Optimal. Leaf size=25 \[ -\tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right )+i \sinh ^{-1}(a x) \]
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Rubi [A] time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5060, 844, 215, 266, 63, 208} \[ -\tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right )+i \sinh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 215
Rule 266
Rule 844
Rule 5060
Rubi steps
\begin {align*} \int \frac {e^{i \tan ^{-1}(a x)}}{x} \, dx &=\int \frac {1+i a x}{x \sqrt {1+a^2 x^2}} \, dx\\ &=(i a) \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx+\int \frac {1}{x \sqrt {1+a^2 x^2}} \, dx\\ &=i \sinh ^{-1}(a x)+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right )\\ &=i \sinh ^{-1}(a x)+\frac {\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^2}\\ &=i \sinh ^{-1}(a x)-\tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 29, normalized size = 1.16 \[ -\log \left (\sqrt {a^2 x^2+1}+1\right )+i \sinh ^{-1}(a x)+\log (x) \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.43, size = 58, normalized size = 2.32 \[ -\log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) - i \, \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) + \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 69, normalized size = 2.76 \[ -\frac {a i \log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right )}{{\left | a \right |}} - \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1} + 1 \right |}\right ) + \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.16, size = 48, normalized size = 1.92 \[ \frac {i a \ln \left (\frac {x \,a^{2}}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{\sqrt {a^{2}}}-\arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 18, normalized size = 0.72 \[ i \, \operatorname {arsinh}\left (a x\right ) - \operatorname {arsinh}\left (\frac {1}{a {\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 32, normalized size = 1.28 \[ -\mathrm {atanh}\left (\sqrt {a^2\,x^2+1}\right )+\frac {a\,\mathrm {asinh}\left (x\,\sqrt {a^2}\right )\,1{}\mathrm {i}}{\sqrt {a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.35, size = 53, normalized size = 2.12 \[ i a \left (\begin {cases} \sqrt {- \frac {1}{a^{2}}} \operatorname {asin}{\left (x \sqrt {- a^{2}} \right )} & \text {for}\: a^{2} < 0 \\\sqrt {\frac {1}{a^{2}}} \operatorname {asinh}{\left (x \sqrt {a^{2}} \right )} & \text {for}\: a^{2} > 0 \end {cases}\right ) - \operatorname {asinh}{\left (\frac {1}{a x} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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