Optimal. Leaf size=66 \[ -\frac {i a \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{15 d}-\frac {i \cos ^5(c+d x) (a+i a \tan (c+d x))^4}{5 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 66, 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 = {3578, 3569}
\begin {gather*} -\frac {i \cos ^5(c+d x) (a+i a \tan (c+d x))^4}{5 d}-\frac {i a \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{15 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3569
Rule 3578
Rubi steps
\begin {align*} \int \cos ^5(c+d x) (a+i a \tan (c+d x))^4 \, dx &=-\frac {i \cos ^5(c+d x) (a+i a \tan (c+d x))^4}{5 d}+\frac {1}{5} a \int \cos ^3(c+d x) (a+i a \tan (c+d x))^3 \, dx\\ &=-\frac {i a \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{15 d}-\frac {i \cos ^5(c+d x) (a+i a \tan (c+d x))^4}{5 d}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 50, normalized size = 0.76 \begin {gather*} \frac {a^4 (4 \cos (c+d x)-i \sin (c+d x)) (-i \cos (4 (c+d x))+\sin (4 (c+d x)))}{15 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 138 vs. \(2 (58 ) = 116\).
time = 0.24, size = 139, normalized size = 2.11
method | result | size |
risch | \(-\frac {i a^{4} {\mathrm e}^{5 i \left (d x +c \right )}}{10 d}-\frac {i a^{4} {\mathrm e}^{3 i \left (d x +c \right )}}{6 d}\) | \(38\) |
derivativedivides | \(\frac {\frac {a^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{5}-4 i a^{4} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )-6 a^{4} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {\left (\cos ^{2}\left (d x +c \right )+2\right ) \sin \left (d x +c \right )}{15}\right )-\frac {4 i a^{4} \left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}}{d}\) | \(139\) |
default | \(\frac {\frac {a^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{5}-4 i a^{4} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )-6 a^{4} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {\left (\cos ^{2}\left (d x +c \right )+2\right ) \sin \left (d x +c \right )}{15}\right )-\frac {4 i a^{4} \left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}}{d}\) | \(139\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 118 vs. \(2 (54) = 108\).
time = 0.28, size = 118, normalized size = 1.79 \begin {gather*} -\frac {12 i \, a^{4} \cos \left (d x + c\right )^{5} - 3 \, a^{4} \sin \left (d x + c\right )^{5} + 4 i \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{4} - 6 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 5 \, \sin \left (d x + c\right )^{3}\right )} a^{4} - {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a^{4}}{15 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 34, normalized size = 0.52 \begin {gather*} \frac {-3 i \, a^{4} e^{\left (5 i \, d x + 5 i \, c\right )} - 5 i \, a^{4} e^{\left (3 i \, d x + 3 i \, c\right )}}{30 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.24, size = 80, normalized size = 1.21 \begin {gather*} \begin {cases} \frac {- 6 i a^{4} d e^{5 i c} e^{5 i d x} - 10 i a^{4} d e^{3 i c} e^{3 i d x}}{60 d^{2}} & \text {for}\: d^{2} \neq 0 \\x \left (\frac {a^{4} e^{5 i c}}{2} + \frac {a^{4} e^{3 i c}}{2}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 915 vs. \(2 (54) = 108\).
time = 1.21, size = 915, normalized size = 13.86 \begin {gather*} \frac {9075 \, a^{4} e^{\left (8 i \, d x + 4 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 36300 \, a^{4} e^{\left (6 i \, d x + 2 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 36300 \, a^{4} e^{\left (2 i \, d x - 2 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 54450 \, a^{4} e^{\left (4 i \, d x\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 9075 \, a^{4} e^{\left (-4 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 9000 \, a^{4} e^{\left (8 i \, d x + 4 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) + 36000 \, a^{4} e^{\left (6 i \, d x + 2 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) + 36000 \, a^{4} e^{\left (2 i \, d x - 2 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) + 54000 \, a^{4} e^{\left (4 i \, d x\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) + 9000 \, a^{4} e^{\left (-4 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 9075 \, a^{4} e^{\left (8 i \, d x + 4 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 36300 \, a^{4} e^{\left (6 i \, d x + 2 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 36300 \, a^{4} e^{\left (2 i \, d x - 2 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 54450 \, a^{4} e^{\left (4 i \, d x\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 9075 \, a^{4} e^{\left (-4 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 9000 \, a^{4} e^{\left (8 i \, d x + 4 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 36000 \, a^{4} e^{\left (6 i \, d x + 2 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 36000 \, a^{4} e^{\left (2 i \, d x - 2 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 54000 \, a^{4} e^{\left (4 i \, d x\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 9000 \, a^{4} e^{\left (-4 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 75 \, a^{4} e^{\left (8 i \, d x + 4 i \, c\right )} \log \left (i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) - 300 \, a^{4} e^{\left (6 i \, d x + 2 i \, c\right )} \log \left (i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) - 300 \, a^{4} e^{\left (2 i \, d x - 2 i \, c\right )} \log \left (i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) - 450 \, a^{4} e^{\left (4 i \, d x\right )} \log \left (i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) - 75 \, a^{4} e^{\left (-4 i \, c\right )} \log \left (i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 75 \, a^{4} e^{\left (8 i \, d x + 4 i \, c\right )} \log \left (-i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 300 \, a^{4} e^{\left (6 i \, d x + 2 i \, c\right )} \log \left (-i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 300 \, a^{4} e^{\left (2 i \, d x - 2 i \, c\right )} \log \left (-i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 450 \, a^{4} e^{\left (4 i \, d x\right )} \log \left (-i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 75 \, a^{4} e^{\left (-4 i \, c\right )} \log \left (-i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) - 768 i \, a^{4} e^{\left (13 i \, d x + 9 i \, c\right )} - 4352 i \, a^{4} e^{\left (11 i \, d x + 7 i \, c\right )} - 9728 i \, a^{4} e^{\left (9 i \, d x + 5 i \, c\right )} - 10752 i \, a^{4} e^{\left (7 i \, d x + 3 i \, c\right )} - 5888 i \, a^{4} e^{\left (5 i \, d x + i \, c\right )} - 1280 i \, a^{4} e^{\left (3 i \, d x - i \, c\right )}}{7680 \, {\left (d e^{\left (8 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 2 i \, c\right )} + 4 \, d e^{\left (2 i \, d x - 2 i \, c\right )} + 6 \, d e^{\left (4 i \, d x\right )} + d e^{\left (-4 i \, c\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.55, size = 130, normalized size = 1.97 \begin {gather*} \frac {2\,a^4\,\left (15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,15{}\mathrm {i}-25\,{\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}+4\right )}{15\,d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,5{}\mathrm {i}-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,10{}\mathrm {i}+5\,\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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