Optimal. Leaf size=29 \[ \frac {i \sqrt {1+a^2 x^2}}{a}+\frac {\sinh ^{-1}(a x)}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5167, 655, 221}
\begin {gather*} \frac {\sinh ^{-1}(a x)}{a}+\frac {i \sqrt {a^2 x^2+1}}{a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 221
Rule 655
Rule 5167
Rubi steps
\begin {align*} \int e^{i \tan ^{-1}(a x)} \, dx &=\int \frac {1+i a x}{\sqrt {1+a^2 x^2}} \, dx\\ &=\frac {i \sqrt {1+a^2 x^2}}{a}+\int \frac {1}{\sqrt {1+a^2 x^2}} \, dx\\ &=\frac {i \sqrt {1+a^2 x^2}}{a}+\frac {\sinh ^{-1}(a x)}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 26, normalized size = 0.90 \begin {gather*} \frac {i \sqrt {1+a^2 x^2}+\sinh ^{-1}(a x)}{a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 48, normalized size = 1.66
method | result | size |
meijerg | \(\frac {\arcsinh \left (a x \right )}{a}+\frac {i \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}\right )}{2 a \sqrt {\pi }}\) | \(41\) |
default | \(\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{\sqrt {a^{2}}}+\frac {i \sqrt {a^{2} x^{2}+1}}{a}\) | \(48\) |
risch | \(\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{\sqrt {a^{2}}}+\frac {i \sqrt {a^{2} x^{2}+1}}{a}\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.25, size = 25, normalized size = 0.86 \begin {gather*} \frac {\operatorname {arsinh}\left (a x\right )}{a} + \frac {i \, \sqrt {a^{2} x^{2} + 1}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.61, size = 37, normalized size = 1.28 \begin {gather*} \frac {i \, \sqrt {a^{2} x^{2} + 1} - \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.74, size = 68, normalized size = 2.34 \begin {gather*} i a \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\\frac {\sqrt {a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases}\right ) + \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.42, size = 41, normalized size = 1.41 \begin {gather*} -\frac {\log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right )}{{\left | a \right |}} + \frac {i \, \sqrt {a^{2} x^{2} + 1}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.04, size = 32, normalized size = 1.10 \begin {gather*} \frac {\mathrm {asinh}\left (x\,\sqrt {a^2}\right )}{\sqrt {a^2}}+\frac {\sqrt {a^2\,x^2+1}\,1{}\mathrm {i}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________