Optimal. Leaf size=139 \[ -\frac {\sqrt {1+a^2 x^2}}{4 x^4}+\frac {i a \sqrt {1+a^2 x^2}}{x^3}+\frac {19 a^2 \sqrt {1+a^2 x^2}}{8 x^2}-\frac {6 i a^3 \sqrt {1+a^2 x^2}}{x}+\frac {4 i a^4 \sqrt {1+a^2 x^2}}{i-a x}-\frac {51}{8} a^4 \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right ) \]
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Rubi [A]
time = 0.53, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5168, 6874,
272, 44, 65, 214, 277, 270, 665} \begin {gather*} \frac {19 a^2 \sqrt {a^2 x^2+1}}{8 x^2}-\frac {\sqrt {a^2 x^2+1}}{4 x^4}+\frac {i a \sqrt {a^2 x^2+1}}{x^3}+\frac {4 i a^4 \sqrt {a^2 x^2+1}}{-a x+i}-\frac {51}{8} a^4 \tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right )-\frac {6 i a^3 \sqrt {a^2 x^2+1}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 214
Rule 270
Rule 272
Rule 277
Rule 665
Rule 5168
Rule 6874
Rubi steps
\begin {align*} \int \frac {e^{-3 i \tan ^{-1}(a x)}}{x^5} \, dx &=\int \frac {(1-i a x)^2}{x^5 (1+i a x) \sqrt {1+a^2 x^2}} \, dx\\ &=\int \left (\frac {1}{x^5 \sqrt {1+a^2 x^2}}-\frac {3 i a}{x^4 \sqrt {1+a^2 x^2}}-\frac {4 a^2}{x^3 \sqrt {1+a^2 x^2}}+\frac {4 i a^3}{x^2 \sqrt {1+a^2 x^2}}+\frac {4 a^4}{x \sqrt {1+a^2 x^2}}-\frac {4 a^5}{(-i+a x) \sqrt {1+a^2 x^2}}\right ) \, dx\\ &=-\left ((3 i a) \int \frac {1}{x^4 \sqrt {1+a^2 x^2}} \, dx\right )-\left (4 a^2\right ) \int \frac {1}{x^3 \sqrt {1+a^2 x^2}} \, dx+\left (4 i a^3\right ) \int \frac {1}{x^2 \sqrt {1+a^2 x^2}} \, dx+\left (4 a^4\right ) \int \frac {1}{x \sqrt {1+a^2 x^2}} \, dx-\left (4 a^5\right ) \int \frac {1}{(-i+a x) \sqrt {1+a^2 x^2}} \, dx+\int \frac {1}{x^5 \sqrt {1+a^2 x^2}} \, dx\\ &=\frac {i a \sqrt {1+a^2 x^2}}{x^3}-\frac {4 i a^3 \sqrt {1+a^2 x^2}}{x}+\frac {4 i a^4 \sqrt {1+a^2 x^2}}{i-a x}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^3 \sqrt {1+a^2 x}} \, dx,x,x^2\right )-\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1+a^2 x}} \, dx,x,x^2\right )+\left (2 i a^3\right ) \int \frac {1}{x^2 \sqrt {1+a^2 x^2}} \, dx+\left (2 a^4\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1+a^2 x^2}}{4 x^4}+\frac {i a \sqrt {1+a^2 x^2}}{x^3}+\frac {2 a^2 \sqrt {1+a^2 x^2}}{x^2}-\frac {6 i a^3 \sqrt {1+a^2 x^2}}{x}+\frac {4 i a^4 \sqrt {1+a^2 x^2}}{i-a x}-\frac {1}{8} \left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1+a^2 x}} \, dx,x,x^2\right )+\left (4 a^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {1+a^2 x^2}\right )+a^4 \text {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1+a^2 x^2}}{4 x^4}+\frac {i a \sqrt {1+a^2 x^2}}{x^3}+\frac {19 a^2 \sqrt {1+a^2 x^2}}{8 x^2}-\frac {6 i a^3 \sqrt {1+a^2 x^2}}{x}+\frac {4 i a^4 \sqrt {1+a^2 x^2}}{i-a x}-4 a^4 \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right )+\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {1+a^2 x^2}\right )+\frac {1}{16} \left (3 a^4\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1+a^2 x^2}}{4 x^4}+\frac {i a \sqrt {1+a^2 x^2}}{x^3}+\frac {19 a^2 \sqrt {1+a^2 x^2}}{8 x^2}-\frac {6 i a^3 \sqrt {1+a^2 x^2}}{x}+\frac {4 i a^4 \sqrt {1+a^2 x^2}}{i-a x}-6 a^4 \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right )+\frac {1}{8} \left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {1+a^2 x^2}\right )\\ &=-\frac {\sqrt {1+a^2 x^2}}{4 x^4}+\frac {i a \sqrt {1+a^2 x^2}}{x^3}+\frac {19 a^2 \sqrt {1+a^2 x^2}}{8 x^2}-\frac {6 i a^3 \sqrt {1+a^2 x^2}}{x}+\frac {4 i a^4 \sqrt {1+a^2 x^2}}{i-a x}-\frac {51}{8} a^4 \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 95, normalized size = 0.68 \begin {gather*} \frac {1}{8} \left (\frac {\sqrt {1+a^2 x^2} \left (2 i+6 a x-11 i a^2 x^2-29 a^3 x^3-80 i a^4 x^4\right )}{x^4 (-i+a x)}+51 a^4 \log (x)-51 a^4 \log \left (1+\sqrt {1+a^2 x^2}\right )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 936 vs. \(2 (117 ) = 234\).
time = 0.12, size = 937, normalized size = 6.74
method | result | size |
risch | \(-\frac {i \left (48 a^{5} x^{5}+19 i a^{4} x^{4}+40 a^{3} x^{3}+17 i a^{2} x^{2}-8 a x -2 i\right )}{8 x^{4} \sqrt {a^{2} x^{2}+1}}+\frac {a^{4} \left (-\frac {32 i \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{a \left (x -\frac {i}{a}\right )}-51 \arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{8}\) | \(125\) |
default | \(5 i a^{3} \left (-\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{2}}+3 i a \left (\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3}+i a \left (\frac {\left (2 \left (x -\frac {i}{a}\right ) a^{2}+2 i a \right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{4 a^{2}}+\frac {\ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )-15 a^{4} \left (\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3}+i a \left (\frac {\left (2 \left (x -\frac {i}{a}\right ) a^{2}+2 i a \right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{4 a^{2}}+\frac {\ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{2 \sqrt {a^{2}}}\right )\right )-3 i a \left (-\frac {\left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}{3 x^{3}}+\frac {2 a^{2} \left (-\frac {\left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}{x}+4 a^{2} \left (\frac {x \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4}+\frac {3 x \sqrt {a^{2} x^{2}+1}}{8}+\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{8 \sqrt {a^{2}}}\right )\right )}{3}\right )-\frac {23 a^{2} \left (-\frac {\left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}{2 x^{2}}+\frac {3 a^{2} \left (\frac {\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}+\sqrt {a^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{2}\right )}{4}+15 a^{4} \left (\frac {\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}+\sqrt {a^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )\right )-\frac {\left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}{4 x^{4}}+a^{2} \left (\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{3}}-2 i a \left (-\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{2}}+3 i a \left (\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3}+i a \left (\frac {\left (2 \left (x -\frac {i}{a}\right ) a^{2}+2 i a \right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{4 a^{2}}+\frac {\ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )\right )+10 i a^{3} \left (-\frac {\left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}{x}+4 a^{2} \left (\frac {x \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4}+\frac {3 x \sqrt {a^{2} x^{2}+1}}{8}+\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{8 \sqrt {a^{2}}}\right )\right )\) | \(937\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.24, size = 146, normalized size = 1.05 \begin {gather*} \frac {-80 i \, a^{5} x^{5} - 80 \, a^{4} x^{4} - 51 \, {\left (a^{5} x^{5} - i \, a^{4} x^{4}\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) + 51 \, {\left (a^{5} x^{5} - i \, a^{4} x^{4}\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) + {\left (-80 i \, a^{4} x^{4} - 29 \, a^{3} x^{3} - 11 i \, a^{2} x^{2} + 6 \, a x + 2 i\right )} \sqrt {a^{2} x^{2} + 1}}{8 \, {\left (a x^{5} - i \, x^{4}\right )}} \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*} i \left (\int \frac {\sqrt {a^{2} x^{2} + 1}}{a^{3} x^{8} - 3 i a^{2} x^{7} - 3 a x^{6} + i x^{5}}\, dx + \int \frac {a^{2} x^{2} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{8} - 3 i a^{2} x^{7} - 3 a x^{6} + i x^{5}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [B]
time = 0.43, size = 139, normalized size = 1.00 \begin {gather*} \frac {a^4\,\mathrm {atan}\left (\sqrt {a^2\,x^2+1}\,1{}\mathrm {i}\right )\,51{}\mathrm {i}}{8}-\frac {\sqrt {a^2\,x^2+1}}{4\,x^4}+\frac {a\,\sqrt {a^2\,x^2+1}\,1{}\mathrm {i}}{x^3}+\frac {19\,a^2\,\sqrt {a^2\,x^2+1}}{8\,x^2}-\frac {a^3\,\sqrt {a^2\,x^2+1}\,6{}\mathrm {i}}{x}+\frac {a^5\,\sqrt {a^2\,x^2+1}\,4{}\mathrm {i}}{\left (-x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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