Optimal. Leaf size=89 \[ \frac {5 a x}{16}-\frac {i a \cos ^6(c+d x)}{6 d}+\frac {5 a \cos (c+d x) \sin (c+d x)}{16 d}+\frac {5 a \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{6 d} \]
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Rubi [A]
time = 0.04, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {3567, 2715, 8}
\begin {gather*} -\frac {i a \cos ^6(c+d x)}{6 d}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5 a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac {5 a x}{16} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 3567
Rubi steps
\begin {align*} \int \cos ^6(c+d x) (a+i a \tan (c+d x)) \, dx &=-\frac {i a \cos ^6(c+d x)}{6 d}+a \int \cos ^6(c+d x) \, dx\\ &=-\frac {i a \cos ^6(c+d x)}{6 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac {1}{6} (5 a) \int \cos ^4(c+d x) \, dx\\ &=-\frac {i a \cos ^6(c+d x)}{6 d}+\frac {5 a \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac {1}{8} (5 a) \int \cos ^2(c+d x) \, dx\\ &=-\frac {i a \cos ^6(c+d x)}{6 d}+\frac {5 a \cos (c+d x) \sin (c+d x)}{16 d}+\frac {5 a \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac {1}{16} (5 a) \int 1 \, dx\\ &=\frac {5 a x}{16}-\frac {i a \cos ^6(c+d x)}{6 d}+\frac {5 a \cos (c+d x) \sin (c+d x)}{16 d}+\frac {5 a \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{6 d}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 56, normalized size = 0.63 \begin {gather*} \frac {a \left (60 c+60 d x-32 i \cos ^6(c+d x)+45 \sin (2 (c+d x))+9 \sin (4 (c+d x))+\sin (6 (c+d x))\right )}{192 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 63, normalized size = 0.71
method | result | size |
derivativedivides | \(\frac {-\frac {i a \left (\cos ^{6}\left (d x +c \right )\right )}{6}+a \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(63\) |
default | \(\frac {-\frac {i a \left (\cos ^{6}\left (d x +c \right )\right )}{6}+a \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(63\) |
risch | \(\frac {5 a x}{16}-\frac {i a \,{\mathrm e}^{6 i \left (d x +c \right )}}{192 d}-\frac {i a \cos \left (4 d x +4 c \right )}{32 d}+\frac {3 a \sin \left (4 d x +4 c \right )}{64 d}-\frac {5 i a \cos \left (2 d x +2 c \right )}{64 d}+\frac {15 a \sin \left (2 d x +2 c \right )}{64 d}\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 82, normalized size = 0.92 \begin {gather*} \frac {15 \, {\left (d x + c\right )} a + \frac {15 \, a \tan \left (d x + c\right )^{5} + 40 \, a \tan \left (d x + c\right )^{3} + 33 \, a \tan \left (d x + c\right ) - 8 i \, a}{\tan \left (d x + c\right )^{6} + 3 \, \tan \left (d x + c\right )^{4} + 3 \, \tan \left (d x + c\right )^{2} + 1}}{48 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 80, normalized size = 0.90 \begin {gather*} \frac {{\left (120 \, a d x e^{\left (4 i \, d x + 4 i \, c\right )} - 2 i \, a e^{\left (10 i \, d x + 10 i \, c\right )} - 15 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} - 60 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} + 30 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i \, a\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{384 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.21, size = 211, normalized size = 2.37 \begin {gather*} \frac {5 a x}{16} + \begin {cases} \frac {\left (- 33554432 i a d^{4} e^{12 i c} e^{6 i d x} - 251658240 i a d^{4} e^{10 i c} e^{4 i d x} - 1006632960 i a d^{4} e^{8 i c} e^{2 i d x} + 503316480 i a d^{4} e^{4 i c} e^{- 2 i d x} + 50331648 i a d^{4} e^{2 i c} e^{- 4 i d x}\right ) e^{- 6 i c}}{6442450944 d^{5}} & \text {for}\: d^{5} e^{6 i c} \neq 0 \\x \left (- \frac {5 a}{16} + \frac {\left (a e^{10 i c} + 5 a e^{8 i c} + 10 a e^{6 i c} + 10 a e^{4 i c} + 5 a e^{2 i c} + a\right ) e^{- 4 i c}}{32}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.61, size = 127, normalized size = 1.43 \begin {gather*} \frac {{\left (120 \, a d x e^{\left (4 i \, d x + 2 i \, c\right )} + 12 i \, a e^{\left (4 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 12 i \, a e^{\left (4 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) - 2 i \, a e^{\left (10 i \, d x + 8 i \, c\right )} - 15 i \, a e^{\left (8 i \, d x + 6 i \, c\right )} - 60 i \, a e^{\left (6 i \, d x + 4 i \, c\right )} + 30 i \, a e^{\left (2 i \, d x\right )} + 3 i \, a e^{\left (-2 i \, c\right )}\right )} e^{\left (-4 i \, d x - 2 i \, c\right )}}{384 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.57, size = 108, normalized size = 1.21 \begin {gather*} \frac {5\,a\,x}{16}+\frac {\frac {5\,a\,{\mathrm {tan}\left (c+d\,x\right )}^4}{16}+\frac {5{}\mathrm {i}\,a\,{\mathrm {tan}\left (c+d\,x\right )}^3}{16}+\frac {25\,a\,{\mathrm {tan}\left (c+d\,x\right )}^2}{48}+\frac {25{}\mathrm {i}\,a\,\mathrm {tan}\left (c+d\,x\right )}{48}+\frac {a}{6}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^5+{\mathrm {tan}\left (c+d\,x\right )}^4\,1{}\mathrm {i}+2\,{\mathrm {tan}\left (c+d\,x\right )}^3+{\mathrm {tan}\left (c+d\,x\right )}^2\,2{}\mathrm {i}+\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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