Integrand size = 10, antiderivative size = 130 \[ \int e^{-\frac {5}{2} \coth ^{-1}(a x)} \, dx=\frac {10 \sqrt [4]{1-\frac {1}{a x}}}{a \sqrt [4]{1+\frac {1}{a x}}}+\frac {\left (1-\frac {1}{a x}\right )^{5/4} x}{\sqrt [4]{1+\frac {1}{a x}}}+\frac {5 \arctan \left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{a}-\frac {5 \text {arctanh}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{a} \] Output:
10*(1-1/a/x)^(1/4)/a/(1+1/a/x)^(1/4)+(1-1/a/x)^(5/4)*x/(1+1/a/x)^(1/4)+5*a rctan((1+1/a/x)^(1/4)/(1-1/a/x)^(1/4))/a-5*arctanh((1+1/a/x)^(1/4)/(1-1/a/ x)^(1/4))/a
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.24 \[ \int e^{-\frac {5}{2} \coth ^{-1}(a x)} \, dx=\frac {8 e^{-\frac {1}{2} \coth ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},2,\frac {3}{4},e^{2 \coth ^{-1}(a x)}\right )}{a} \] Input:
Integrate[E^((-5*ArcCoth[a*x])/2),x]
Output:
(8*Hypergeometric2F1[-1/4, 2, 3/4, E^(2*ArcCoth[a*x])])/(a*E^(ArcCoth[a*x] /2))
Time = 0.44 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6720, 105, 105, 104, 25, 827, 216, 219}
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.
| \(\displaystyle \int e^{-\frac {5}{2} \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6720 |
| \(\displaystyle -\int \frac {\left (1-\frac {1}{a x}\right )^{5/4} x^2}{\left (1+\frac {1}{a x}\right )^{5/4}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {5 \int \frac {\sqrt [4]{1-\frac {1}{a x}} x}{\left (1+\frac {1}{a x}\right )^{5/4}}d\frac {1}{x}}{2 a}+\frac {x \left (1-\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{\frac {1}{a x}+1}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {5 \left (\int \frac {x}{\left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}+\frac {4 \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}\right )}{2 a}+\frac {x \left (1-\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{\frac {1}{a x}+1}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {5 \left (4 \int -\frac {1}{\left (1-\frac {1}{x^4}\right ) x^2}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}+\frac {4 \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}\right )}{2 a}+\frac {x \left (1-\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{\frac {1}{a x}+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5 \left (\frac {4 \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}-4 \int \frac {1}{\left (1-\frac {1}{x^4}\right ) x^2}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{2 a}+\frac {x \left (1-\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{\frac {1}{a x}+1}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {5 \left (4 \left (\frac {1}{2} \int \frac {1}{1+\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {1}{2} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )+\frac {4 \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}\right )}{2 a}+\frac {x \left (1-\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{\frac {1}{a x}+1}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {5 \left (4 \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )+\frac {4 \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}\right )}{2 a}+\frac {x \left (1-\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{\frac {1}{a x}+1}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5 \left (4 \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )\right )+\frac {4 \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}\right )}{2 a}+\frac {x \left (1-\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{\frac {1}{a x}+1}}\) |
Input:
Int[E^((-5*ArcCoth[a*x])/2),x]
Output:
((1 - 1/(a*x))^(5/4)*x)/(1 + 1/(a*x))^(1/4) + (5*((4*(1 - 1/(a*x))^(1/4))/ (1 + 1/(a*x))^(1/4) + 4*(ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)]/2 - ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)]/2)))/(2*a)
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_)), x_Symbol] :> -Subst[Int[(1 + x/a)^(n/2)/( x^2*(1 - x/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, n}, x] && !IntegerQ[n]
\[\int \left (\frac {a x -1}{a x +1}\right )^{\frac {5}{4}}d x\]
Input:
int(((a*x-1)/(a*x+1))^(5/4),x)
Output:
int(((a*x-1)/(a*x+1))^(5/4),x)
Time = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.66 \[ \int e^{-\frac {5}{2} \coth ^{-1}(a x)} \, dx=\frac {2 \, {\left (a x + 9\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 10 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) - 5 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + 5 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{2 \, a} \] Input:
integrate(((a*x-1)/(a*x+1))^(5/4),x, algorithm="fricas")
Output:
1/2*(2*(a*x + 9)*((a*x - 1)/(a*x + 1))^(1/4) - 10*arctan(((a*x - 1)/(a*x + 1))^(1/4)) - 5*log(((a*x - 1)/(a*x + 1))^(1/4) + 1) + 5*log(((a*x - 1)/(a *x + 1))^(1/4) - 1))/a
\[ \int e^{-\frac {5}{2} \coth ^{-1}(a x)} \, dx=\int \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}}\, dx \] Input:
integrate(((a*x-1)/(a*x+1))**(5/4),x)
Output:
Integral(((a*x - 1)/(a*x + 1))**(5/4), x)
Time = 0.11 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.02 \[ \int e^{-\frac {5}{2} \coth ^{-1}(a x)} \, dx=-\frac {1}{2} \, a {\left (\frac {4 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{\frac {{\left (a x - 1\right )} a^{2}}{a x + 1} - a^{2}} + \frac {10 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{2}} + \frac {5 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{2}} - \frac {5 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{a^{2}} - \frac {16 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a^{2}}\right )} \] Input:
integrate(((a*x-1)/(a*x+1))^(5/4),x, algorithm="maxima")
Output:
-1/2*a*(4*((a*x - 1)/(a*x + 1))^(1/4)/((a*x - 1)*a^2/(a*x + 1) - a^2) + 10 *arctan(((a*x - 1)/(a*x + 1))^(1/4))/a^2 + 5*log(((a*x - 1)/(a*x + 1))^(1/ 4) + 1)/a^2 - 5*log(((a*x - 1)/(a*x + 1))^(1/4) - 1)/a^2 - 16*((a*x - 1)/( a*x + 1))^(1/4)/a^2)
Time = 0.15 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.99 \[ \int e^{-\frac {5}{2} \coth ^{-1}(a x)} \, dx=-\frac {1}{2} \, a {\left (\frac {10 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{2}} + \frac {5 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{2}} - \frac {5 \, \log \left ({\left | \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1 \right |}\right )}{a^{2}} - \frac {16 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a^{2}} + \frac {4 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a^{2} {\left (\frac {a x - 1}{a x + 1} - 1\right )}}\right )} \] Input:
integrate(((a*x-1)/(a*x+1))^(5/4),x, algorithm="giac")
Output:
-1/2*a*(10*arctan(((a*x - 1)/(a*x + 1))^(1/4))/a^2 + 5*log(((a*x - 1)/(a*x + 1))^(1/4) + 1)/a^2 - 5*log(abs(((a*x - 1)/(a*x + 1))^(1/4) - 1))/a^2 - 16*((a*x - 1)/(a*x + 1))^(1/4)/a^2 + 4*((a*x - 1)/(a*x + 1))^(1/4)/(a^2*(( a*x - 1)/(a*x + 1) - 1)))
Time = 23.78 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.79 \[ \int e^{-\frac {5}{2} \coth ^{-1}(a x)} \, dx=\frac {2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}}{a-\frac {a\,\left (a\,x-1\right )}{a\,x+1}}+\frac {8\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}}{a}-\frac {5\,\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{a}+\frac {\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\,1{}\mathrm {i}\right )\,5{}\mathrm {i}}{a} \] Input:
int(((a*x - 1)/(a*x + 1))^(5/4),x)
Output:
(2*((a*x - 1)/(a*x + 1))^(1/4))/(a - (a*(a*x - 1))/(a*x + 1)) + (atan(((a* x - 1)/(a*x + 1))^(1/4)*1i)*5i)/a + (8*((a*x - 1)/(a*x + 1))^(1/4))/a - (5 *atan(((a*x - 1)/(a*x + 1))^(1/4)))/a
\[ \int e^{-\frac {5}{2} \coth ^{-1}(a x)} \, dx=-\left (\int \frac {\left (a x -1\right )^{\frac {1}{4}}}{\left (a x +1\right )^{\frac {1}{4}} a x +\left (a x +1\right )^{\frac {1}{4}}}d x \right )+\left (\int \frac {\left (a x -1\right )^{\frac {1}{4}} x}{\left (a x +1\right )^{\frac {1}{4}} a x +\left (a x +1\right )^{\frac {1}{4}}}d x \right ) a \] Input:
int(((a*x-1)/(a*x+1))^(5/4),x)
Output:
- int((a*x - 1)**(1/4)/((a*x + 1)**(1/4)*a*x + (a*x + 1)**(1/4)),x) + int (((a*x - 1)**(1/4)*x)/((a*x + 1)**(1/4)*a*x + (a*x + 1)**(1/4)),x)*a