Integrand size = 22, antiderivative size = 135 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {6 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{a}+\frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {33 c^3 \csc ^{-1}(a x)}{2 a}-\frac {6 c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \] Output:
6*c^3*(1-1/a^2/x^2)^(1/2)/a+32*c^3*(a-1/x)/a^2/(1-1/a^2/x^2)^(1/2)-1/2*c^3 *(1-1/a^2/x^2)^(1/2)/a^2/x+c^3*(1-1/a^2/x^2)^(1/2)*x+33/2*c^3*arccsc(a*x)/ a-6*c^3*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 = 0.54 (sec) , antiderivative size = 663, normalized size of antiderivative = 4.91 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {c^3 \left (420 a^2 \sqrt {1+\frac {1}{a x}} x^2-3465 a^3 \sqrt {1+\frac {1}{a x}} x^3+16800 a^4 \sqrt {1+\frac {1}{a x}} x^4+17955 a^5 \sqrt {1+\frac {1}{a x}} x^5-32340 a^6 \sqrt {1+\frac {1}{a x}} x^6+630 a^7 \sqrt {1+\frac {1}{a x}} x^7+44730 a^5 \sqrt {1-\frac {1}{a x}} x^5 \arcsin \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )+44730 a^6 \sqrt {1-\frac {1}{a x}} x^6 \arcsin \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )-2520 a^5 \sqrt {1-\frac {1}{a x}} x^5 \arcsin \left (\frac {1}{a x}\right )-2520 a^6 \sqrt {1-\frac {1}{a x}} x^6 \arcsin \left (\frac {1}{a x}\right )-3780 a^6 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {1+\frac {1}{a x}} x^6 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )+126 \sqrt {2} a^2 x^2 (-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 )+90 \sqrt {2} a x (-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 )-70 \sqrt {2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {9}{2},\frac {11}{2},\frac {1}{2} \left (1-\frac {1}{a x}\right )\right )+280 \sqrt {2} 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 )-350 \sqrt {2} a^2 x^2 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {9}{2},\frac {11}{2},\frac {1}{2} \left (1-\frac {1}{a x}\right )\right )+350 \sqrt {2} a^4 x^4 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {9}{2},\frac {11}{2},\frac {1}{2} \left (1-\frac {1}{a x}\right )\right )-280 \sqrt {2} a^5 x^5 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {9}{2},\frac {11}{2},\frac {1}{2} \left (1-\frac {1}{a x}\right )\right )+70 \sqrt {2} a^6 x^6 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {9}{2},\frac {11}{2},\frac {1}{2} \left (1-\frac {1}{a x}\right )\right )\right )}{630 a^6 \sqrt {1-\frac {1}{a x}} x^5 (1+a x)} \] Input:
Integrate[(c - c/(a*x))^3/E^(3*ArcCoth[a*x]),x]
Output:
(c^3*(420*a^2*Sqrt[1 + 1/(a*x)]*x^2 - 3465*a^3*Sqrt[1 + 1/(a*x)]*x^3 + 168 00*a^4*Sqrt[1 + 1/(a*x)]*x^4 + 17955*a^5*Sqrt[1 + 1/(a*x)]*x^5 - 32340*a^6 *Sqrt[1 + 1/(a*x)]*x^6 + 630*a^7*Sqrt[1 + 1/(a*x)]*x^7 + 44730*a^5*Sqrt[1 - 1/(a*x)]*x^5*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] + 44730*a^6*Sqrt[1 - 1/(a *x)]*x^6*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] - 2520*a^5*Sqrt[1 - 1/(a*x)]*x^ 5*ArcSin[1/(a*x)] - 2520*a^6*Sqrt[1 - 1/(a*x)]*x^6*ArcSin[1/(a*x)] - 3780* a^6*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[1 + 1/(a*x)]*x^6*ArcTanh[Sqrt[1 - 1/(a^2*x^ 2)]] + 126*Sqrt[2]*a^2*x^2*(-1 + a*x)^3*(1 + a*x)*Hypergeometric2F1[3/2, 5 /2, 7/2, (1 - 1/(a*x))/2] + 90*Sqrt[2]*a*x*(-1 + a*x)^4*(1 + a*x)*Hypergeo metric2F1[3/2, 7/2, 9/2, (1 - 1/(a*x))/2] - 70*Sqrt[2]*Hypergeometric2F1[3 /2, 9/2, 11/2, (1 - 1/(a*x))/2] + 280*Sqrt[2]*a*x*Hypergeometric2F1[3/2, 9 /2, 11/2, (1 - 1/(a*x))/2] - 350*Sqrt[2]*a^2*x^2*Hypergeometric2F1[3/2, 9/ 2, 11/2, (1 - 1/(a*x))/2] + 350*Sqrt[2]*a^4*x^4*Hypergeometric2F1[3/2, 9/2 , 11/2, (1 - 1/(a*x))/2] - 280*Sqrt[2]*a^5*x^5*Hypergeometric2F1[3/2, 9/2, 11/2, (1 - 1/(a*x))/2] + 70*Sqrt[2]*a^6*x^6*Hypergeometric2F1[3/2, 9/2, 1 1/2, (1 - 1/(a*x))/2]))/(630*a^6*Sqrt[1 - 1/(a*x)]*x^5*(1 + a*x))
Time = 1.23 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.02, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {6731, 27, 528, 2338, 2340, 27, 2340, 25, 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.
| \(\displaystyle \int \left (c-\frac {c}{a x}\right )^3 e^{-3 \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6731 |
| \(\displaystyle -\frac {\int \frac {c^6 \left (a-\frac {1}{x}\right )^6 x^2}{a^6 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^3 \int \frac {\left (a-\frac {1}{x}\right )^6 x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{a^6}\) |
| \(\Big \downarrow \) 528 |
| \(\displaystyle -\frac {c^3 \left (a^2 \int \frac {\left (a^4-\frac {6 a^3}{x}-\frac {16 a^2}{x^2}+\frac {6 a}{x^3}-\frac {1}{x^4}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {32 a^4 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^6}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle -\frac {c^3 \left (a^2 \left (a^4 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )-\int \frac {\left (6 a^3+\frac {16 a^2}{x}-\frac {6 a}{x^2}+\frac {1}{x^3}\right ) x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )-\frac {32 a^4 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^6}\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle -\frac {c^3 \left (a^2 \left (\frac {1}{2} a^2 \int -\frac {3 \left (4 a+\frac {11}{x}-\frac {4}{x^2 a}\right ) x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}+a^4 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )-\frac {32 a^4 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^3 \left (a^2 \left (-\frac {3}{2} a^2 \int \frac {\left (4 a+\frac {11}{x}-\frac {4}{x^2 a}\right ) x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}+a^4 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )-\frac {32 a^4 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^6}\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle -\frac {c^3 \left (a^2 \left (-\frac {3}{2} a^2 \left (4 a \sqrt {1-\frac {1}{a^2 x^2}}-a^2 \int -\frac {\left (4 a+\frac {11}{x}\right ) x}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}+a^4 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )-\frac {32 a^4 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^6}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {c^3 \left (a^2 \left (-\frac {3}{2} a^2 \left (a^2 \int \frac {\left (4 a+\frac {11}{x}\right ) x}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+4 a \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}+a^4 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )-\frac {32 a^4 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^3 \left (a^2 \left (-\frac {3}{2} a^2 \left (\int \frac {\left (4 a+\frac {11}{x}\right ) x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+4 a \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}+a^4 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )-\frac {32 a^4 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^6}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle -\frac {c^3 \left (a^2 \left (-\frac {3}{2} a^2 \left (4 a \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+11 \int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+4 a \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}+a^4 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )-\frac {32 a^4 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^6}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {c^3 \left (a^2 \left (-\frac {3}{2} a^2 \left (4 a \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+4 a \sqrt {1-\frac {1}{a^2 x^2}}+11 a \arcsin \left (\frac {1}{a x}\right )\right )+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}+a^4 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )-\frac {32 a^4 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^6}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {c^3 \left (a^2 \left (-\frac {3}{2} a^2 \left (2 a \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}+4 a \sqrt {1-\frac {1}{a^2 x^2}}+11 a \arcsin \left (\frac {1}{a x}\right )\right )+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}+a^4 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )-\frac {32 a^4 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^6}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {c^3 \left (a^2 \left (-\frac {3}{2} a^2 \left (-4 a^3 \int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}+4 a \sqrt {1-\frac {1}{a^2 x^2}}+11 a \arcsin \left (\frac {1}{a x}\right )\right )+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}+a^4 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )-\frac {32 a^4 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^6}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {c^3 \left (a^2 \left (-\frac {3}{2} a^2 \left (-4 a \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )+4 a \sqrt {1-\frac {1}{a^2 x^2}}+11 a \arcsin \left (\frac {1}{a x}\right )\right )+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}+a^4 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )-\frac {32 a^4 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )}{a^6}\) |
Input:
Int[(c - c/(a*x))^3/E^(3*ArcCoth[a*x]),x]
Output:
-((c^3*((-32*a^4*(a - x^(-1)))/Sqrt[1 - 1/(a^2*x^2)] + a^2*((a^2*Sqrt[1 - 1/(a^2*x^2)])/(2*x) - a^4*Sqrt[1 - 1/(a^2*x^2)]*x - (3*a^2*(4*a*Sqrt[1 - 1 /(a^2*x^2)] + 11*a*ArcSin[1/(a*x)] - 4*a*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]))/ 2)))/a^6)
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 = 181, normalized size of antiderivative = 1.34
| method | result | size |
| risch | \(\frac {\left (a x +1\right ) \left (12 a x -1\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}}{2 x^{2} a^{3}}+\frac {\left (\frac {33 a^{2} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{2}-\frac {6 a^{3} \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+\frac {32 a \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{x +\frac {1}{a}}+a^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a^{3} \left (a x -1\right )}\) | \(181\) |
| default | \(-\frac {\left (-12 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{5} x^{5}+12 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{3} x^{3}-57 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{4} x^{4}-33 a^{4} \sqrt {a^{2}}\, x^{4} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+12 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+32 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}+23 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{2} x^{2}-78 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{3} x^{3}-66 a^{3} \sqrt {a^{2}}\, x^{3} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+24 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+10 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x -33 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{2} x^{2}-33 a^{2} \sqrt {a^{2}}\, x^{2} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+12 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right ) c^{3} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{2 a^{3} \sqrt {a^{2}}\, x^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )}\) | \(450\) |
Input:
int((c-c/a/x)^3*((a*x-1)/(a*x+1))^(3/2),x,method=_RETURNVERBOSE)
Output:
1/2*(a*x+1)*(12*a*x-1)/x^2*c^3/a^3*((a*x-1)/(a*x+1))^(1/2)+(33/2*a^2*arcta n(1/(a^2*x^2-1)^(1/2))-6*a^3*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2))/(a^2) ^(1/2)+32*a/(x+1/a)*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2)+a^2*((a*x-1)*(a*x+1) )^(1/2))*c^3/a^3/(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 = 146, normalized size of antiderivative = 1.08 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=-\frac {66 \, a^{2} c^{3} x^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + 12 \, a^{2} c^{3} x^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 12 \, a^{2} c^{3} x^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (2 \, a^{3} c^{3} x^{3} + 78 \, a^{2} c^{3} x^{2} + 11 \, a c^{3} x - c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, a^{3} x^{2}} \] Input:
integrate((c-c/a/x)^3*((a*x-1)/(a*x+1))^(3/2),x, algorithm="fricas")
Output:
-1/2*(66*a^2*c^3*x^2*arctan(sqrt((a*x - 1)/(a*x + 1))) + 12*a^2*c^3*x^2*lo g(sqrt((a*x - 1)/(a*x + 1)) + 1) - 12*a^2*c^3*x^2*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (2*a^3*c^3*x^3 + 78*a^2*c^3*x^2 + 11*a*c^3*x - c^3)*sqrt((a*x - 1)/(a*x + 1)))/(a^3*x^2)
\[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {c^{3} \left (\int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{4} + x^{3}}\, dx + \int \left (- \frac {4 a \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{3} + x^{2}}\right )\, dx + \int \frac {6 a^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{2} + x}\, dx + \int \left (- \frac {4 a^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \frac {a^{4} x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx\right )}{a^{3}} \] Input:
integrate((c-c/a/x)**3*((a*x-1)/(a*x+1))**(3/2),x)
Output:
c**3*(Integral(sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x**4 + x**3), x) + Int egral(-4*a*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x**3 + x**2), x) + Integra l(6*a**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x**2 + x), x) + Integral(-4* a**3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x) + Integral(a**4*x*sqr t(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x))/a**3
Time = 0.11 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.67 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=-{\left (\frac {33 \, c^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {6 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {6 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {32 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2}} + \frac {11 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 6 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 13 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - \frac {{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} + a^{2}}\right )} a \] Input:
integrate((c-c/a/x)^3*((a*x-1)/(a*x+1))^(3/2),x, algorithm="maxima")
Output:
-(33*c^3*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 + 6*c^3*log(sqrt((a*x - 1)/ (a*x + 1)) + 1)/a^2 - 6*c^3*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2 - 32*c^ 3*sqrt((a*x - 1)/(a*x + 1))/a^2 + (11*c^3*((a*x - 1)/(a*x + 1))^(5/2) - 6* c^3*((a*x - 1)/(a*x + 1))^(3/2) - 13*c^3*sqrt((a*x - 1)/(a*x + 1)))/((a*x - 1)*a^2/(a*x + 1) - (a*x - 1)^2*a^2/(a*x + 1)^2 - (a*x - 1)^3*a^2/(a*x + 1)^3 + a^2))*a
\[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\int { {\left (c - \frac {c}{a x}\right )}^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((c-c/a/x)^3*((a*x-1)/(a*x+1))^(3/2),x, algorithm="giac")
Output:
undef
Time = 13.51 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.41 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {13\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}+6\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-11\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{a+\frac {a\,\left (a\,x-1\right )}{a\,x+1}-\frac {a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}}+\frac {32\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a}-\frac {33\,c^3\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}+\frac {c^3\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,12{}\mathrm {i}}{a} \] Input:
int((c - c/(a*x))^3*((a*x - 1)/(a*x + 1))^(3/2),x)
Output:
(13*c^3*((a*x - 1)/(a*x + 1))^(1/2) + 6*c^3*((a*x - 1)/(a*x + 1))^(3/2) - 11*c^3*((a*x - 1)/(a*x + 1))^(5/2))/(a + (a*(a*x - 1))/(a*x + 1) - (a*(a*x - 1)^2)/(a*x + 1)^2 - (a*(a*x - 1)^3)/(a*x + 1)^3) + (32*c^3*((a*x - 1)/( a*x + 1))^(1/2))/a - (33*c^3*atan(((a*x - 1)/(a*x + 1))^(1/2)))/a + (c^3*a tan(((a*x - 1)/(a*x + 1))^(1/2)*1i)*12i)/a
Time = 0.17 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.86 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {c^{3} \left (-66 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}-1\right ) a^{3} x^{3}-66 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}-1\right ) a^{2} x^{2}+66 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}+1\right ) a^{3} x^{3}+66 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}+1\right ) a^{2} x^{2}+2 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}+78 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}+11 \sqrt {a x +1}\, \sqrt {a x -1}\, a x -\sqrt {a x +1}\, \sqrt {a x -1}-24 \,\mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{3} x^{3}-24 \,\mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{2} x^{2}+53 a^{3} x^{3}+53 a^{2} x^{2}\right )}{2 a^{3} x^{2} \left (a x +1\right )} \] Input:
int((c-c/a/x)^3*((a*x-1)/(a*x+1))^(3/2),x)
Output:
(c**3*( - 66*atan(sqrt(a*x - 1) + sqrt(a*x + 1) - 1)*a**3*x**3 - 66*atan(s qrt(a*x - 1) + sqrt(a*x + 1) - 1)*a**2*x**2 + 66*atan(sqrt(a*x - 1) + sqrt (a*x + 1) + 1)*a**3*x**3 + 66*atan(sqrt(a*x - 1) + sqrt(a*x + 1) + 1)*a**2 *x**2 + 2*sqrt(a*x + 1)*sqrt(a*x - 1)*a**3*x**3 + 78*sqrt(a*x + 1)*sqrt(a* x - 1)*a**2*x**2 + 11*sqrt(a*x + 1)*sqrt(a*x - 1)*a*x - sqrt(a*x + 1)*sqrt (a*x - 1) - 24*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a**3*x**3 - 24 *log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a**2*x**2 + 53*a**3*x**3 + 5 3*a**2*x**2))/(2*a**3*x**2*(a*x + 1))