Optimal. Leaf size=71 \[ \frac {3 \sin (c+d x)}{5 a^2 d}-\frac {\sin ^3(c+d x)}{5 a^2 d}+\frac {2 i \cos ^3(c+d x)}{5 d \left (a^2+i a^2 \tan (c+d x)\right )} \]
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Rubi [A]
time = 0.04, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3581, 2713}
\begin {gather*} -\frac {\sin ^3(c+d x)}{5 a^2 d}+\frac {3 \sin (c+d x)}{5 a^2 d}+\frac {2 i \cos ^3(c+d x)}{5 d \left (a^2+i a^2 \tan (c+d x)\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2713
Rule 3581
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^2} \, dx &=\frac {2 i \cos ^3(c+d x)}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {3 \int \cos ^3(c+d x) \, dx}{5 a^2}\\ &=\frac {2 i \cos ^3(c+d x)}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}-\frac {3 \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 a^2 d}\\ &=\frac {3 \sin (c+d x)}{5 a^2 d}-\frac {\sin ^3(c+d x)}{5 a^2 d}+\frac {2 i \cos ^3(c+d x)}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 68, normalized size = 0.96 \begin {gather*} \frac {\sec (c+d x) (-12 i+4 i \cos (2 (c+d x))-3 \sec (c+d x) \sin (3 (c+d x))+5 \tan (c+d x))}{20 a^2 d (-i+\tan (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.47, size = 108, normalized size = 1.52
method | result | size |
risch | \(\frac {i {\mathrm e}^{-3 i \left (d x +c \right )}}{8 a^{2} d}+\frac {i {\mathrm e}^{-5 i \left (d x +c \right )}}{40 a^{2} d}+\frac {i \cos \left (d x +c \right )}{4 a^{2} d}+\frac {\sin \left (d x +c \right )}{2 a^{2} d}\) | \(67\) |
derivativedivides | \(\frac {\frac {2}{8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+8 i}-\frac {2 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {5 i}{2 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {4}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {3}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {7}{4 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{a^{2} d}\) | \(108\) |
default | \(\frac {\frac {2}{8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+8 i}-\frac {2 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {5 i}{2 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {4}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {3}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {7}{4 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{a^{2} d}\) | \(108\) |
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 = 0.36, size = 52, normalized size = 0.73 \begin {gather*} \frac {{\left (-5 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 15 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 5 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-5 i \, d x - 5 i \, c\right )}}{40 \, a^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 163 vs. \(2 (60) = 120\).
time = 0.22, size = 163, normalized size = 2.30 \begin {gather*} \begin {cases} \frac {\left (- 2560 i a^{6} d^{3} e^{10 i c} e^{i d x} + 7680 i a^{6} d^{3} e^{8 i c} e^{- i d x} + 2560 i a^{6} d^{3} e^{6 i c} e^{- 3 i d x} + 512 i a^{6} d^{3} e^{4 i c} e^{- 5 i d x}\right ) e^{- 9 i c}}{20480 a^{8} d^{4}} & \text {for}\: a^{8} d^{4} e^{9 i c} \neq 0 \\\frac {x \left (e^{6 i c} + 3 e^{4 i c} + 3 e^{2 i c} + 1\right ) e^{- 5 i c}}{8 a^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.68, size = 93, normalized size = 1.31 \begin {gather*} \frac {\frac {5}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right )}} + \frac {35 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 90 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 70 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 21}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{5}}}{20 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.89, size = 90, normalized size = 1.27 \begin {gather*} -\frac {2\,\left (-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,10{}\mathrm {i}+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-2{}\mathrm {i}\right )}{5\,a^2\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\mathrm {i}\right )}^5\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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