Integrand size = 14, antiderivative size = 110 \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^3} \, dx=-\frac {a (1-a x)^{3/4} \sqrt [4]{1+a x}}{4 x}-\frac {(1-a x)^{3/4} (1+a x)^{5/4}}{2 x^2}-\frac {1}{4} a^2 \arctan \left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac {1}{4} a^2 \text {arctanh}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right ) \] Output:
-1/4*a*(-a*x+1)^(3/4)*(a*x+1)^(1/4)/x-1/2*(-a*x+1)^(3/4)*(a*x+1)^(5/4)/x^2 -1/4*a^2*arctan((a*x+1)^(1/4)/(-a*x+1)^(1/4))-1/4*a^2*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 = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.64 \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^3} \, dx=-\frac {(1-a x)^{3/4} \left (6+15 a x+9 a^2 x^2+2 a^2 x^2 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},\frac {1-a x}{1+a x}\right )\right )}{12 x^2 (1+a x)^{3/4}} \] Input:
Integrate[E^(ArcTanh[a*x]/2)/x^3,x]
Output:
-1/12*((1 - a*x)^(3/4)*(6 + 15*a*x + 9*a^2*x^2 + 2*a^2*x^2*Hypergeometric2 F1[3/4, 1, 7/4, (1 - a*x)/(1 + a*x)]))/(x^2*(1 + a*x)^(3/4))
Time = 0.43 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6676, 107, 105, 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.
\(\displaystyle \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^3} \, dx\) |
\(\Big \downarrow \) 6676 |
\(\displaystyle \int \frac {\sqrt [4]{a x+1}}{x^3 \sqrt [4]{1-a x}}dx\) |
\(\Big \downarrow \) 107 |
\(\displaystyle \frac {1}{4} a \int \frac {\sqrt [4]{a x+1}}{x^2 \sqrt [4]{1-a x}}dx-\frac {(1-a x)^{3/4} (a x+1)^{5/4}}{2 x^2}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {1}{4} a \left (\frac {1}{2} a \int \frac {1}{x \sqrt [4]{1-a x} (a x+1)^{3/4}}dx-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{x}\right )-\frac {(1-a x)^{3/4} (a x+1)^{5/4}}{2 x^2}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {1}{4} a \left (2 a \int \frac {1}{\frac {a x+1}{1-a x}-1}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{x}\right )-\frac {(1-a x)^{3/4} (a x+1)^{5/4}}{2 x^2}\) |
\(\Big \downarrow \) 756 |
\(\displaystyle \frac {1}{4} a \left (2 a \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a x+1}}{\sqrt {1-a x}}}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {1}{2} \int \frac {1}{\frac {\sqrt {a x+1}}{\sqrt {1-a x}}+1}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{x}\right )-\frac {(1-a x)^{3/4} (a x+1)^{5/4}}{2 x^2}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{4} a \left (2 a \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a x+1}}{\sqrt {1-a x}}}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )\right )-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{x}\right )-\frac {(1-a x)^{3/4} (a x+1)^{5/4}}{2 x^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{4} a \left (2 a \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )\right )-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{x}\right )-\frac {(1-a x)^{3/4} (a x+1)^{5/4}}{2 x^2}\) |
Input:
Int[E^(ArcTanh[a*x]/2)/x^3,x]
Output:
-1/2*((1 - a*x)^(3/4)*(1 + a*x)^(5/4))/x^2 + (a*(-(((1 - a*x)^(3/4)*(1 + a *x)^(1/4))/x) + 2*a*(-1/2*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - ArcTan h[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]/2)))/4
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_))^(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[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[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]
\[\int \frac {\sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}}{x^{3}}d x\]
Input:
int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^3,x)
Output:
int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^3,x)
Time = 0.09 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.31 \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^3} \, dx=-\frac {2 \, a^{2} x^{2} \arctan \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) + a^{2} x^{2} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) - a^{2} x^{2} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) - 2 \, {\left (3 \, a^{2} x^{2} - a x - 2\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{8 \, x^{2}} \] Input:
integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^3,x, algorithm="fricas")
Output:
-1/8*(2*a^2*x^2*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + a^2*x^2*log( sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - a^2*x^2*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) - 2*(3*a^2*x^2 - a*x - 2)*sqrt(-sqrt(-a^2*x^2 + 1)/( a*x - 1)))/x^2
\[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^3} \, dx=\int \frac {\sqrt {\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}}}{x^{3}}\, dx \] Input:
integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/x**3,x)
Output:
Integral(sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1))/x**3, x)
\[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^3} \, dx=\int { \frac {\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{x^{3}} \,d x } \] Input:
integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^3,x, algorithm="maxima")
Output:
integrate(sqrt((a*x + 1)/sqrt(-a^2*x^2 + 1))/x^3, x)
\[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^3} \, dx=\int { \frac {\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{x^{3}} \,d x } \] Input:
integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^3,x, algorithm="giac")
Output:
integrate(sqrt((a*x + 1)/sqrt(-a^2*x^2 + 1))/x^3, x)
Timed out. \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^3} \, dx=\int \frac {\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}}{x^3} \,d x \] Input:
int(((a*x + 1)/(1 - a^2*x^2)^(1/2))^(1/2)/x^3,x)
Output:
int(((a*x + 1)/(1 - a^2*x^2)^(1/2))^(1/2)/x^3, x)
\[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^3} \, dx=\int \frac {\sqrt {a x +1}}{\left (-a^{2} x^{2}+1\right )^{\frac {1}{4}} x^{3}}d x \] Input:
int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^3,x)
Output:
int(sqrt(a*x + 1)/(( - a**2*x**2 + 1)**(1/4)*x**3),x)