Optimal. Leaf size=25 \[ -\tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right )-i \sinh ^{-1}(a x) \]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
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\\ &=-\left ((i a) \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx\right )+\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*}
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.46, 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.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.14, size = 68, normalized size = 2.72 \[ \frac {a i \log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right )}{{\left | a \right |}} - \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1} + 1 \right |}\right ) + \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.16, size = 121, normalized size = 4.84 \[ \sqrt {a^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )-\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}-\frac {i a \ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{\sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.47, size = 26, normalized size = 1.04 \[ -i \, a {\left (\frac {\operatorname {arsinh}\left (a x\right )}{a} - \frac {i \, \operatorname {arsinh}\left (\frac {1}{a {\left | x \right |}}\right )}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - i \int \frac {\sqrt {a^{2} x^{2} + 1}}{a x^{2} - i x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________