Optimal. Leaf size=113 \[ \frac {3 a^2 \sqrt {a^2 x^2+1}}{8 x^2}-\frac {\sqrt {a^2 x^2+1}}{4 x^4}+\frac {i a \sqrt {a^2 x^2+1}}{3 x^3}-\frac {3}{8} a^4 \tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right )-\frac {2 i a^3 \sqrt {a^2 x^2+1}}{3 x} \]
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Rubi [A] time = 0.09, 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 = {5060, 835, 807, 266, 63, 208} \[ -\frac {2 i a^3 \sqrt {a^2 x^2+1}}{3 x}+\frac {3 a^2 \sqrt {a^2 x^2+1}}{8 x^2}+\frac {i a \sqrt {a^2 x^2+1}}{3 x^3}-\frac {\sqrt {a^2 x^2+1}}{4 x^4}-\frac {3}{8} a^4 \tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 835
Rule 5060
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 ) \operatorname {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 ) \operatorname {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 = 0.06, size = 76, normalized size = 0.67 \[ \frac {1}{24} \left (9 a^4 \log (x)-9 a^4 \log \left (\sqrt {a^2 x^2+1}+1\right )+\frac {\sqrt {a^2 x^2+1} \left (-16 i a^3 x^3+9 a^2 x^2+8 i a x-6\right )}{x^4}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.46, size = 101, normalized size = 0.89 \[ -\frac {9 \, a^{4} x^{4} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) - 9 \, a^{4} x^{4} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) + 16 i \, a^{4} x^{4} - {\left (-16 i \, a^{3} x^{3} + 9 \, a^{2} x^{2} + 8 i \, a x - 6\right )} \sqrt {a^{2} x^{2} + 1}}{24 \, x^{4}} \]
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 \[ \mathit {undef} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 259, normalized size = 2.29 \[ \frac {i a \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 x^{3}}-\frac {3 a^{4} \arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )}{8}+\frac {3 a^{4} \sqrt {a^{2} x^{2}+1}}{8}-\frac {i a^{3} \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{x}+i a^{5} x \sqrt {a^{2} x^{2}+1}+\frac {i a^{5} \ln \left (\frac {x \,a^{2}}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{\sqrt {a^{2}}}-a^{4} \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}-\frac {i a^{5} \ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{\sqrt {a^{2}}}-\frac {\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4 x^{4}}+\frac {5 a^{2} \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{8 x^{2}} \]
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 \[ \int \frac {\sqrt {a^{2} x^{2} + 1}}{{\left (i \, a x + 1\right )} x^{5}}\,{d x} \]
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 \[ \frac {a^4\,\mathrm {atan}\left (\sqrt {a^2\,x^2+1}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{8}-\frac {\sqrt {a^2\,x^2+1}}{4\,x^4}+\frac {a\,\sqrt {a^2\,x^2+1}\,1{}\mathrm {i}}{3\,x^3}+\frac {3\,a^2\,\sqrt {a^2\,x^2+1}}{8\,x^2}-\frac {a^3\,\sqrt {a^2\,x^2+1}\,2{}\mathrm {i}}{3\,x} \]
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 \[ - i \int \frac {\sqrt {a^{2} x^{2} + 1}}{a x^{6} - i x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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