Integrand size = 10, antiderivative size = 90 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4} \, dx=-\frac {\sqrt {1-a^2 x^2}}{3 x^3}-\frac {a \sqrt {1-a^2 x^2}}{2 x^2}-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {1}{2} a^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \]
-1/2*a^3*arctanh((-a^2*x^2+1)^(1/2))-1/3*(-a^2*x^2+1)^(1/2)/x^3-1/2*a*(-a^ 2*x^2+1)^(1/2)/x^2-2/3*a^2*(-a^2*x^2+1)^(1/2)/x
Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.74 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4} \, dx=\frac {1}{6} \left (-\frac {\sqrt {1-a^2 x^2} \left (2+3 a x+4 a^2 x^2\right )}{x^3}+3 a^3 \log (x)-3 a^3 \log \left (1+\sqrt {1-a^2 x^2}\right )\right ) \]
(-((Sqrt[1 - a^2*x^2]*(2 + 3*a*x + 4*a^2*x^2))/x^3) + 3*a^3*Log[x] - 3*a^3 *Log[1 + Sqrt[1 - a^2*x^2]])/6
Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {6674, 539, 25, 27, 539, 25, 27, 534, 243, 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.
\(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{x^4} \, dx\) |
\(\Big \downarrow \) 6674 |
\(\displaystyle \int \frac {a x+1}{x^4 \sqrt {1-a^2 x^2}}dx\) |
\(\Big \downarrow \) 539 |
\(\displaystyle -\frac {1}{3} \int -\frac {a (2 a x+3)}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{3} \int \frac {a (2 a x+3)}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} a \int \frac {2 a x+3}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\) |
\(\Big \downarrow \) 539 |
\(\displaystyle \frac {1}{3} a \left (-\frac {1}{2} \int -\frac {a (3 a x+4)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {3 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{3} a \left (\frac {1}{2} \int \frac {a (3 a x+4)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {3 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} a \left (\frac {1}{2} a \int \frac {3 a x+4}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {3 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\) |
\(\Big \downarrow \) 534 |
\(\displaystyle \frac {1}{3} a \left (\frac {1}{2} a \left (3 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {4 \sqrt {1-a^2 x^2}}{x}\right )-\frac {3 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{3} a \left (\frac {1}{2} a \left (\frac {3}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {4 \sqrt {1-a^2 x^2}}{x}\right )-\frac {3 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} a \left (\frac {1}{2} a \left (-\frac {3 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {4 \sqrt {1-a^2 x^2}}{x}\right )-\frac {3 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{3} a \left (\frac {1}{2} a \left (-3 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {4 \sqrt {1-a^2 x^2}}{x}\right )-\frac {3 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\) |
-1/3*Sqrt[1 - a^2*x^2]/x^3 + (a*((-3*Sqrt[1 - a^2*x^2])/(2*x^2) + (a*((-4* Sqrt[1 - a^2*x^2])/x - 3*a*ArcTanh[Sqrt[1 - a^2*x^2]]))/2))/3
3.1.9.3.1 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_)^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[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] + Simp[1/(a*(m + 1)) Int[x^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, p}, x] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x )^m*((1 + a*x)^((n + 1)/2)/((1 - a*x)^((n - 1)/2)*Sqrt[1 - a^2*x^2])), x] / ; FreeQ[{a, c, m}, x] && IntegerQ[(n - 1)/2]
Time = 0.07 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.74
method | result | size |
risch | \(\frac {4 a^{4} x^{4}+3 a^{3} x^{3}-2 a^{2} x^{2}-3 a x -2}{6 x^{3} \sqrt {-a^{2} x^{2}+1}}-\frac {a^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\) | \(67\) |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}-\frac {2 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}+a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )\) | \(77\) |
meijerg | \(-\frac {a^{3} \left (-\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{8 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}}{a^{2} x^{2}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )-\frac {\left (1-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}+\frac {\sqrt {\pi }}{x^{2} a^{2}}\right )}{2 \sqrt {\pi }}-\frac {\left (2 a^{2} x^{2}+1\right ) \sqrt {-a^{2} x^{2}+1}}{3 x^{3}}\) | \(133\) |
1/6*(4*a^4*x^4+3*a^3*x^3-2*a^2*x^2-3*a*x-2)/x^3/(-a^2*x^2+1)^(1/2)-1/2*a^3 *arctanh(1/(-a^2*x^2+1)^(1/2))
Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.68 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4} \, dx=\frac {3 \, a^{3} x^{3} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (4 \, a^{2} x^{2} + 3 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, x^{3}} \]
1/6*(3*a^3*x^3*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (4*a^2*x^2 + 3*a*x + 2)*s qrt(-a^2*x^2 + 1))/x^3
Result contains complex when optimal does not.
Time = 1.80 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.04 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4} \, dx=a \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases} \]
a*Piecewise((-a**2*acosh(1/(a*x))/2 + a/(2*x*sqrt(-1 + 1/(a**2*x**2))) - 1 /(2*a*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (I*a**2*asin( 1/(a*x))/2 - I*a*sqrt(1 - 1/(a**2*x**2))/(2*x), True)) + Piecewise((-2*I*a **2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x **2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x **3), True))
Time = 0.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.97 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4} \, dx=-\frac {1}{2} \, a^{3} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{2}}{3 \, x} - \frac {\sqrt {-a^{2} x^{2} + 1} a}{2 \, x^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{3 \, x^{3}} \]
-1/2*a^3*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) - 2/3*sqrt(-a^2*x^2 + 1)*a^2/x - 1/2*sqrt(-a^2*x^2 + 1)*a/x^2 - 1/3*sqrt(-a^2*x^2 + 1)/x^3
Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (74) = 148\).
Time = 0.28 (sec) , antiderivative size = 210, normalized size of antiderivative = 2.33 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4} \, dx=\frac {{\left (a^{4} + \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{2}}{x} + \frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{x^{2}}\right )} a^{6} x^{3}}{24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} {\left | a \right |}} - \frac {a^{4} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, {\left | a \right |}} - \frac {\frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4}}{x} + \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{2}}{x^{2}} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{x^{3}}}{24 \, a^{2} {\left | a \right |}} \]
1/24*(a^4 + 3*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^2/x + 9*(sqrt(-a^2*x^2 + 1 )*abs(a) + a)^2/x^2)*a^6*x^3/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*abs(a)) - 1/2*a^4*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs( a) - 1/24*(9*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^4/x + 3*(sqrt(-a^2*x^2 + 1) *abs(a) + a)^2*a^2/x^2 + (sqrt(-a^2*x^2 + 1)*abs(a) + a)^3/x^3)/(a^2*abs(a ))
Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.87 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4} \, dx=-\frac {\sqrt {1-a^2\,x^2}}{3\,x^3}-\frac {a\,\sqrt {1-a^2\,x^2}}{2\,x^2}-\frac {2\,a^2\,\sqrt {1-a^2\,x^2}}{3\,x}+\frac {a^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} \]