Optimal. Leaf size=113 \[ -\frac {\sqrt {1+a^2 x^2}}{4 x^4}+\frac {i a \sqrt {1+a^2 x^2}}{3 x^3}+\frac {3 a^2 \sqrt {1+a^2 x^2}}{8 x^2}-\frac {2 i a^3 \sqrt {1+a^2 x^2}}{3 x}-\frac {3}{8} a^4 \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right ) \]
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Rubi [A]
time = 0.07, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5168, 849, 821,
272, 65, 214} \begin {gather*} \frac {3 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}}{3 x^3}-\frac {3}{8} a^4 \tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right )-\frac {2 i a^3 \sqrt {a^2 x^2+1}}{3 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 849
Rule 5168
Rubi steps
\begin {align*} \int \frac {e^{-i \tan ^{-1}(a x)}}{x^5} \, dx &=\int \frac {1-i a x}{x^5 \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1+a^2 x^2}}{4 x^4}-\frac {1}{4} \int \frac {4 i a+3 a^2 x}{x^4 \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1+a^2 x^2}}{4 x^4}+\frac {i a \sqrt {1+a^2 x^2}}{3 x^3}+\frac {1}{12} \int \frac {-9 a^2+8 i a^3 x}{x^3 \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1+a^2 x^2}}{4 x^4}+\frac {i a \sqrt {1+a^2 x^2}}{3 x^3}+\frac {3 a^2 \sqrt {1+a^2 x^2}}{8 x^2}-\frac {1}{24} \int \frac {-16 i a^3-9 a^4 x}{x^2 \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1+a^2 x^2}}{4 x^4}+\frac {i a \sqrt {1+a^2 x^2}}{3 x^3}+\frac {3 a^2 \sqrt {1+a^2 x^2}}{8 x^2}-\frac {2 i a^3 \sqrt {1+a^2 x^2}}{3 x}+\frac {1}{8} \left (3 a^4\right ) \int \frac {1}{x \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1+a^2 x^2}}{4 x^4}+\frac {i a \sqrt {1+a^2 x^2}}{3 x^3}+\frac {3 a^2 \sqrt {1+a^2 x^2}}{8 x^2}-\frac {2 i a^3 \sqrt {1+a^2 x^2}}{3 x}+\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}}{3 x^3}+\frac {3 a^2 \sqrt {1+a^2 x^2}}{8 x^2}-\frac {2 i a^3 \sqrt {1+a^2 x^2}}{3 x}+\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}}{3 x^3}+\frac {3 a^2 \sqrt {1+a^2 x^2}}{8 x^2}-\frac {2 i a^3 \sqrt {1+a^2 x^2}}{3 x}-\frac {3}{8} a^4 \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 76, normalized size = 0.67 \begin {gather*} \frac {1}{24} \left (\frac {\sqrt {1+a^2 x^2} \left (-6+8 i a x+9 a^2 x^2-16 i a^3 x^3\right )}{x^4}+9 a^4 \log (x)-9 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. 289 vs. \(2 (91 ) = 182\).
time = 0.08, size = 290, normalized size = 2.57
method | result | size |
risch | \(-\frac {i \left (16 a^{5} x^{5}+9 i a^{4} x^{4}+8 a^{3} x^{3}+3 i a^{2} x^{2}-8 a x -6 i\right )}{24 x^{4} \sqrt {a^{2} x^{2}+1}}-\frac {3 a^{4} \arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )}{8}\) | \(77\) |
default | \(-a^{4} \left (\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}+\frac {i a \ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{\sqrt {a^{2}}}\right )+\frac {i a \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 x^{3}}-\frac {5 a^{2} \left (-\frac {\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{2 x^{2}}+\frac {a^{2} \left (\sqrt {a^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{2}\right )}{4}+a^{4} \left (\sqrt {a^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )\right )-\frac {\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4 x^{4}}+i a^{3} \left (-\frac {\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{x}+2 a^{2} \left (\frac {x \sqrt {a^{2} x^{2}+1}}{2}+\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{2 \sqrt {a^{2}}}\right )\right )\) | \(290\) |
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 = 2.49, size = 101, normalized size = 0.89 \begin {gather*} -\frac {9 \, a^{4} x^{4} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) - 9 \, a^{4} x^{4} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) + 16 i \, a^{4} x^{4} - {\left (-16 i \, a^{3} x^{3} + 9 \, a^{2} x^{2} + 8 i \, a x - 6\right )} \sqrt {a^{2} x^{2} + 1}}{24 \, x^{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*} - i \int \frac {\sqrt {a^{2} x^{2} + 1}}{a x^{6} - i x^{5}}\, dx \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 = 0.03, size = 95, normalized size = 0.84 \begin {gather*} \frac {a^4\,\mathrm {atan}\left (\sqrt {a^2\,x^2+1}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{8}-\frac {\sqrt {a^2\,x^2+1}}{4\,x^4}+\frac {a\,\sqrt {a^2\,x^2+1}\,1{}\mathrm {i}}{3\,x^3}+\frac {3\,a^2\,\sqrt {a^2\,x^2+1}}{8\,x^2}-\frac {a^3\,\sqrt {a^2\,x^2+1}\,2{}\mathrm {i}}{3\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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