Optimal. Leaf size=41 \[ \frac {2 i \sqrt {1-i a x}}{a \sqrt {1+i a x}}-\frac {\sinh ^{-1}(a x)}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5181, 49, 41,
221} \begin {gather*} -\frac {\sinh ^{-1}(a x)}{a}+\frac {2 i \sqrt {1-i a x}}{a \sqrt {1+i a x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 41
Rule 49
Rule 221
Rule 5181
Rubi steps
\begin {align*} \int \frac {e^{-2 i \tan ^{-1}(a x)}}{\sqrt {1+a^2 x^2}} \, dx &=\int \frac {\sqrt {1-i a x}}{(1+i a x)^{3/2}} \, dx\\ &=\frac {2 i \sqrt {1-i a x}}{a \sqrt {1+i a x}}-\int \frac {1}{\sqrt {1-i a x} \sqrt {1+i a x}} \, dx\\ &=\frac {2 i \sqrt {1-i a x}}{a \sqrt {1+i a x}}-\int \frac {1}{\sqrt {1+a^2 x^2}} \, dx\\ &=\frac {2 i \sqrt {1-i a x}}{a \sqrt {1+i a x}}-\frac {\sinh ^{-1}(a x)}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 56, normalized size = 1.37 \begin {gather*} \frac {2 \left (\sqrt {1+a^2 x^2}+(-1-i a x) \text {ArcSin}\left (\frac {\sqrt {1-i a x}}{\sqrt {2}}\right )\right )}{a (-i+a x)} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 148 vs. \(2 (34 ) = 68\).
time = 0.08, size = 149, normalized size = 3.63
method | result | size |
default | \(-\frac {\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{a \left (x -\frac {i}{a}\right )^{2}}-i a \left (\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}+\frac {i a \ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{\sqrt {a^{2}}}\right )}{a^{2}}\) | \(149\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.47, size = 33, normalized size = 0.80 \begin {gather*} -\frac {\operatorname {arsinh}\left (a x\right )}{a} + \frac {2 i \, \sqrt {a^{2} x^{2} + 1}}{i \, a^{2} x + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.30, size = 54, normalized size = 1.32 \begin {gather*} \frac {2 \, a x + {\left (a x - i\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) + 2 \, \sqrt {a^{2} x^{2} + 1} - 2 i}{a^{2} x - i \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {\sqrt {a^{2} x^{2} + 1}}{a^{2} x^{2} - 2 i a x - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.49, size = 56, normalized size = 1.37 \begin {gather*} -\frac {\mathrm {asinh}\left (x\,\sqrt {a^2}\right )}{\sqrt {a^2}}-\frac {2\,\sqrt {a^2\,x^2+1}}{\left (-x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________