Optimal. Leaf size=25 \[ i \sinh ^{-1}(a x)-\tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right ) \]
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Rubi [A]
time = 0.03, 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 = {5168, 858, 221,
272, 65, 214} \begin {gather*} -\tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right )+i \sinh ^{-1}(a x) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 221
Rule 272
Rule 858
Rule 5168
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} \text {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right )\\ &=i \sinh ^{-1}(a x)+\frac {\text {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 = 0.01, size = 29, normalized size = 1.16 \begin {gather*} i \sinh ^{-1}(a x)+\log (x)-\log \left (1+\sqrt {1+a^2 x^2}\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. 47 vs. \(2 (22 ) = 44\).
time = 0.07, size = 48, normalized size = 1.92
method | result | size |
default | \(\frac {i a \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{\sqrt {a^{2}}}-\arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )\) | \(48\) |
meijerg | \(\frac {-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {a^{2} x^{2}+1}}{2}\right )+\left (-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (a^{2}\right )\right ) \sqrt {\pi }}{2 \sqrt {\pi }}+i \arcsinh \left (a x \right )\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 18, normalized size = 0.72 \begin {gather*} i \, \operatorname {arsinh}\left (a x\right ) - \operatorname {arsinh}\left (\frac {1}{a {\left | x \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 58 vs. \(2 (21) = 42\).
time = 1.35, size = 58, normalized size = 2.32 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.09, size = 53, normalized size = 2.12 \begin {gather*} 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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 68 vs. \(2 (21) = 42\).
time = 0.43, size = 68, normalized size = 2.72 \begin {gather*} -\frac {i \, a \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 ) \end {gather*}
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 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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