Optimal. Leaf size=68 \[ \frac {i \sec ^3(c+d x)}{5 d (a+i a \tan (c+d x))^4}+\frac {i \sec ^3(c+d x)}{15 a d (a+i a \tan (c+d x))^3} \]
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Rubi [A]
time = 0.06, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3583, 3569}
\begin {gather*} \frac {i \sec ^3(c+d x)}{15 a d (a+i a \tan (c+d x))^3}+\frac {i \sec ^3(c+d x)}{5 d (a+i a \tan (c+d x))^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 3569
Rule 3583
Rubi steps
\begin {align*} \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^4} \, dx &=\frac {i \sec ^3(c+d x)}{5 d (a+i a \tan (c+d x))^4}+\frac {\int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^3} \, dx}{5 a}\\ &=\frac {i \sec ^3(c+d x)}{5 d (a+i a \tan (c+d x))^4}+\frac {i \sec ^3(c+d x)}{15 a d (a+i a \tan (c+d x))^3}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 40, normalized size = 0.59 \begin {gather*} -\frac {\sec ^3(c+d x) (-4 i+\tan (c+d x))}{15 a^4 d (-i+\tan (c+d x))^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 90, normalized size = 1.32
method | result | size |
risch | \(\frac {i {\mathrm e}^{-3 i \left (d x +c \right )}}{6 a^{4} d}+\frac {i {\mathrm e}^{-5 i \left (d x +c \right )}}{10 a^{4} d}\) | \(38\) |
derivativedivides | \(\frac {-\frac {8 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {6 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {2}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {28}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {16}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}}{a^{4} d}\) | \(90\) |
default | \(\frac {-\frac {8 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {6 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {2}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {28}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {16}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}}{a^{4} d}\) | \(90\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 53, normalized size = 0.78 \begin {gather*} \frac {3 i \, \cos \left (5 \, d x + 5 \, c\right ) + 5 i \, \cos \left (3 \, d x + 3 \, c\right ) + 3 \, \sin \left (5 \, d x + 5 \, c\right ) + 5 \, \sin \left (3 \, d x + 3 \, c\right )}{30 \, a^{4} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 30, normalized size = 0.44 \begin {gather*} \frac {{\left (5 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\right )} e^{\left (-5 i \, d x - 5 i \, c\right )}}{30 \, a^{4} 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. 182 vs. \(2 (54) = 108\).
time = 1.38, size = 182, normalized size = 2.68 \begin {gather*} \begin {cases} - \frac {\tan {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{15 a^{4} d \tan ^{4}{\left (c + d x \right )} - 60 i a^{4} d \tan ^{3}{\left (c + d x \right )} - 90 a^{4} d \tan ^{2}{\left (c + d x \right )} + 60 i a^{4} d \tan {\left (c + d x \right )} + 15 a^{4} d} + \frac {4 i \sec ^{3}{\left (c + d x \right )}}{15 a^{4} d \tan ^{4}{\left (c + d x \right )} - 60 i a^{4} d \tan ^{3}{\left (c + d x \right )} - 90 a^{4} d \tan ^{2}{\left (c + d x \right )} + 60 i a^{4} d \tan {\left (c + d x \right )} + 15 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \sec ^{3}{\left (c \right )}}{\left (i a \tan {\left (c \right )} + a\right )^{4}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.73, size = 73, normalized size = 1.07 \begin {gather*} \frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 15 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 25 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 5 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4\right )}}{15 \, a^{4} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.66, size = 133, normalized size = 1.96 \begin {gather*} \frac {2\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,15{}\mathrm {i}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,25{}\mathrm {i}-5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+4{}\mathrm {i}\right )}{15\,a^4\,d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,1{}\mathrm {i}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,10{}\mathrm {i}-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,5{}\mathrm {i}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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