Integrand size = 14, antiderivative size = 168 \[ \int \frac {e^{\frac {3}{2} \text {arctanh}(a x)}}{x^5} \, dx=-\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{4 x^4}-\frac {3 a \sqrt [4]{1-a x} (1+a x)^{3/4}}{8 x^3}-\frac {15 a^2 \sqrt [4]{1-a x} (1+a x)^{3/4}}{32 x^2}-\frac {63 a^3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{64 x}+\frac {123}{64} a^4 \arctan \left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac {123}{64} a^4 \text {arctanh}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right ) \]
-1/4*(-a*x+1)^(1/4)*(a*x+1)^(3/4)/x^4-3/8*a*(-a*x+1)^(1/4)*(a*x+1)^(3/4)/x ^3-15/32*a^2*(-a*x+1)^(1/4)*(a*x+1)^(3/4)/x^2-63/64*a^3*(-a*x+1)^(1/4)*(a* x+1)^(3/4)/x+123/64*a^4*arctan((a*x+1)^(1/4)/(-a*x+1)^(1/4))-123/64*a^4*ar ctanh((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 = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.51 \[ \int \frac {e^{\frac {3}{2} \text {arctanh}(a x)}}{x^5} \, dx=-\frac {\sqrt [4]{1-a x} \left (16+40 a x+54 a^2 x^2+93 a^3 x^3+63 a^4 x^4+246 a^4 x^4 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},1,\frac {5}{4},\frac {1-a x}{1+a x}\right )\right )}{64 x^4 \sqrt [4]{1+a x}} \]
-1/64*((1 - a*x)^(1/4)*(16 + 40*a*x + 54*a^2*x^2 + 93*a^3*x^3 + 63*a^4*x^4 + 246*a^4*x^4*Hypergeometric2F1[1/4, 1, 5/4, (1 - a*x)/(1 + a*x)]))/(x^4* (1 + a*x)^(1/4))
Time = 0.34 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6676, 110, 27, 168, 27, 168, 27, 168, 27, 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.
\(\displaystyle \int \frac {e^{\frac {3}{2} \text {arctanh}(a x)}}{x^5} \, dx\) |
\(\Big \downarrow \) 6676 |
\(\displaystyle \int \frac {(a x+1)^{3/4}}{x^5 (1-a x)^{3/4}}dx\) |
\(\Big \downarrow \) 110 |
\(\displaystyle \frac {1}{4} \int \frac {3 a (2 a x+3)}{2 x^4 (1-a x)^{3/4} \sqrt [4]{a x+1}}dx-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{4 x^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3}{8} a \int \frac {2 a x+3}{x^4 (1-a x)^{3/4} \sqrt [4]{a x+1}}dx-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{4 x^4}\) |
\(\Big \downarrow \) 168 |
\(\displaystyle \frac {3}{8} a \left (-\frac {1}{3} \int -\frac {3 a (4 a x+5)}{2 x^3 (1-a x)^{3/4} \sqrt [4]{a x+1}}dx-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{x^3}\right )-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{4 x^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3}{8} a \left (\frac {1}{2} a \int \frac {4 a x+5}{x^3 (1-a x)^{3/4} \sqrt [4]{a x+1}}dx-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{x^3}\right )-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{4 x^4}\) |
\(\Big \downarrow \) 168 |
\(\displaystyle \frac {3}{8} a \left (\frac {1}{2} a \left (-\frac {1}{2} \int -\frac {a (10 a x+21)}{2 x^2 (1-a x)^{3/4} \sqrt [4]{a x+1}}dx-\frac {5 \sqrt [4]{1-a x} (a x+1)^{3/4}}{2 x^2}\right )-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{x^3}\right )-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{4 x^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3}{8} a \left (\frac {1}{2} a \left (\frac {1}{4} a \int \frac {10 a x+21}{x^2 (1-a x)^{3/4} \sqrt [4]{a x+1}}dx-\frac {5 \sqrt [4]{1-a x} (a x+1)^{3/4}}{2 x^2}\right )-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{x^3}\right )-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{4 x^4}\) |
\(\Big \downarrow \) 168 |
\(\displaystyle \frac {3}{8} a \left (\frac {1}{2} a \left (\frac {1}{4} a \left (-\int -\frac {41 a}{2 x (1-a x)^{3/4} \sqrt [4]{a x+1}}dx-\frac {21 \sqrt [4]{1-a x} (a x+1)^{3/4}}{x}\right )-\frac {5 \sqrt [4]{1-a x} (a x+1)^{3/4}}{2 x^2}\right )-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{x^3}\right )-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{4 x^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3}{8} a \left (\frac {1}{2} a \left (\frac {1}{4} a \left (\frac {41}{2} a \int \frac {1}{x (1-a x)^{3/4} \sqrt [4]{a x+1}}dx-\frac {21 \sqrt [4]{1-a x} (a x+1)^{3/4}}{x}\right )-\frac {5 \sqrt [4]{1-a x} (a x+1)^{3/4}}{2 x^2}\right )-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{x^3}\right )-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{4 x^4}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {3}{8} a \left (\frac {1}{2} a \left (\frac {1}{4} a \left (82 a \int -\frac {\sqrt {a x+1}}{\sqrt {1-a x} \left (1-\frac {a x+1}{1-a x}\right )}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {21 \sqrt [4]{1-a x} (a x+1)^{3/4}}{x}\right )-\frac {5 \sqrt [4]{1-a x} (a x+1)^{3/4}}{2 x^2}\right )-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{x^3}\right )-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{4 x^4}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {3}{8} a \left (\frac {1}{2} a \left (\frac {1}{4} a \left (-82 a \int \frac {\sqrt {a x+1}}{\sqrt {1-a x} \left (1-\frac {a x+1}{1-a x}\right )}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {21 \sqrt [4]{1-a x} (a x+1)^{3/4}}{x}\right )-\frac {5 \sqrt [4]{1-a x} (a x+1)^{3/4}}{2 x^2}\right )-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{x^3}\right )-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{4 x^4}\) |
\(\Big \downarrow \) 827 |
\(\displaystyle \frac {3}{8} a \left (\frac {1}{2} a \left (\frac {1}{4} a \left (82 a \left (\frac {1}{2} \int \frac {1}{\frac {\sqrt {a x+1}}{\sqrt {1-a x}}+1}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a x+1}}{\sqrt {1-a x}}}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {21 \sqrt [4]{1-a x} (a x+1)^{3/4}}{x}\right )-\frac {5 \sqrt [4]{1-a x} (a x+1)^{3/4}}{2 x^2}\right )-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{x^3}\right )-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{4 x^4}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {3}{8} a \left (\frac {1}{2} a \left (\frac {1}{4} a \left (82 a \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a x+1}}{\sqrt {1-a x}}}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {21 \sqrt [4]{1-a x} (a x+1)^{3/4}}{x}\right )-\frac {5 \sqrt [4]{1-a x} (a x+1)^{3/4}}{2 x^2}\right )-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{x^3}\right )-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{4 x^4}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {3}{8} a \left (\frac {1}{2} a \left (\frac {1}{4} a \left (82 a \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )\right )-\frac {21 \sqrt [4]{1-a x} (a x+1)^{3/4}}{x}\right )-\frac {5 \sqrt [4]{1-a x} (a x+1)^{3/4}}{2 x^2}\right )-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{x^3}\right )-\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{4 x^4}\) |
-1/4*((1 - a*x)^(1/4)*(1 + a*x)^(3/4))/x^4 + (3*a*(-(((1 - a*x)^(1/4)*(1 + a*x)^(3/4))/x^3) + (a*((-5*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(2*x^2) + (a* ((-21*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/x + 82*a*(ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]/2 - ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]/2)))/4))/2))/8
3.1.79.3.1 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[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 0] && (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_)^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^(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]
\[\int \frac {{\left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}^{\frac {3}{2}}}{x^{5}}d x\]
Time = 0.27 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.98 \[ \int \frac {e^{\frac {3}{2} \text {arctanh}(a x)}}{x^5} \, dx=\frac {246 \, a^{4} x^{4} \arctan \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) - 123 \, a^{4} x^{4} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) + 123 \, a^{4} x^{4} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) - 2 \, {\left (63 \, a^{3} x^{3} + 30 \, a^{2} x^{2} + 24 \, a x + 16\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{128 \, x^{4}} \]
1/128*(246*a^4*x^4*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 123*a^4*x ^4*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + 123*a^4*x^4*log(sqrt(-sq rt(-a^2*x^2 + 1)/(a*x - 1)) - 1) - 2*(63*a^3*x^3 + 30*a^2*x^2 + 24*a*x + 1 6)*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x^4
\[ \int \frac {e^{\frac {3}{2} \text {arctanh}(a x)}}{x^5} \, dx=\int \frac {\left (\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}}{x^{5}}\, dx \]
\[ \int \frac {e^{\frac {3}{2} \text {arctanh}(a x)}}{x^5} \, dx=\int { \frac {\left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}}{x^{5}} \,d x } \]
\[ \int \frac {e^{\frac {3}{2} \text {arctanh}(a x)}}{x^5} \, dx=\int { \frac {\left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}}{x^{5}} \,d x } \]
Timed out. \[ \int \frac {e^{\frac {3}{2} \text {arctanh}(a x)}}{x^5} \, dx=\int \frac {{\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{3/2}}{x^5} \,d x \]