Optimal. Leaf size=23 \[ -x-\frac {2 i \log (i-a-b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5201, 45}
\begin {gather*} -x-\frac {2 i \log (-a-b x+i)}{b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 5201
Rubi steps
\begin {align*} \int e^{-2 i \tan ^{-1}(a+b x)} \, dx &=\int \frac {1-i a-i b x}{1+i a+i b x} \, dx\\ &=\int \left (-1-\frac {2 i}{-i+a+b x}\right ) \, dx\\ &=-x-\frac {2 i \log (i-a-b x)}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 32, normalized size = 1.39 \begin {gather*} -x+\frac {2 \text {ArcTan}(a+b x)}{b}-\frac {i \log \left (1+(a+b x)^2\right )}{b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.07, size = 22, normalized size = 0.96
method | result | size |
default | \(-x -\frac {2 i \ln \left (-b x -a +i\right )}{b}\) | \(22\) |
risch | \(-x -\frac {i \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{b}+\frac {2 \arctan \left (b x +a \right )}{b}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 19, normalized size = 0.83 \begin {gather*} -x - \frac {2 i \, \log \left (i \, b x + i \, a + 1\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.55, size = 22, normalized size = 0.96 \begin {gather*} -\frac {b x + 2 i \, \log \left (\frac {b x + a - i}{b}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.07, size = 15, normalized size = 0.65 \begin {gather*} - x - \frac {2 i \log {\left (a + b x - i \right )}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.45, size = 37, normalized size = 1.61 \begin {gather*} \frac {i \, {\left (i \, b x + i \, a + 1\right )}}{b} + \frac {2 i \, \log \left (\frac {1}{\sqrt {{\left (b x + a\right )}^{2} + 1} {\left | b \right |}}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.06, size = 21, normalized size = 0.91 \begin {gather*} -x-\frac {\ln \left (x+\frac {a-\mathrm {i}}{b}\right )\,2{}\mathrm {i}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________