Optimal. Leaf size=90 \[ -\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a \left (a+\frac {1}{x}\right )}-\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} x}{a}+\frac {1}{2} \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {9 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.59, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6304, 6874,
272, 44, 65, 214, 270, 665} \begin {gather*} \frac {1}{2} x^2 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {3 x \sqrt {1-\frac {1}{a^2 x^2}}}{a}-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a \left (a+\frac {1}{x}\right )}+\frac {9 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 65
Rule 214
Rule 270
Rule 272
Rule 665
Rule 6304
Rule 6874
Rubi steps
\begin {align*} \int e^{-3 \coth ^{-1}(a x)} x \, dx &=-\text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x^3 \left (1+\frac {x}{a}\right ) \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\text {Subst}\left (\int \left (\frac {1}{x^3 \sqrt {1-\frac {x^2}{a^2}}}-\frac {3}{a x^2 \sqrt {1-\frac {x^2}{a^2}}}+\frac {4}{a^2 x \sqrt {1-\frac {x^2}{a^2}}}-\frac {4}{a^2 (a+x) \sqrt {1-\frac {x^2}{a^2}}}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {4 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^2}+\frac {4 \text {Subst}\left (\int \frac {1}{(a+x) \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^2}+\frac {3 \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a}-\text {Subst}\left (\int \frac {1}{x^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a \left (a+\frac {1}{x}\right )}-\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} x}{a}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )-\frac {2 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{a^2}\\ &=-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a \left (a+\frac {1}{x}\right )}-\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} x}{a}+\frac {1}{2} \sqrt {1-\frac {1}{a^2 x^2}} x^2+4 \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{4 a^2}\\ &=-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a \left (a+\frac {1}{x}\right )}-\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} x}{a}+\frac {1}{2} \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a^2}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )\\ &=-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a \left (a+\frac {1}{x}\right )}-\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} x}{a}+\frac {1}{2} \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {9 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 66, normalized size = 0.73 \begin {gather*} \frac {\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (-14-5 a x+a^2 x^2\right )}{1+a x}+9 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{2 a^2} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(420\) vs.
\(2(78)=156\).
time = 0.10, size = 421, normalized size = 4.68
method | result | size |
risch | \(\frac {\left (a x -6\right ) \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{2 a^{2}}+\frac {\left (\frac {9 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{2 a \sqrt {a^{2}}}-\frac {4 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{a^{3} \left (x +\frac {1}{a}\right )}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{a x -1}\) | \(136\) |
default | \(\frac {\left (\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}-10 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}+2 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}-\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+10 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+4 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-20 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a x +\sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x -2 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x +20 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -10 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}-\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a +10 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right )\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{2 a^{2} \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \left (a x -1\right )}\) | \(421\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 151, normalized size = 1.68 \begin {gather*} -\frac {1}{2} \, a {\left (\frac {2 \, {\left (7 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 5 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {2 \, {\left (a x - 1\right )} a^{3}}{a x + 1} - \frac {{\left (a x - 1\right )}^{2} a^{3}}{{\left (a x + 1\right )}^{2}} - a^{3}} - \frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{3}} + \frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{3}} + \frac {8 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{3}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.33, size = 75, normalized size = 0.83 \begin {gather*} \frac {{\left (a^{2} x^{2} - 5 \, a x - 14\right )} \sqrt {\frac {a x - 1}{a x + 1}} + 9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{2 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.22, size = 120, normalized size = 1.33 \begin {gather*} \frac {9\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a^2}-\frac {4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a^2}-\frac {5\,\sqrt {\frac {a\,x-1}{a\,x+1}}-7\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{a^2+\frac {a^2\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {2\,a^2\,\left (a\,x-1\right )}{a\,x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________