Optimal. Leaf size=59 \[ -\frac {x}{4 a^2}+\frac {3 i}{4 a^2 d (1+i \tan (c+d x))}-\frac {i}{4 d (a+i a \tan (c+d x))^2} \]
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Rubi [A]
time = 0.06, antiderivative size = 59, 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 = {3621, 3607, 8}
\begin {gather*} \frac {3 i}{4 a^2 d (1+i \tan (c+d x))}-\frac {x}{4 a^2}-\frac {i}{4 d (a+i a \tan (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3607
Rule 3621
Rubi steps
\begin {align*} \int \frac {\tan ^2(c+d x)}{(a+i a \tan (c+d x))^2} \, dx &=-\frac {i}{4 d (a+i a \tan (c+d x))^2}+\frac {\int \frac {a-2 i a \tan (c+d x)}{a+i a \tan (c+d x)} \, dx}{2 a^2}\\ &=\frac {3 i}{4 a^2 d (1+i \tan (c+d x))}-\frac {i}{4 d (a+i a \tan (c+d x))^2}-\frac {\int 1 \, dx}{4 a^2}\\ &=-\frac {x}{4 a^2}+\frac {3 i}{4 a^2 d (1+i \tan (c+d x))}-\frac {i}{4 d (a+i a \tan (c+d x))^2}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 68, normalized size = 1.15 \begin {gather*} \frac {\sec ^2(c+d x) (-4 i+(i+4 d x) \cos (2 (c+d x))+(1+4 i d x) \sin (2 (c+d x)))}{16 a^2 d (-i+\tan (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 62, normalized size = 1.05
method | result | size |
risch | \(-\frac {x}{4 a^{2}}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{4 a^{2} d}-\frac {i {\mathrm e}^{-4 i \left (d x +c \right )}}{16 a^{2} d}\) | \(44\) |
derivativedivides | \(\frac {\frac {i}{4 \left (\tan \left (d x +c \right )-i\right )^{2}}+\frac {i \ln \left (\tan \left (d x +c \right )-i\right )}{8}+\frac {3}{4 \left (\tan \left (d x +c \right )-i\right )}-\frac {i \ln \left (\tan \left (d x +c \right )+i\right )}{8}}{d \,a^{2}}\) | \(62\) |
default | \(\frac {\frac {i}{4 \left (\tan \left (d x +c \right )-i\right )^{2}}+\frac {i \ln \left (\tan \left (d x +c \right )-i\right )}{8}+\frac {3}{4 \left (\tan \left (d x +c \right )-i\right )}-\frac {i \ln \left (\tan \left (d x +c \right )+i\right )}{8}}{d \,a^{2}}\) | \(62\) |
norman | \(\frac {\frac {i \left (\tan ^{2}\left (d x +c \right )\right )}{d a}-\frac {x}{4 a}-\frac {x \left (\tan ^{2}\left (d x +c \right )\right )}{2 a}-\frac {x \left (\tan ^{4}\left (d x +c \right )\right )}{4 a}+\frac {i}{2 d a}+\frac {\tan \left (d x +c \right )}{4 d a}+\frac {3 \left (\tan ^{3}\left (d x +c \right )\right )}{4 d a}}{a \left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}\) | \(108\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 43, normalized size = 0.73 \begin {gather*} -\frac {{\left (4 \, d x e^{\left (4 i \, d x + 4 i \, c\right )} - 4 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{16 \, a^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.16, size = 117, normalized size = 1.98 \begin {gather*} \begin {cases} \frac {\left (16 i a^{2} d e^{4 i c} e^{- 2 i d x} - 4 i a^{2} d e^{2 i c} e^{- 4 i d x}\right ) e^{- 6 i c}}{64 a^{4} d^{2}} & \text {for}\: a^{4} d^{2} e^{6 i c} \neq 0 \\x \left (\frac {\left (- e^{4 i c} + 2 e^{2 i c} - 1\right ) e^{- 4 i c}}{4 a^{2}} + \frac {1}{4 a^{2}}\right ) & \text {otherwise} \end {cases} - \frac {x}{4 a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.68, size = 72, normalized size = 1.22 \begin {gather*} -\frac {\frac {2 i \, \log \left (-i \, \tan \left (d x + c\right ) + 1\right )}{a^{2}} - \frac {2 i \, \log \left (-i \, \tan \left (d x + c\right ) - 1\right )}{a^{2}} + \frac {3 i \, \tan \left (d x + c\right )^{2} - 6 \, \tan \left (d x + c\right ) + 5 i}{a^{2} {\left (\tan \left (d x + c\right ) - i\right )}^{2}}}{16 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.97, size = 39, normalized size = 0.66 \begin {gather*} -\frac {x}{4\,a^2}-\frac {\frac {3\,\mathrm {tan}\left (c+d\,x\right )}{4}-\frac {1}{2}{}\mathrm {i}}{a^2\,d\,{\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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