7*ArcTanh[Sqrt[1 - 1 /(a^2*x^2)]] + 44*Sqrt[2]*a*x*(-1 + a*x)^5*(1 + a*x)*Hypergeometric2F1[3/2 , 9/2, 11/2, (1 - 1/(a*x))/2] + 36*Sqrt[2]*(-1 + a*x)^6*(1 + a*x)*Hypergeo metric2F1[3/2, 11/2, 13/2, (1 - 1/(a*x))/2])))/(13860*a^7*Sqrt[1 - 1/(a*x) ]*x^6*(1 + a*x))
Time = 1.48 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.09, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {6731, 27, 528, 2338, 2340, 25, 2340, 25, 2340, 27, 538, 223, 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.
| \(\displaystyle \int \left (c-\frac {c}{a x}\right )^4 e^{-3 \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6731 |
| \(\displaystyle -\frac {\int \frac {c^7 \left (a-\frac {1}{x}\right )^7 x^2}{a^7 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^4 \int \frac {\left (a-\frac {1}{x}\right )^7 x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{a^7}\) |
| \(\Big \downarrow \) 528 |
| \(\displaystyle -\frac {c^4 \left (a^2 \int \frac {\left (a^5-\frac {7 a^4}{x}-\frac {42 a^3}{x^2}+\frac {22 a^2}{x^3}-\frac {7 a}{x^4}+\frac {1}{x^5}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {64 a^5 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^7}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle -\frac {c^4 \left (a^2 \left (a^5 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )-\int \frac {\left (7 a^4+\frac {42 a^3}{x}-\frac {22 a^2}{x^2}+\frac {7 a}{x^3}-\frac {1}{x^4}\right ) x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )-\frac {64 a^5 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^7}\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle -\frac {c^4 \left (a^2 \left (\frac {1}{3} a^2 \int -\frac {\left (21 a^2+\frac {126 a}{x}-\frac {68}{x^2}+\frac {21}{x^3 a}\right ) x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}+a^5 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )-\frac {64 a^5 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {c^4 \left (a^2 \left (-\frac {1}{3} a^2 \int \frac {\left (21 a^2+\frac {126 a}{x}-\frac {68}{x^2}+\frac {21}{x^3 a}\right ) x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}+a^5 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )-\frac {64 a^5 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^7}\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle -\frac {c^4 \left (a^2 \left (-\frac {1}{3} a^2 \left (-\frac {1}{2} a^2 \int -\frac {\left (42+\frac {273}{a x}-\frac {136}{a^2 x^2}\right ) x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {21 a \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}+a^5 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )-\frac {64 a^5 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {c^4 \left (a^2 \left (-\frac {1}{3} a^2 \left (\frac {1}{2} a^2 \int \frac {\left (42+\frac {273}{a x}-\frac {136}{a^2 x^2}\right ) x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {21 a \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}+a^5 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )-\frac {64 a^5 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^7}\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle -\frac {c^4 \left (a^2 \left (-\frac {1}{3} a^2 \left (\frac {1}{2} a^2 \left (136 \sqrt {1-\frac {1}{a^2 x^2}}-a^2 \int -\frac {21 \left (2 a+\frac {13}{x}\right ) x}{a^3 \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )-\frac {21 a \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}+a^5 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )-\frac {64 a^5 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^4 \left (a^2 \left (-\frac {1}{3} a^2 \left (\frac {1}{2} a^2 \left (\frac {21 \int \frac {\left (2 a+\frac {13}{x}\right ) x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{a}+136 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {21 a \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}+a^5 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )-\frac {64 a^5 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^7}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle -\frac {c^4 \left (a^2 \left (-\frac {1}{3} a^2 \left (\frac {1}{2} a^2 \left (\frac {21 \left (13 \int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+2 a \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )}{a}+136 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {21 a \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}+a^5 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )-\frac {64 a^5 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^7}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {c^4 \left (a^2 \left (-\frac {1}{3} a^2 \left (\frac {1}{2} a^2 \left (\frac {21 \left (2 a \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+13 a \arcsin \left (\frac {1}{a x}\right )\right )}{a}+136 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {21 a \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}+a^5 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )-\frac {64 a^5 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^7}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {c^4 \left (a^2 \left (-\frac {1}{3} a^2 \left (\frac {1}{2} a^2 \left (\frac {21 \left (a \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}+13 a \arcsin \left (\frac {1}{a x}\right )\right )}{a}+136 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {21 a \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}+a^5 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )-\frac {64 a^5 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^7}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {c^4 \left (a^2 \left (-\frac {1}{3} a^2 \left (\frac {1}{2} a^2 \left (\frac {21 \left (13 a \arcsin \left (\frac {1}{a x}\right )-2 a^3 \int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}+136 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {21 a \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}+a^5 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )-\frac {64 a^5 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^7}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {c^4 \left (a^2 \left (-\frac {1}{3} a^2 \left (\frac {1}{2} a^2 \left (\frac {21 \left (13 a \arcsin \left (\frac {1}{a x}\right )-2 a \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a}+136 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {21 a \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}+a^5 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )-\frac {64 a^5 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^7}\) |
Input:
Int[(c - c/(a*x))^4/E^(3*ArcCoth[a*x]),x]
Output:
-((c^4*((-64*a^5*(a - x^(-1)))/Sqrt[1 - 1/(a^2*x^2)] + a^2*(-1/3*(a^2*Sqrt [1 - 1/(a^2*x^2)])/x^2 - a^5*Sqrt[1 - 1/(a^2*x^2)]*x - (a^2*((-21*a*Sqrt[1 - 1/(a^2*x^2)])/(2*x) + (a^2*(136*Sqrt[1 - 1/(a^2*x^2)] + (21*(13*a*ArcSi n[1/(a*x)] - 2*a*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]))/a))/2))/3)))/a^7)
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[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[(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)^(3/2), x_Sy mbol] :> Simp[(-2^(n - 1))*c^(m + n - 2)*((c + d*x)/(b*d^(m - 1)*Sqrt[a + b *x^2])), x] + Simp[c^2/a Int[(x^m/Sqrt[a + b*x^2])*ExpandToSum[((c + d*x) ^(n - 1) - (2^(n - 1)*c^(m + n - 1))/(d^m*x^m))/(c - d*x), x], x], x] /; Fr eeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && EqQ[b*c^2 + a*d^2, 0]
Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp [c Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d Int[1/Sqrt[a + b*x^2], x] , x] /; FreeQ[{a, b, c, d}, x]
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[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 )*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m + q + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) *Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ [Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
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 = 191, normalized size of antiderivative = 1.20
| method | result | size |
| risch | \(\frac {\left (a x +1\right ) \left (136 a^{2} x^{2}-21 a x +2\right ) c^{4} \sqrt {\frac {a x -1}{a x +1}}}{6 x^{3} a^{4}}+\frac {\left (\frac {91 a^{3} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{2}-\frac {7 a^{4} \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+\frac {64 a^{2} \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{x +\frac {1}{a}}+a^{3} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\right ) c^{4} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a^{4} \left (a x -1\right )}\) | \(191\) |
| default | \(-\frac {\left (-138 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{6} x^{6}+138 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}-549 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{5} x^{5}-273 \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {a^{2}}\, a^{5} x^{5}+138 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}+96 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}-96 \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}+255 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{3} x^{3}-684 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{4} x^{4}-546 a^{4} \sqrt {a^{2}}\, x^{4} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+276 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+192 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{3} x^{3}+192 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{4} x^{4}-192 \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}+98 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{2} x^{2}-273 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{3} x^{3}-273 a^{3} \sqrt {a^{2}}\, x^{3} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+138 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+96 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}-96 \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}-17 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x +2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right ) c^{4} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{6 a^{4} \sqrt {a^{2}}\, x^{3} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )}\) | \(672\) |
Input:
int((c-c/a/x)^4*((a*x-1)/(a*x+1))^(3/2),x,method=_RETURNVERBOSE)
Output:
1/6*(a*x+1)*(136*a^2*x^2-21*a*x+2)/x^3*c^4/a^4*((a*x-1)/(a*x+1))^(1/2)+(91 /2*a^3*arctan(1/(a^2*x^2-1)^(1/2))-7*a^4*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2-1)^ (1/2))/(a^2)^(1/2)+64*a^2/(x+1/a)*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2)+a^3*(( a*x-1)*(a*x+1))^(1/2))*c^4/a^4/(a*x-1)*((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a *x+1))^(1/2)
Time = 0.10 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.99 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=-\frac {546 \, a^{3} c^{4} x^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + 42 \, a^{3} c^{4} x^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 42 \, a^{3} c^{4} x^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (6 \, a^{4} c^{4} x^{4} + 526 \, a^{3} c^{4} x^{3} + 115 \, a^{2} c^{4} x^{2} - 19 \, a c^{4} x + 2 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, a^{4} x^{3}} \] Input:
integrate((c-c/a/x)^4*((a*x-1)/(a*x+1))^(3/2),x, algorithm="fricas")
Output:
-1/6*(546*a^3*c^4*x^3*arctan(sqrt((a*x - 1)/(a*x + 1))) + 42*a^3*c^4*x^3*l og(sqrt((a*x - 1)/(a*x + 1)) + 1) - 42*a^3*c^4*x^3*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (6*a^4*c^4*x^4 + 526*a^3*c^4*x^3 + 115*a^2*c^4*x^2 - 19*a*c^ 4*x + 2*c^4)*sqrt((a*x - 1)/(a*x + 1)))/(a^4*x^3)
\[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {c^{4} \left (\int \left (- \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{5} + x^{4}}\right )\, dx + \int \frac {5 a \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{4} + x^{3}}\, dx + \int \left (- \frac {10 a^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{3} + x^{2}}\right )\, dx + \int \frac {10 a^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{2} + x}\, dx + \int \left (- \frac {5 a^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \frac {a^{5} x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx\right )}{a^{4}} \] Input:
integrate((c-c/a/x)**4*((a*x-1)/(a*x+1))**(3/2),x)
Output:
c**4*(Integral(-sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x**5 + x**4), x) + In tegral(5*a*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x**4 + x**3), x) + Integra l(-10*a**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x**3 + x**2), x) + Integra l(10*a**3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x**2 + x), x) + Integral(-5 *a**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x) + Integral(a**5*x*sq rt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x))/a**4
Time = 0.12 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.55 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=-\frac {1}{3} \, {\left (\frac {273 \, c^{4} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {21 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {21 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {192 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2}} + \frac {153 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 91 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 169 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 123 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {2 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {2 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + a^{2}}\right )} a \] Input:
integrate((c-c/a/x)^4*((a*x-1)/(a*x+1))^(3/2),x, algorithm="maxima")
Output:
-1/3*(273*c^4*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 + 21*c^4*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 21*c^4*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2 - 192*c^4*sqrt((a*x - 1)/(a*x + 1))/a^2 + (153*c^4*((a*x - 1)/(a*x + 1))^( 7/2) + 91*c^4*((a*x - 1)/(a*x + 1))^(5/2) - 169*c^4*((a*x - 1)/(a*x + 1))^ (3/2) - 123*c^4*sqrt((a*x - 1)/(a*x + 1)))/(2*(a*x - 1)*a^2/(a*x + 1) - 2* (a*x - 1)^3*a^2/(a*x + 1)^3 - (a*x - 1)^4*a^2/(a*x + 1)^4 + a^2))*a
\[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\int { {\left (c - \frac {c}{a x}\right )}^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((c-c/a/x)^4*((a*x-1)/(a*x+1))^(3/2),x, algorithm="giac")
Output:
undef
Time = 0.08 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.33 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {41\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}+\frac {169\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}-\frac {91\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{3}-51\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{a+\frac {2\,a\,\left (a\,x-1\right )}{a\,x+1}-\frac {2\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}}+\frac {64\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a}-\frac {91\,c^4\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}+\frac {c^4\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,14{}\mathrm {i}}{a} \] Input:
int((c - c/(a*x))^4*((a*x - 1)/(a*x + 1))^(3/2),x)
Output:
(41*c^4*((a*x - 1)/(a*x + 1))^(1/2) + (169*c^4*((a*x - 1)/(a*x + 1))^(3/2) )/3 - (91*c^4*((a*x - 1)/(a*x + 1))^(5/2))/3 - 51*c^4*((a*x - 1)/(a*x + 1) )^(7/2))/(a + (2*a*(a*x - 1))/(a*x + 1) - (2*a*(a*x - 1)^3)/(a*x + 1)^3 - (a*(a*x - 1)^4)/(a*x + 1)^4) + (64*c^4*((a*x - 1)/(a*x + 1))^(1/2))/a - (9 1*c^4*atan(((a*x - 1)/(a*x + 1))^(1/2)))/a + (c^4*atan(((a*x - 1)/(a*x + 1 ))^(1/2)*1i)*14i)/a
Time = 0.19 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.70 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {c^{4} \left (-546 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}-1\right ) a^{4} x^{4}-546 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}-1\right ) a^{3} x^{3}+546 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}+1\right ) a^{4} x^{4}+546 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}+1\right ) a^{3} x^{3}+6 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{4} x^{4}+526 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}+115 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}-19 \sqrt {a x +1}\, \sqrt {a x -1}\, a x +2 \sqrt {a x +1}\, \sqrt {a x -1}-84 \,\mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{4} x^{4}-84 \,\mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{3} x^{3}+284 a^{4} x^{4}+284 a^{3} x^{3}\right )}{6 a^{4} x^{3} \left (a x +1\right )} \] Input:
int((c-c/a/x)^4*((a*x-1)/(a*x+1))^(3/2),x)
Output:
(c**4*( - 546*atan(sqrt(a*x - 1) + sqrt(a*x + 1) - 1)*a**4*x**4 - 546*atan (sqrt(a*x - 1) + sqrt(a*x + 1) - 1)*a**3*x**3 + 546*atan(sqrt(a*x - 1) + s qrt(a*x + 1) + 1)*a**4*x**4 + 546*atan(sqrt(a*x - 1) + sqrt(a*x + 1) + 1)* a**3*x**3 + 6*sqrt(a*x + 1)*sqrt(a*x - 1)*a**4*x**4 + 526*sqrt(a*x + 1)*sq rt(a*x - 1)*a**3*x**3 + 115*sqrt(a*x + 1)*sqrt(a*x - 1)*a**2*x**2 - 19*sqr t(a*x + 1)*sqrt(a*x - 1)*a*x + 2*sqrt(a*x + 1)*sqrt(a*x - 1) - 84*log((sqr t(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a**4*x**4 - 84*log((sqrt(a*x - 1) + s qrt(a*x + 1))/sqrt(2))*a**3*x**3 + 284*a**4*x**4 + 284*a**3*x**3))/(6*a**4 *x**3*(a*x + 1))