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 = 0.47, 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 [B] time = 0.16, size = 240, normalized size = 2.12 \[ -\frac {3}{8} \, a^{4} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1} + 1 \right |}\right ) + \frac {3}{8} \, a^{4} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1} - 1 \right |}\right ) - \frac {9 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{7} a^{4} - 33 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{5} a^{4} - 48 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{4} a^{3} i {\left | a \right |} - 33 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{3} a^{4} + 64 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} a^{3} i {\left | a \right |} + 9 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )} a^{4} - 16 \, a^{3} i {\left | a \right |}}{12 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} - 1\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 97, normalized size = 0.86 \[ i a \left (-\frac {\sqrt {a^{2} x^{2}+1}}{3 x^{3}}+\frac {2 a^{2} \sqrt {a^{2} x^{2}+1}}{3 x}\right )-\frac {\sqrt {a^{2} x^{2}+1}}{4 x^{4}}-\frac {3 a^{2} \left (-\frac {\sqrt {a^{2} x^{2}+1}}{2 x^{2}}+\frac {a^{2} \arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )}{2}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 86, normalized size = 0.76 \[ -\frac {3}{8} \, a^{4} \operatorname {arsinh}\left (\frac {1}{a {\left | x \right |}}\right ) + \frac {2 i \, \sqrt {a^{2} x^{2} + 1} a^{3}}{3 \, x} + \frac {3 \, \sqrt {a^{2} x^{2} + 1} a^{2}}{8 \, x^{2}} - \frac {i \, \sqrt {a^{2} x^{2} + 1} a}{3 \, x^{3}} - \frac {\sqrt {a^{2} x^{2} + 1}}{4 \, x^{4}} \]
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 [A] time = 4.30, size = 122, normalized size = 1.08 \[ \frac {2 i a^{4} \sqrt {1 + \frac {1}{a^{2} x^{2}}}}{3} - \frac {3 a^{4} \operatorname {asinh}{\left (\frac {1}{a x} \right )}}{8} + \frac {3 a^{3}}{8 x \sqrt {1 + \frac {1}{a^{2} x^{2}}}} - \frac {i a^{2} \sqrt {1 + \frac {1}{a^{2} x^{2}}}}{3 x^{2}} + \frac {a}{8 x^{3} \sqrt {1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{4 a x^{5} \sqrt {1 + \frac {1}{a^{2} x^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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