Optimal. Leaf size=67 \[ \frac {i \sqrt {1-i a x}}{3 a (1+i a x)^{3/2}}+\frac {i \sqrt {1-i a x}}{3 a \sqrt {1+i a x}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5181, 47, 37}
\begin {gather*} \frac {i \sqrt {1-i a x}}{3 a \sqrt {1+i a x}}+\frac {i \sqrt {1-i a x}}{3 a (1+i a x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 47
Rule 5181
Rubi steps
\begin {align*} \int \frac {e^{-2 i \tan ^{-1}(a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx &=\int \frac {1}{\sqrt {1-i a x} (1+i a x)^{5/2}} \, dx\\ &=\frac {i \sqrt {1-i a x}}{3 a (1+i a x)^{3/2}}+\frac {1}{3} \int \frac {1}{\sqrt {1-i a x} (1+i a x)^{3/2}} \, dx\\ &=\frac {i \sqrt {1-i a x}}{3 a (1+i a x)^{3/2}}+\frac {i \sqrt {1-i a x}}{3 a \sqrt {1+i a x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 48, normalized size = 0.72 \begin {gather*} \frac {\sqrt {1-i a x} (2+i a x)}{3 a \sqrt {1+i a x} (-i+a x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.06, size = 93, normalized size = 1.39
method | result | size |
default | \(-\frac {\frac {i \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{3 a \left (x -\frac {i}{a}\right )^{2}}-\frac {\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{3 \left (x -\frac {i}{a}\right )}}{a^{2}}\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.47, size = 58, normalized size = 0.87 \begin {gather*} -\frac {i \, \sqrt {a^{2} x^{2} + 1}}{3 \, {\left (a^{3} x^{2} - 2 i \, a^{2} x - a\right )}} + \frac {i \, \sqrt {a^{2} x^{2} + 1}}{3 i \, a^{2} x + 3 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.66, size = 51, normalized size = 0.76 \begin {gather*} \frac {a^{2} x^{2} - 2 i \, a x + \sqrt {a^{2} x^{2} + 1} {\left (a x - 2 i\right )} - 1}{3 \, {\left (a^{3} x^{2} - 2 i \, a^{2} x - a\right )}} \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 {1}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} - 2 i a x \sqrt {a^{2} x^{2} + 1} - \sqrt {a^{2} x^{2} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.43, size = 67, normalized size = 1.00 \begin {gather*} -\frac {2 \, {\left (2 \, a^{2} - 3 \, {\left (\sqrt {a^{2} + \frac {1}{x^{2}}} - \frac {1}{x}\right )}^{2} + 3 \, a {\left (i \, \sqrt {a^{2} + \frac {1}{x^{2}}} - \frac {i}{x}\right )}\right )}}{3 \, {\left (-i \, a + \sqrt {a^{2} + \frac {1}{x^{2}}} - \frac {1}{x}\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.06, size = 31, normalized size = 0.46 \begin {gather*} -\frac {\sqrt {a^2\,x^2+1}\,\left (a\,x-2{}\mathrm {i}\right )}{3\,a\,{\left (1+a\,x\,1{}\mathrm {i}\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________