Optimal. Leaf size=114 \[ -\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{3 a^3}+\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}{8 a^2}-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x^3}{3 a}+\frac {1}{4} \sqrt {1-\frac {1}{a^2 x^2}} x^4+\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a^4} \]
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Rubi [A]
time = 0.09, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6304, 849, 821,
272, 65, 214} \begin {gather*} \frac {3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{8 a^2}+\frac {1}{4} x^4 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {x^3 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a}+\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a^4}-\frac {2 x \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 849
Rule 6304
Rubi steps
\begin {align*} \int e^{-\coth ^{-1}(a x)} x^3 \, dx &=-\text {Subst}\left (\int \frac {1-\frac {x}{a}}{x^5 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{4} \sqrt {1-\frac {1}{a^2 x^2}} x^4+\frac {1}{4} \text {Subst}\left (\int \frac {\frac {4}{a}-\frac {3 x}{a^2}}{x^4 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x^3}{3 a}+\frac {1}{4} \sqrt {1-\frac {1}{a^2 x^2}} x^4-\frac {1}{12} \text {Subst}\left (\int \frac {\frac {9}{a^2}-\frac {8 x}{a^3}}{x^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}{8 a^2}-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x^3}{3 a}+\frac {1}{4} \sqrt {1-\frac {1}{a^2 x^2}} x^4+\frac {1}{24} \text {Subst}\left (\int \frac {\frac {16}{a^3}-\frac {9 x}{a^4}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{3 a^3}+\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}{8 a^2}-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x^3}{3 a}+\frac {1}{4} \sqrt {1-\frac {1}{a^2 x^2}} x^4-\frac {3 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{8 a^4}\\ &=-\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{3 a^3}+\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}{8 a^2}-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x^3}{3 a}+\frac {1}{4} \sqrt {1-\frac {1}{a^2 x^2}} x^4-\frac {3 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{16 a^4}\\ &=-\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{3 a^3}+\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}{8 a^2}-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x^3}{3 a}+\frac {1}{4} \sqrt {1-\frac {1}{a^2 x^2}} x^4+\frac {3 \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a^2}\\ &=-\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{3 a^3}+\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}{8 a^2}-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x^3}{3 a}+\frac {1}{4} \sqrt {1-\frac {1}{a^2 x^2}} x^4+\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a^4}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 68, normalized size = 0.60 \begin {gather*} \frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (-16+9 a x-8 a^2 x^2+6 a^3 x^3\right )+9 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{24 a^4} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(192\) vs.
\(2(94)=188\).
time = 0.09, size = 193, normalized size = 1.69
method | result | size |
risch | \(\frac {\left (6 a^{3} x^{3}-8 a^{2} x^{2}+9 a x -16\right ) \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{24 a^{4}}+\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{8 a^{3} \sqrt {a^{2}}\, \left (a x -1\right )}\) | \(117\) |
default | \(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \left (6 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x +15 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x -8 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+24 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right )-15 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a -24 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\right )}{24 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{4} \sqrt {a^{2}}}\) | \(193\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 203 vs.
\(2 (94) = 188\).
time = 0.26, size = 203, normalized size = 1.78 \begin {gather*} -\frac {1}{24} \, a {\left (\frac {2 \, {\left (39 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 31 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 49 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 9 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {4 \, {\left (a x - 1\right )} a^{5}}{a x + 1} - \frac {6 \, {\left (a x - 1\right )}^{2} a^{5}}{{\left (a x + 1\right )}^{2}} + \frac {4 \, {\left (a x - 1\right )}^{3} a^{5}}{{\left (a x + 1\right )}^{3}} - \frac {{\left (a x - 1\right )}^{4} a^{5}}{{\left (a x + 1\right )}^{4}} - a^{5}} - \frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{5}} + \frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{5}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 91, normalized size = 0.80 \begin {gather*} \frac {{\left (6 \, a^{4} x^{4} - 2 \, a^{3} x^{3} + a^{2} x^{2} - 7 \, a x - 16\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 )}{24 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \sqrt {\frac {a x - 1}{a x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.21, size = 172, normalized size = 1.51 \begin {gather*} \frac {3\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4\,a^4}-\frac {\frac {3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{4}-\frac {49\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{12}+\frac {31\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{12}-\frac {13\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{4}}{a^4+\frac {6\,a^4\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {4\,a^4\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {a^4\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {4\,a^4\,\left (a\,x-1\right )}{a\,x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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