Integrand size = 14, antiderivative size = 194 \[ \int \frac {e^{\frac {5}{2} \text {arctanh}(a x)}}{x^5} \, dx=\frac {2467 a^4 \sqrt [4]{1+a x}}{192 \sqrt [4]{1-a x}}-\frac {\sqrt [4]{1+a x}}{4 x^4 \sqrt [4]{1-a x}}-\frac {17 a \sqrt [4]{1+a x}}{24 x^3 \sqrt [4]{1-a x}}-\frac {113 a^2 \sqrt [4]{1+a x}}{96 x^2 \sqrt [4]{1-a x}}-\frac {521 a^3 \sqrt [4]{1+a x}}{192 x \sqrt [4]{1-a x}}-\frac {475}{64} a^4 \arctan \left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac {475}{64} a^4 \text {arctanh}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right ) \]
2467/192*a^4*(a*x+1)^(1/4)/(-a*x+1)^(1/4)-1/4*(a*x+1)^(1/4)/x^4/(-a*x+1)^( 1/4)-17/24*a*(a*x+1)^(1/4)/x^3/(-a*x+1)^(1/4)-113/96*a^2*(a*x+1)^(1/4)/x^2 /(-a*x+1)^(1/4)-521/192*a^3*(a*x+1)^(1/4)/x/(-a*x+1)^(1/4)-475/64*a^4*arct an((a*x+1)^(1/4)/(-a*x+1)^(1/4))-475/64*a^4*arctanh((a*x+1)^(1/4)/(-a*x+1) ^(1/4))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.51 \[ \int \frac {e^{\frac {5}{2} \text {arctanh}(a x)}}{x^5} \, dx=\frac {-48-184 a x-362 a^2 x^2-747 a^3 x^3+1946 a^4 x^4+2467 a^5 x^5+950 a^4 x^4 (-1+a x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},\frac {1-a x}{1+a x}\right )}{192 x^4 \sqrt [4]{1-a x} (1+a x)^{3/4}} \]
(-48 - 184*a*x - 362*a^2*x^2 - 747*a^3*x^3 + 1946*a^4*x^4 + 2467*a^5*x^5 + 950*a^4*x^4*(-1 + a*x)*Hypergeometric2F1[3/4, 1, 7/4, (1 - a*x)/(1 + a*x) ])/(192*x^4*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))
Time = 0.35 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.04, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.071, Rules used = {6676, 109, 27, 168, 27, 168, 27, 168, 27, 172, 27, 104, 756, 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.
\(\displaystyle \int \frac {e^{\frac {5}{2} \text {arctanh}(a x)}}{x^5} \, dx\) |
\(\Big \downarrow \) 6676 |
\(\displaystyle \int \frac {(a x+1)^{5/4}}{x^5 (1-a x)^{5/4}}dx\) |
\(\Big \downarrow \) 109 |
\(\displaystyle -\frac {1}{4} \int -\frac {a (16 a x+17)}{2 x^4 (1-a x)^{5/4} (a x+1)^{3/4}}dx-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{8} a \int \frac {16 a x+17}{x^4 (1-a x)^{5/4} (a x+1)^{3/4}}dx-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\) |
\(\Big \downarrow \) 168 |
\(\displaystyle \frac {1}{8} a \left (-\frac {1}{3} \int -\frac {a (102 a x+113)}{2 x^3 (1-a x)^{5/4} (a x+1)^{3/4}}dx-\frac {17 \sqrt [4]{a x+1}}{3 x^3 \sqrt [4]{1-a x}}\right )-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \int \frac {102 a x+113}{x^3 (1-a x)^{5/4} (a x+1)^{3/4}}dx-\frac {17 \sqrt [4]{a x+1}}{3 x^3 \sqrt [4]{1-a x}}\right )-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\) |
\(\Big \downarrow \) 168 |
\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (-\frac {1}{2} \int -\frac {a (452 a x+521)}{2 x^2 (1-a x)^{5/4} (a x+1)^{3/4}}dx-\frac {113 \sqrt [4]{a x+1}}{2 x^2 \sqrt [4]{1-a x}}\right )-\frac {17 \sqrt [4]{a x+1}}{3 x^3 \sqrt [4]{1-a x}}\right )-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (\frac {1}{4} a \int \frac {452 a x+521}{x^2 (1-a x)^{5/4} (a x+1)^{3/4}}dx-\frac {113 \sqrt [4]{a x+1}}{2 x^2 \sqrt [4]{1-a x}}\right )-\frac {17 \sqrt [4]{a x+1}}{3 x^3 \sqrt [4]{1-a x}}\right )-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\) |
\(\Big \downarrow \) 168 |
\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (\frac {1}{4} a \left (-\int -\frac {a (1042 a x+1425)}{2 x (1-a x)^{5/4} (a x+1)^{3/4}}dx-\frac {521 \sqrt [4]{a x+1}}{x \sqrt [4]{1-a x}}\right )-\frac {113 \sqrt [4]{a x+1}}{2 x^2 \sqrt [4]{1-a x}}\right )-\frac {17 \sqrt [4]{a x+1}}{3 x^3 \sqrt [4]{1-a x}}\right )-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (\frac {1}{4} a \left (\frac {1}{2} a \int \frac {1042 a x+1425}{x (1-a x)^{5/4} (a x+1)^{3/4}}dx-\frac {521 \sqrt [4]{a x+1}}{x \sqrt [4]{1-a x}}\right )-\frac {113 \sqrt [4]{a x+1}}{2 x^2 \sqrt [4]{1-a x}}\right )-\frac {17 \sqrt [4]{a x+1}}{3 x^3 \sqrt [4]{1-a x}}\right )-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\) |
\(\Big \downarrow \) 172 |
\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (\frac {1}{4} a \left (\frac {1}{2} a \left (\frac {4934 \sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {2 \int -\frac {1425 a}{2 x \sqrt [4]{1-a x} (a x+1)^{3/4}}dx}{a}\right )-\frac {521 \sqrt [4]{a x+1}}{x \sqrt [4]{1-a x}}\right )-\frac {113 \sqrt [4]{a x+1}}{2 x^2 \sqrt [4]{1-a x}}\right )-\frac {17 \sqrt [4]{a x+1}}{3 x^3 \sqrt [4]{1-a x}}\right )-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (\frac {1}{4} a \left (\frac {1}{2} a \left (1425 \int \frac {1}{x \sqrt [4]{1-a x} (a x+1)^{3/4}}dx+\frac {4934 \sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {521 \sqrt [4]{a x+1}}{x \sqrt [4]{1-a x}}\right )-\frac {113 \sqrt [4]{a x+1}}{2 x^2 \sqrt [4]{1-a x}}\right )-\frac {17 \sqrt [4]{a x+1}}{3 x^3 \sqrt [4]{1-a x}}\right )-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (\frac {1}{4} a \left (\frac {1}{2} a \left (5700 \int \frac {1}{\frac {a x+1}{1-a x}-1}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}+\frac {4934 \sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {521 \sqrt [4]{a x+1}}{x \sqrt [4]{1-a x}}\right )-\frac {113 \sqrt [4]{a x+1}}{2 x^2 \sqrt [4]{1-a x}}\right )-\frac {17 \sqrt [4]{a x+1}}{3 x^3 \sqrt [4]{1-a x}}\right )-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\) |
\(\Big \downarrow \) 756 |
\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (\frac {1}{4} a \left (\frac {1}{2} a \left (5700 \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a x+1}}{\sqrt {1-a x}}}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {1}{2} \int \frac {1}{\frac {\sqrt {a x+1}}{\sqrt {1-a x}}+1}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+\frac {4934 \sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {521 \sqrt [4]{a x+1}}{x \sqrt [4]{1-a x}}\right )-\frac {113 \sqrt [4]{a x+1}}{2 x^2 \sqrt [4]{1-a x}}\right )-\frac {17 \sqrt [4]{a x+1}}{3 x^3 \sqrt [4]{1-a x}}\right )-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (\frac {1}{4} a \left (\frac {1}{2} a \left (5700 \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a x+1}}{\sqrt {1-a x}}}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )\right )+\frac {4934 \sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {521 \sqrt [4]{a x+1}}{x \sqrt [4]{1-a x}}\right )-\frac {113 \sqrt [4]{a x+1}}{2 x^2 \sqrt [4]{1-a x}}\right )-\frac {17 \sqrt [4]{a x+1}}{3 x^3 \sqrt [4]{1-a x}}\right )-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (\frac {1}{4} a \left (\frac {1}{2} a \left (5700 \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )\right )+\frac {4934 \sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {521 \sqrt [4]{a x+1}}{x \sqrt [4]{1-a x}}\right )-\frac {113 \sqrt [4]{a x+1}}{2 x^2 \sqrt [4]{1-a x}}\right )-\frac {17 \sqrt [4]{a x+1}}{3 x^3 \sqrt [4]{1-a x}}\right )-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\) |
-1/4*(1 + a*x)^(1/4)/(x^4*(1 - a*x)^(1/4)) + (a*((-17*(1 + a*x)^(1/4))/(3* x^3*(1 - a*x)^(1/4)) + (a*((-113*(1 + a*x)^(1/4))/(2*x^2*(1 - a*x)^(1/4)) + (a*((-521*(1 + a*x)^(1/4))/(x*(1 - a*x)^(1/4)) + (a*((4934*(1 + a*x)^(1/ 4))/(1 - a*x)^(1/4) + 5700*(-1/2*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]/2)))/2))/4))/6))/8
3.1.89.3.1 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_))/((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[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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] && ILtQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> With[{mnp = Simplify[m + n + p]}, 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] + 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*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)*(mnp + 3)*x, x], x], x] /; ILtQ[mnp + 2, 0] && (SumSimplerQ[m, 1] | | ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) && !(NeQ[p, -1] && SumSimplerQ[p, 1 ])))] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && 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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x) ^m*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c, m, n}, x] && !Int egerQ[(n - 1)/2]
\[\int \frac {{\left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}^{\frac {5}{2}}}{x^{5}}d x\]
Time = 0.27 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.83 \[ \int \frac {e^{\frac {5}{2} \text {arctanh}(a x)}}{x^5} \, dx=-\frac {2850 \, a^{4} x^{4} \arctan \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) + 1425 \, a^{4} x^{4} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) - 1425 \, a^{4} x^{4} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) - 2 \, {\left (2467 \, a^{4} x^{4} - 521 \, a^{3} x^{3} - 226 \, a^{2} x^{2} - 136 \, a x - 48\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{384 \, x^{4}} \]
-1/384*(2850*a^4*x^4*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 1425*a^ 4*x^4*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - 1425*a^4*x^4*log(sqrt (-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) - 2*(2467*a^4*x^4 - 521*a^3*x^3 - 226 *a^2*x^2 - 136*a*x - 48)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x^4
\[ \int \frac {e^{\frac {5}{2} \text {arctanh}(a x)}}{x^5} \, dx=\int \frac {\left (\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}}{x^{5}}\, dx \]
\[ \int \frac {e^{\frac {5}{2} \text {arctanh}(a x)}}{x^5} \, dx=\int { \frac {\left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}}{x^{5}} \,d x } \]
\[ \int \frac {e^{\frac {5}{2} \text {arctanh}(a x)}}{x^5} \, dx=\int { \frac {\left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}}{x^{5}} \,d x } \]
Timed out. \[ \int \frac {e^{\frac {5}{2} \text {arctanh}(a x)}}{x^5} \, dx=\int \frac {{\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{5/2}}{x^5} \,d x \]