Integrand size = 22, antiderivative size = 157 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=-\frac {2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {3 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{a \sqrt {c-\frac {c}{a x}}}+\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x}{\left (c-\frac {c}{a x}\right )^{3/2}}-\frac {3 c^{5/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a} \]
-2/3*c^4*(1-1/a^2/x^2)^(3/2)/a/(c-c/a/x)^(3/2)+c^4*(1-1/a^2/x^2)^(3/2)*x/( c-c/a/x)^(3/2)-3*c^(5/2)*arctanh(c^(1/2)*(1-1/a^2/x^2)^(1/2)/(c-c/a/x)^(1/ 2))/a+3*c^3*(1-1/a^2/x^2)^(1/2)/a/(c-c/a/x)^(1/2)
Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.57 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\frac {c^2 \sqrt {c-\frac {c}{a x}} \left (\sqrt {1+\frac {1}{a x}} \left (-2+10 a x+3 a^2 x^2\right )-9 a x \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )\right )}{3 a^2 \sqrt {1-\frac {1}{a x}} x} \]
(c^2*Sqrt[c - c/(a*x)]*(Sqrt[1 + 1/(a*x)]*(-2 + 10*a*x + 3*a^2*x^2) - 9*a* x*ArcTanh[Sqrt[1 + 1/(a*x)]]))/(3*a^2*Sqrt[1 - 1/(a*x)]*x)
Time = 0.34 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.73, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6731, 585, 27, 100, 27, 90, 60, 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.
| \(\displaystyle \int \left (c-\frac {c}{a x}\right )^{5/2} e^{\coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6731 |
| \(\displaystyle -c \int \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{3/2} x^2d\frac {1}{x}\) |
| \(\Big \downarrow \) 585 |
| \(\displaystyle -\frac {c^2 \sqrt {c-\frac {c}{a x}} \int \frac {\left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}} x^2}{a^2}d\frac {1}{x}}{\sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^2 \sqrt {c-\frac {c}{a x}} \int \left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}} x^2d\frac {1}{x}}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle -\frac {c^2 \sqrt {c-\frac {c}{a x}} \left (\int -\frac {1}{2} \left (3 a-\frac {2}{x}\right ) \sqrt {1+\frac {1}{a x}} xd\frac {1}{x}-a^2 x \left (\frac {1}{a x}+1\right )^{3/2}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^2 \sqrt {c-\frac {c}{a x}} \left (a^2 x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right )-\frac {1}{2} \int \left (3 a-\frac {2}{x}\right ) \sqrt {1+\frac {1}{a x}} xd\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle -\frac {c^2 \sqrt {c-\frac {c}{a x}} \left (\frac {1}{2} \left (\frac {4}{3} a \left (\frac {1}{a x}+1\right )^{3/2}-3 a \int \sqrt {1+\frac {1}{a x}} xd\frac {1}{x}\right )-a^2 x \left (\frac {1}{a x}+1\right )^{3/2}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle -\frac {c^2 \sqrt {c-\frac {c}{a x}} \left (\frac {1}{2} \left (\frac {4}{3} a \left (\frac {1}{a x}+1\right )^{3/2}-3 a \left (\int \frac {x}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+2 \sqrt {\frac {1}{a x}+1}\right )\right )-a^2 x \left (\frac {1}{a x}+1\right )^{3/2}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {c^2 \sqrt {c-\frac {c}{a x}} \left (\frac {1}{2} \left (\frac {4}{3} a \left (\frac {1}{a x}+1\right )^{3/2}-3 a \left (2 a \int \frac {1}{\frac {a}{x^2}-a}d\sqrt {1+\frac {1}{a x}}+2 \sqrt {\frac {1}{a x}+1}\right )\right )-a^2 x \left (\frac {1}{a x}+1\right )^{3/2}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {c^2 \left (\frac {1}{2} \left (\frac {4}{3} a \left (\frac {1}{a x}+1\right )^{3/2}-3 a \left (2 \sqrt {\frac {1}{a x}+1}-2 \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right )\right )\right )-a^2 x \left (\frac {1}{a x}+1\right )^{3/2}\right ) \sqrt {c-\frac {c}{a x}}}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
-((c^2*Sqrt[c - c/(a*x)]*(-(a^2*(1 + 1/(a*x))^(3/2)*x) + ((4*a*(1 + 1/(a*x ))^(3/2))/3 - 3*a*(2*Sqrt[1 + 1/(a*x)] - 2*ArcTanh[Sqrt[1 + 1/(a*x)]]))/2) )/(a^2*Sqrt[1 - 1/(a*x)]))
3.5.40.3.1 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_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
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_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_) , x_Symbol] :> Simp[a^p*c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^F racPart[n]) Int[(e*x)^m*(1 - d*(x/c))^p*(1 + d*(x/c))^(n + p), x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[a, 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 = 0.08 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.84
| method | result | size |
| default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{2} \left (-6 a^{\frac {5}{2}} x^{2} \sqrt {\left (a x +1\right ) x}+9 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{2} x^{2}-20 a^{\frac {3}{2}} x \sqrt {\left (a x +1\right ) x}+4 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}\right )}{6 \sqrt {\frac {a x -1}{a x +1}}\, x \,a^{\frac {5}{2}} \sqrt {\left (a x +1\right ) x}}\) | \(132\) |
| risch | \(\frac {\left (3 a^{3} x^{3}+13 a^{2} x^{2}+8 a x -2\right ) c^{2} \sqrt {\frac {c \left (a x -1\right )}{a x}}}{3 x \,a^{2} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}-\frac {3 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right ) c^{2} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{2 \sqrt {a^{2} c}\, \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(168\) |
-1/6/((a*x-1)/(a*x+1))^(1/2)*(c*(a*x-1)/a/x)^(1/2)/x*c^2/a^(5/2)*(-6*a^(5/ 2)*x^2*((a*x+1)*x)^(1/2)+9*ln(1/2*(2*((a*x+1)*x)^(1/2)*a^(1/2)+2*a*x+1)/a^ (1/2))*a^2*x^2-20*a^(3/2)*x*((a*x+1)*x)^(1/2)+4*((a*x+1)*x)^(1/2)*a^(1/2)) /((a*x+1)*x)^(1/2)
Time = 0.29 (sec) , antiderivative size = 381, normalized size of antiderivative = 2.43 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\left [\frac {9 \, {\left (a^{2} c^{2} x^{2} - a c^{2} x\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (3 \, a^{3} c^{2} x^{3} + 13 \, a^{2} c^{2} x^{2} + 8 \, a c^{2} x - 2 \, c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{12 \, {\left (a^{3} x^{2} - a^{2} x\right )}}, \frac {9 \, {\left (a^{2} c^{2} x^{2} - a c^{2} x\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (3 \, a^{3} c^{2} x^{3} + 13 \, a^{2} c^{2} x^{2} + 8 \, a c^{2} x - 2 \, c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{6 \, {\left (a^{3} x^{2} - a^{2} x\right )}}\right ] \]
[1/12*(9*(a^2*c^2*x^2 - a*c^2*x)*sqrt(c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*( 2*a^3*x^3 + 3*a^2*x^2 + a*x)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(3*a^3*c^2*x^3 + 13*a^2*c^2*x^2 + 8*a*c^2 *x - 2*c^2)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2 - a^2*x), 1/6*(9*(a^2*c^2*x^2 - a*c^2*x)*sqrt(-c)*arctan(2*(a^2*x^2 + a*x)*s qrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a *c*x - c)) + 2*(3*a^3*c^2*x^3 + 13*a^2*c^2*x^2 + 8*a*c^2*x - 2*c^2)*sqrt(( a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2 - a^2*x)]
Timed out. \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\text {Timed out} \]
\[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\int { \frac {{\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
\[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\int { \frac {{\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
Timed out. \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\int \frac {{\left (c-\frac {c}{a\,x}\right )}^{5/2}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]