Optimal. Leaf size=64 \[ \frac {4 a \sqrt {a^2 x^2+1}}{-a x+i}-\frac {\sqrt {a^2 x^2+1}}{x}+3 i a \tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right ) \]
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Rubi [A] time = 0.55, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5060, 6742, 264, 266, 63, 208, 651} \[ \frac {4 a \sqrt {a^2 x^2+1}}{-a x+i}-\frac {\sqrt {a^2 x^2+1}}{x}+3 i a \tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 264
Rule 266
Rule 651
Rule 5060
Rule 6742
Rubi steps
\begin {align*} \int \frac {e^{-3 i \tan ^{-1}(a x)}}{x^2} \, dx &=\int \frac {(1-i a x)^2}{x^2 (1+i a x) \sqrt {1+a^2 x^2}} \, dx\\ &=\int \left (\frac {1}{x^2 \sqrt {1+a^2 x^2}}-\frac {3 i a}{x \sqrt {1+a^2 x^2}}+\frac {4 i a^2}{(-i+a x) \sqrt {1+a^2 x^2}}\right ) \, dx\\ &=-\left ((3 i a) \int \frac {1}{x \sqrt {1+a^2 x^2}} \, dx\right )+\left (4 i a^2\right ) \int \frac {1}{(-i+a x) \sqrt {1+a^2 x^2}} \, dx+\int \frac {1}{x^2 \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1+a^2 x^2}}{x}+\frac {4 a \sqrt {1+a^2 x^2}}{i-a x}-\frac {1}{2} (3 i a) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1+a^2 x^2}}{x}+\frac {4 a \sqrt {1+a^2 x^2}}{i-a x}-\frac {(3 i) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {1+a^2 x^2}\right )}{a}\\ &=-\frac {\sqrt {1+a^2 x^2}}{x}+\frac {4 a \sqrt {1+a^2 x^2}}{i-a x}+3 i a \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 61, normalized size = 0.95 \[ \sqrt {a^2 x^2+1} \left (-\frac {1}{x}-\frac {4 a}{a x-i}\right )+3 i a \log \left (\sqrt {a^2 x^2+1}+1\right )-3 i a \log (x) \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.41, size = 109, normalized size = 1.70 \[ -\frac {5 \, a^{2} x^{2} - 5 i \, a x + 3 \, {\left (-i \, a^{2} x^{2} - a x\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) + 3 \, {\left (i \, a^{2} x^{2} + a x\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) + \sqrt {a^{2} x^{2} + 1} {\left (5 \, a x - i\right )}}{a x^{2} - i \, x} \]
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 = 0.17, size = 305, normalized size = 4.77 \[ i a \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}+3 i a \arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )-i a \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}-\frac {\left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}{x}+a^{2} x \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}+\frac {3 a^{2} x \sqrt {a^{2} x^{2}+1}}{2}+\frac {3 a^{2} \ln \left (\frac {x \,a^{2}}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{2 \sqrt {a^{2}}}-3 i a \sqrt {a^{2} x^{2}+1}-\frac {3 a^{2} \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\, x}{2}-\frac {3 a^{2} \ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{2 \sqrt {a^{2}}}+\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a^{2} \left (x -\frac {i}{a}\right )^{3}} \]
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 \[ \int \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (i \, a x + 1\right )}^{3} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 76, normalized size = 1.19 \[ a\,\mathrm {atanh}\left (\sqrt {a^2\,x^2+1}\right )\,3{}\mathrm {i}-\frac {\sqrt {a^2\,x^2+1}}{x}+\frac {4\,a^2\,\sqrt {a^2\,x^2+1}}{\left (-x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]
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 \[ i \left (\int \frac {\sqrt {a^{2} x^{2} + 1}}{a^{3} x^{5} - 3 i a^{2} x^{4} - 3 a x^{3} + i x^{2}}\, dx + \int \frac {a^{2} x^{2} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{5} - 3 i a^{2} x^{4} - 3 a x^{3} + i x^{2}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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