Integrand size = 24, antiderivative size = 199 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {2 \left (1-\frac {1}{a x}\right )^{5/2}}{a \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {\left (1-\frac {1}{a x}\right )^{5/2} x}{\sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{\sqrt {2} a \left (c-\frac {c}{a x}\right )^{5/2}} \]
-(1-1/a/x)^(5/2)*arctanh((1+1/a/x)^(1/2))/a/(c-c/a/x)^(5/2)-1/2*(1-1/a/x)^ (5/2)*arctanh(1/2*(1+1/a/x)^(1/2)*2^(1/2))/a/(c-c/a/x)^(5/2)*2^(1/2)+2*(1- 1/a/x)^(5/2)/a/(c-c/a/x)^(5/2)/(1+1/a/x)^(1/2)+(1-1/a/x)^(5/2)*x/(c-c/a/x) ^(5/2)/(1+1/a/x)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.45 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {\sqrt {1-\frac {1}{a x}} \left (a x+\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {a+\frac {1}{x}}{2 a}\right )+\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1+\frac {1}{a x}\right )\right )}{a c^2 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}}} \]
(Sqrt[1 - 1/(a*x)]*(a*x + Hypergeometric2F1[-1/2, 1, 1/2, (a + x^(-1))/(2* a)] + Hypergeometric2F1[-1/2, 1, 1/2, 1 + 1/(a*x)]))/(a*c^2*Sqrt[1 + 1/(a* x)]*Sqrt[c - c/(a*x)])
Time = 0.37 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.61, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6731, 585, 27, 114, 27, 169, 174, 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.
| \(\displaystyle \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6731 |
| \(\displaystyle -\frac {\int \frac {\sqrt {c-\frac {c}{a x}} x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{c^3}\) |
| \(\Big \downarrow \) 585 |
| \(\displaystyle -\frac {\sqrt {c-\frac {c}{a x}} \int \frac {a x^2}{\left (a-\frac {1}{x}\right ) \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}}{c^3 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \sqrt {c-\frac {c}{a x}} \int \frac {x^2}{\left (a-\frac {1}{x}\right ) \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}}{c^3 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {a \sqrt {c-\frac {c}{a x}} \left (-\frac {\int \frac {\left (a-\frac {3}{x}\right ) x}{2 a \left (a-\frac {1}{x}\right ) \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}}{a}-\frac {x}{a \sqrt {\frac {1}{a x}+1}}\right )}{c^3 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \sqrt {c-\frac {c}{a x}} \left (-\frac {\int \frac {\left (a-\frac {3}{x}\right ) x}{\left (a-\frac {1}{x}\right ) \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}}{2 a^2}-\frac {x}{a \sqrt {\frac {1}{a x}+1}}\right )}{c^3 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a \sqrt {c-\frac {c}{a x}} \left (-\frac {\int \frac {\left (a-\frac {2}{x}\right ) x}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {4}{\sqrt {\frac {1}{a x}+1}}}{2 a^2}-\frac {x}{a \sqrt {\frac {1}{a x}+1}}\right )}{c^3 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle -\frac {a \sqrt {c-\frac {c}{a x}} \left (-\frac {-\int \frac {1}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\int \frac {x}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {4}{\sqrt {\frac {1}{a x}+1}}}{2 a^2}-\frac {x}{a \sqrt {\frac {1}{a x}+1}}\right )}{c^3 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {a \sqrt {c-\frac {c}{a x}} \left (-\frac {-2 a \int \frac {1}{2 a-\frac {a}{x^2}}d\sqrt {1+\frac {1}{a x}}+2 a \int \frac {1}{\frac {a}{x^2}-a}d\sqrt {1+\frac {1}{a x}}+\frac {4}{\sqrt {\frac {1}{a x}+1}}}{2 a^2}-\frac {x}{a \sqrt {\frac {1}{a x}+1}}\right )}{c^3 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {a \left (-\frac {-2 \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right )+\frac {4}{\sqrt {\frac {1}{a x}+1}}}{2 a^2}-\frac {x}{a \sqrt {\frac {1}{a x}+1}}\right ) \sqrt {c-\frac {c}{a x}}}{c^3 \sqrt {1-\frac {1}{a x}}}\) |
-((a*Sqrt[c - c/(a*x)]*(-(x/(a*Sqrt[1 + 1/(a*x)])) - (4/Sqrt[1 + 1/(a*x)] - 2*ArcTanh[Sqrt[1 + 1/(a*x)]] - Sqrt[2]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2] ])/(2*a^2)))/(c^3*Sqrt[1 - 1/(a*x)]))
3.5.87.3.1 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] :> 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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[((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 = 0.32 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.30
| method | result | size |
| risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{2} \sqrt {\frac {c \left (a x -1\right )}{a x}}}+\frac {\left (-\frac {\ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{2 a^{3} \sqrt {a^{2} c}}+\frac {\sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-\left (x +\frac {1}{a}\right ) a c}}{a^{5} c \left (x +\frac {1}{a}\right )}-\frac {\sqrt {2}\, \ln \left (\frac {4 c +3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+3 \left (x -\frac {1}{a}\right ) a c +2 c}}{x -\frac {1}{a}}\right )}{4 a^{4} \sqrt {c}}\right ) a^{2} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) a c x}}{c^{2} x \sqrt {\frac {c \left (a x -1\right )}{a x}}}\) | \(258\) |
| default | \(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (4 \sqrt {\left (a x +1\right ) x}\, a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, x -a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) x +8 \sqrt {\left (a x +1\right ) x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}-2 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{2} \sqrt {\frac {1}{a}}\, x -2 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}-\sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) \sqrt {a}\right )}{4 \left (a x -1\right )^{2} a^{\frac {3}{2}} c^{3} \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}}\) | \(264\) |
1/a*(a*x+1)/c^2*((a*x-1)/(a*x+1))^(1/2)/(c*(a*x-1)/a/x)^(1/2)+(-1/2/a^3*ln ((1/2*a*c+a^2*c*x)/(a^2*c)^(1/2)+(a^2*c*x^2+a*c*x)^(1/2))/(a^2*c)^(1/2)+1/ a^5/c/(x+1/a)*(a^2*c*(x+1/a)^2-(x+1/a)*a*c)^(1/2)-1/4/a^4/c^(1/2)*2^(1/2)* ln((4*c+3*(x-1/a)*a*c+2*2^(1/2)*c^(1/2)*(a^2*c*(x-1/a)^2+3*(x-1/a)*a*c+2*c )^(1/2))/(x-1/a)))*a^2/c^2*((a*x-1)/(a*x+1))^(1/2)/x/(c*(a*x-1)/a/x)^(1/2) *((a*x+1)*a*c*x)^(1/2)
Time = 0.32 (sec) , antiderivative size = 524, normalized size of antiderivative = 2.63 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\left [\frac {\sqrt {2} {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt {2} {\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 2 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 8 \, {\left (a^{2} x^{2} + 2 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{8 \, {\left (a^{2} c^{3} x - a c^{3}\right )}}, \frac {\sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {2} {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) + 2 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 4 \, {\left (a^{2} x^{2} + 2 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} c^{3} x - a c^{3}\right )}}\right ] \]
[1/8*(sqrt(2)*(a*x - 1)*sqrt(c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c* x - 4*sqrt(2)*(3*a^3*x^3 + 4*a^2*x^2 + a*x)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 2*(a *x - 1)*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)) + 8*(a^2*x^2 + 2*a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x) ))/(a^2*c^3*x - a*c^3), 1/4*(sqrt(2)*(a*x - 1)*sqrt(-c)*arctan(2*sqrt(2)*( a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/ (3*a^2*c*x^2 - 2*a*c*x - c)) + 2*(a*x - 1)*sqrt(-c)*arctan(2*(a^2*x^2 + a* x)*sqrt(-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)) + 4*(a^2*x^2 + 2*a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^2*c^3*x - a*c^3)]
Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}}} \,d x } \]
Exception generated. \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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
Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\int \frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{{\left (c-\frac {c}{a\,x}\right )}^{5/2}} \,d x \]