Optimal. Leaf size=33 \[ \frac {x}{2 a}+\frac {i}{2 d (a+i a \tan (c+d x))} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3479, 8} \[ \frac {x}{2 a}+\frac {i}{2 d (a+i a \tan (c+d x))} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3479
Rubi steps
\begin {align*} \int \frac {1}{a+i a \tan (c+d x)} \, dx &=\frac {i}{2 d (a+i a \tan (c+d x))}+\frac {\int 1 \, dx}{2 a}\\ &=\frac {x}{2 a}+\frac {i}{2 d (a+i a \tan (c+d x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 45, normalized size = 1.36 \[ \frac {(2 d x-i) \tan (c+d x)-2 i d x+1}{4 a d (\tan (c+d x)-i)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.56, size = 32, normalized size = 0.97 \[ \frac {{\left (2 \, d x e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{4 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.40, size = 60, normalized size = 1.82 \[ -\frac {\frac {i \, \log \left (\tan \left (d x + c\right ) - i\right )}{a} - \frac {i \, \log \left (-i \, \tan \left (d x + c\right ) + 1\right )}{a} + \frac {-i \, \tan \left (d x + c\right ) - 3}{a {\left (\tan \left (d x + c\right ) - i\right )}}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.11, size = 59, normalized size = 1.79 \[ \frac {i \ln \left (\tan \left (d x +c \right )+i\right )}{4 d a}-\frac {i \ln \left (\tan \left (d x +c \right )-i\right )}{4 d a}+\frac {1}{2 d a \left (\tan \left (d x +c \right )-i\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.36, size = 29, normalized size = 0.88 \[ \frac {x}{2\,a}+\frac {1{}\mathrm {i}}{2\,a\,d\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.15, size = 61, normalized size = 1.85 \[ \begin {cases} \frac {i e^{- 2 i c} e^{- 2 i d x}}{4 a d} & \text {for}\: 4 a d e^{2 i c} \neq 0 \\x \left (\frac {\left (e^{2 i c} + 1\right ) e^{- 2 i c}}{2 a} - \frac {1}{2 a}\right ) & \text {otherwise} \end {cases} + \frac {x}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________