Integrand size = 12, antiderivative size = 114 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{x^4} \, dx=\frac {4 a^3 (1+a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}-\frac {3 a \sqrt {1-a^2 x^2}}{2 x^2}-\frac {14 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {11}{2} a^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \] Output:
4*a^3*(a*x+1)/(-a^2*x^2+1)^(1/2)-1/3*(-a^2*x^2+1)^(1/2)/x^3-3/2*a*(-a^2*x^ 2+1)^(1/2)/x^2-14/3*a^2*(-a^2*x^2+1)^(1/2)/x-11/2*a^3*arctanh((-a^2*x^2+1) ^(1/2))
Time = 0.10 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.71 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{x^4} \, dx=\frac {1}{6} \left (\frac {\sqrt {1-a^2 x^2} \left (2+7 a x+19 a^2 x^2-52 a^3 x^3\right )}{x^3 (-1+a x)}+33 a^3 \log (x)-33 a^3 \log \left (1+\sqrt {1-a^2 x^2}\right )\right ) \] Input:
Integrate[E^(3*ArcTanh[a*x])/x^4,x]
Output:
((Sqrt[1 - a^2*x^2]*(2 + 7*a*x + 19*a^2*x^2 - 52*a^3*x^3))/(x^3*(-1 + a*x) ) + 33*a^3*Log[x] - 33*a^3*Log[1 + Sqrt[1 - a^2*x^2]])/6
Time = 0.91 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6674, 2353, 2009}
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.
\(\displaystyle \int \frac {e^{3 \text {arctanh}(a x)}}{x^4} \, dx\) |
\(\Big \downarrow \) 6674 |
\(\displaystyle \int \frac {(a x+1)^2}{x^4 (1-a x) \sqrt {1-a^2 x^2}}dx\) |
\(\Big \downarrow \) 2353 |
\(\displaystyle \int \left (\frac {4 a^2}{x^2 \sqrt {1-a^2 x^2}}+\frac {1}{x^4 \sqrt {1-a^2 x^2}}+\frac {3 a}{x^3 \sqrt {1-a^2 x^2}}-\frac {4 a^4}{(a x-1) \sqrt {1-a^2 x^2}}+\frac {4 a^3}{x \sqrt {1-a^2 x^2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {14 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {3 a \sqrt {1-a^2 x^2}}{2 x^2}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}-\frac {11}{2} a^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )+\frac {4 a^3 \sqrt {1-a^2 x^2}}{1-a x}\) |
Input:
Int[E^(3*ArcTanh[a*x])/x^4,x]
Output:
-1/3*Sqrt[1 - a^2*x^2]/x^3 - (3*a*Sqrt[1 - a^2*x^2])/(2*x^2) - (14*a^2*Sqr t[1 - a^2*x^2])/(3*x) + (4*a^3*Sqrt[1 - a^2*x^2])/(1 - a*x) - (11*a^3*ArcT anh[Sqrt[1 - a^2*x^2]])/2
Int[(Px_)*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2) ^(p_), x_Symbol] :> Int[ExpandIntegrand[Px*(e*x)^m*(c + d*x)^n*(a + b*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && PolyQ[Px, x] && (Integer Q[p] || (IntegerQ[2*p] && IntegerQ[m] && ILtQ[n, 0]))
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 = 0.15 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.97
method | result | size |
risch | \(\frac {28 a^{4} x^{4}+9 a^{3} x^{3}-26 a^{2} x^{2}-9 a x -2}{6 x^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {a^{3} \left (-11 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {8 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{a \left (x -\frac {1}{a}\right )}\right )}{2}\) | \(111\) |
default | \(-\frac {1}{3 x^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {13 a^{2} \left (-\frac {1}{x \sqrt {-a^{2} x^{2}+1}}+\frac {2 a^{2} x}{\sqrt {-a^{2} x^{2}+1}}\right )}{3}+a^{3} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )+3 a \left (-\frac {1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 a^{2} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )\) | \(146\) |
meijerg | \(-\frac {-8 a^{4} x^{4}+4 a^{2} x^{2}+1}{3 x^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {a^{3} \left (\frac {\left (2-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}-\sqrt {\pi }+\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}-\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )\right )}{\sqrt {\pi }}-\frac {3 a^{2} \left (-2 a^{2} x^{2}+1\right )}{x \sqrt {-a^{2} x^{2}+1}}-\frac {3 a^{3} \left (\frac {\sqrt {\pi }}{2 x^{2} a^{2}}-\frac {3 \left (\frac {5}{3}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{4}-\frac {\sqrt {\pi }\, \left (-20 a^{2} x^{2}+8\right )}{16 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \left (-24 a^{2} x^{2}+8\right )}{16 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{2}\right )}{\sqrt {\pi }}\) | \(256\) |
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/x^4,x,method=_RETURNVERBOSE)
Output:
1/6*(28*a^4*x^4+9*a^3*x^3-26*a^2*x^2-9*a*x-2)/x^3/(-a^2*x^2+1)^(1/2)+1/2*a ^3*(-11*arctanh(1/(-a^2*x^2+1)^(1/2))-8/a/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1 /a))^(1/2))
Time = 0.12 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.92 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{x^4} \, dx=\frac {24 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 33 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (52 \, a^{3} x^{3} - 19 \, a^{2} x^{2} - 7 \, a x - 2\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, {\left (a x^{4} - x^{3}\right )}} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/x^4,x, algorithm="fricas")
Output:
1/6*(24*a^4*x^4 - 24*a^3*x^3 + 33*(a^4*x^4 - a^3*x^3)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (52*a^3*x^3 - 19*a^2*x^2 - 7*a*x - 2)*sqrt(-a^2*x^2 + 1))/(a *x^4 - x^3)
\[ \int \frac {e^{3 \text {arctanh}(a x)}}{x^4} \, dx=\int \frac {\left (a x + 1\right )^{3}}{x^{4} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/x**4,x)
Output:
Integral((a*x + 1)**3/(x**4*(-(a*x - 1)*(a*x + 1))**(3/2)), x)
Time = 0.04 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.07 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{x^4} \, dx=\frac {26 \, a^{4} x}{3 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {11}{2} \, a^{3} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {11 \, a^{3}}{2 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {13 \, a^{2}}{3 \, \sqrt {-a^{2} x^{2} + 1} x} - \frac {3 \, a}{2 \, \sqrt {-a^{2} x^{2} + 1} x^{2}} - \frac {1}{3 \, \sqrt {-a^{2} x^{2} + 1} x^{3}} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/x^4,x, algorithm="maxima")
Output:
26/3*a^4*x/sqrt(-a^2*x^2 + 1) - 11/2*a^3*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) + 11/2*a^3/sqrt(-a^2*x^2 + 1) - 13/3*a^2/(sqrt(-a^2*x^2 + 1)*x) - 3/2*a/(sqrt(-a^2*x^2 + 1)*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. 265 vs. \(2 (96) = 192\).
Time = 0.16 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.32 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{x^4} \, dx=-\frac {{\left (a^{4} + \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{2}}{x} + \frac {48 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{x^{2}} - \frac {249 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{2} x^{3}}\right )} a^{6} x^{3}}{24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} - \frac {11 \, a^{4} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, {\left | a \right |}} - \frac {\frac {57 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4}}{x} + \frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{2}}{x^{2}} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{x^{3}}}{24 \, a^{2} {\left | a \right |}} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/x^4,x, algorithm="giac")
Output:
-1/24*(a^4 + 8*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^2/x + 48*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/x^2 - 249*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3/(a^2*x^3))*a ^6*x^3/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*((sqrt(-a^2*x^2 + 1)*abs(a) + a) /(a^2*x) - 1)*abs(a)) - 11/2*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*(57*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^4 /x + 9*(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 = 0.06 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.11 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{x^4} \, dx=\frac {4\,a^4\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {3\,a\,\sqrt {1-a^2\,x^2}}{2\,x^2}-\frac {14\,a^2\,\sqrt {1-a^2\,x^2}}{3\,x}-\frac {\sqrt {1-a^2\,x^2}}{3\,x^3}+\frac {a^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,11{}\mathrm {i}}{2} \] Input:
int((a*x + 1)^3/(x^4*(1 - a^2*x^2)^(3/2)),x)
Output:
(a^3*atan((1 - a^2*x^2)^(1/2)*1i)*11i)/2 - (1 - a^2*x^2)^(1/2)/(3*x^3) - ( 3*a*(1 - a^2*x^2)^(1/2))/(2*x^2) - (14*a^2*(1 - a^2*x^2)^(1/2))/(3*x) + (4 *a^4*(1 - a^2*x^2)^(1/2))/((x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)*(-a^2)^(1/2))
Time = 0.17 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.11 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{x^4} \, dx=\frac {a^{3} \left (132 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{4}-132 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{3}+\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{7}+8 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{6}+48 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{5}-306 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{4}+48 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{2}+8 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )+1\right )}{24 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{3} \left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )-1\right )} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/x^4,x)
Output:
(a**3*(132*log(tan(asin(a*x)/2))*tan(asin(a*x)/2)**4 - 132*log(tan(asin(a* x)/2))*tan(asin(a*x)/2)**3 + tan(asin(a*x)/2)**7 + 8*tan(asin(a*x)/2)**6 + 48*tan(asin(a*x)/2)**5 - 306*tan(asin(a*x)/2)**4 + 48*tan(asin(a*x)/2)**2 + 8*tan(asin(a*x)/2) + 1))/(24*tan(asin(a*x)/2)**3*(tan(asin(a*x)/2) - 1) )