Optimal. Leaf size=124 \[ -\frac {5 i a^3 \cos ^7(c+d x)}{63 d}+\frac {5 a^3 \sin (c+d x)}{9 d}-\frac {5 a^3 \sin ^3(c+d x)}{9 d}+\frac {a^3 \sin ^5(c+d x)}{3 d}-\frac {5 a^3 \sin ^7(c+d x)}{63 d}-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^2}{9 d} \]
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Rubi [A]
time = 0.06, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3577, 3567,
2713} \begin {gather*} -\frac {5 a^3 \sin ^7(c+d x)}{63 d}+\frac {a^3 \sin ^5(c+d x)}{3 d}-\frac {5 a^3 \sin ^3(c+d x)}{9 d}+\frac {5 a^3 \sin (c+d x)}{9 d}-\frac {5 i a^3 \cos ^7(c+d x)}{63 d}-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^2}{9 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2713
Rule 3567
Rule 3577
Rubi steps
\begin {align*} \int \cos ^9(c+d x) (a+i a \tan (c+d x))^3 \, dx &=-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^2}{9 d}+\frac {1}{9} \left (5 a^2\right ) \int \cos ^7(c+d x) (a+i a \tan (c+d x)) \, dx\\ &=-\frac {5 i a^3 \cos ^7(c+d x)}{63 d}-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^2}{9 d}+\frac {1}{9} \left (5 a^3\right ) \int \cos ^7(c+d x) \, dx\\ &=-\frac {5 i a^3 \cos ^7(c+d x)}{63 d}-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^2}{9 d}-\frac {\left (5 a^3\right ) \text {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (c+d x)\right )}{9 d}\\ &=-\frac {5 i a^3 \cos ^7(c+d x)}{63 d}+\frac {5 a^3 \sin (c+d x)}{9 d}-\frac {5 a^3 \sin ^3(c+d x)}{9 d}+\frac {a^3 \sin ^5(c+d x)}{3 d}-\frac {5 a^3 \sin ^7(c+d x)}{63 d}-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^2}{9 d}\\ \end {align*}
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Mathematica [A]
time = 0.62, size = 116, normalized size = 0.94 \begin {gather*} \frac {a^3 (210+567 \cos (2 (c+d x))-162 \cos (4 (c+d x))-7 \cos (6 (c+d x))-378 i \sin (2 (c+d x))+216 i \sin (4 (c+d x))+14 i \sin (6 (c+d x))) (-i \cos (3 (c+2 d x))+\sin (3 (c+2 d x)))}{2016 d (\cos (d x)+i \sin (d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 166, normalized size = 1.34
method | result | size |
risch | \(-\frac {i a^{3} {\mathrm e}^{9 i \left (d x +c \right )}}{576 d}-\frac {3 i a^{3} {\mathrm e}^{7 i \left (d x +c \right )}}{224 d}-\frac {3 i a^{3} {\mathrm e}^{5 i \left (d x +c \right )}}{64 d}-\frac {9 i a^{3} \cos \left (d x +c \right )}{64 d}+\frac {21 a^{3} \sin \left (d x +c \right )}{64 d}-\frac {19 i a^{3} \cos \left (3 d x +3 c \right )}{192 d}+\frac {7 a^{3} \sin \left (3 d x +3 c \right )}{64 d}\) | \(120\) |
derivativedivides | \(\frac {-i a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )-3 a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{8}\left (d x +c \right )\right )}{9}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{63}\right )-\frac {i a^{3} \left (\cos ^{9}\left (d x +c \right )\right )}{3}+\frac {a^{3} \left (\frac {128}{35}+\cos ^{8}\left (d x +c \right )+\frac {8 \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\cos ^{2}\left (d x +c \right )\right )}{35}\right ) \sin \left (d x +c \right )}{9}}{d}\) | \(166\) |
default | \(\frac {-i a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )-3 a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{8}\left (d x +c \right )\right )}{9}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{63}\right )-\frac {i a^{3} \left (\cos ^{9}\left (d x +c \right )\right )}{3}+\frac {a^{3} \left (\frac {128}{35}+\cos ^{8}\left (d x +c \right )+\frac {8 \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\cos ^{2}\left (d x +c \right )\right )}{35}\right ) \sin \left (d x +c \right )}{9}}{d}\) | \(166\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 145, normalized size = 1.17 \begin {gather*} -\frac {105 i \, a^{3} \cos \left (d x + c\right )^{9} + 5 i \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{3} - 3 \, {\left (35 \, \sin \left (d x + c\right )^{9} - 135 \, \sin \left (d x + c\right )^{7} + 189 \, \sin \left (d x + c\right )^{5} - 105 \, \sin \left (d x + c\right )^{3}\right )} a^{3} - {\left (35 \, \sin \left (d x + c\right )^{9} - 180 \, \sin \left (d x + c\right )^{7} + 378 \, \sin \left (d x + c\right )^{5} - 420 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )\right )} a^{3}}{315 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 104, normalized size = 0.84 \begin {gather*} \frac {{\left (-7 i \, a^{3} e^{\left (12 i \, d x + 12 i \, c\right )} - 54 i \, a^{3} e^{\left (10 i \, d x + 10 i \, c\right )} - 189 i \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} - 420 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 945 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 378 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 21 i \, a^{3}\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{4032 \, 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. 275 vs. \(2 (112) = 224\).
time = 0.46, size = 275, normalized size = 2.22 \begin {gather*} \begin {cases} \frac {\left (- 270582939648 i a^{3} d^{6} e^{13 i c} e^{9 i d x} - 2087354105856 i a^{3} d^{6} e^{11 i c} e^{7 i d x} - 7305739370496 i a^{3} d^{6} e^{9 i c} e^{5 i d x} - 16234976378880 i a^{3} d^{6} e^{7 i c} e^{3 i d x} - 36528696852480 i a^{3} d^{6} e^{5 i c} e^{i d x} + 14611478740992 i a^{3} d^{6} e^{3 i c} e^{- i d x} + 811748818944 i a^{3} d^{6} e^{i c} e^{- 3 i d x}\right ) e^{- 4 i c}}{155855773237248 d^{7}} & \text {for}\: d^{7} e^{4 i c} \neq 0 \\\frac {x \left (a^{3} e^{12 i c} + 6 a^{3} e^{10 i c} + 15 a^{3} e^{8 i c} + 20 a^{3} e^{6 i c} + 15 a^{3} e^{4 i c} + 6 a^{3} e^{2 i c} + a^{3}\right ) e^{- 3 i c}}{64} & \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. 1039 vs. \(2 (106) = 212\).
time = 0.87, size = 1039, normalized size = 8.38 \begin {gather*} \frac {119511 \, a^{3} e^{\left (11 i \, d x + 5 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 478044 \, a^{3} e^{\left (9 i \, d x + 3 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 717066 \, a^{3} e^{\left (7 i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 478044 \, a^{3} e^{\left (5 i \, d x - i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 119511 \, a^{3} e^{\left (3 i \, d x - 3 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 128898 \, a^{3} e^{\left (11 i \, d x + 5 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) + 515592 \, a^{3} e^{\left (9 i \, d x + 3 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) + 773388 \, a^{3} e^{\left (7 i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) + 515592 \, a^{3} e^{\left (5 i \, d x - i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) + 128898 \, a^{3} e^{\left (3 i \, d x - 3 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 119511 \, a^{3} e^{\left (11 i \, d x + 5 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 478044 \, a^{3} e^{\left (9 i \, d x + 3 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 717066 \, a^{3} e^{\left (7 i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 478044 \, a^{3} e^{\left (5 i \, d x - i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 119511 \, a^{3} e^{\left (3 i \, d x - 3 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 128898 \, a^{3} e^{\left (11 i \, d x + 5 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 515592 \, a^{3} e^{\left (9 i \, d x + 3 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 773388 \, a^{3} e^{\left (7 i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 515592 \, a^{3} e^{\left (5 i \, d x - i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 128898 \, a^{3} e^{\left (3 i \, d x - 3 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) + 9387 \, a^{3} e^{\left (11 i \, d x + 5 i \, c\right )} \log \left (i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 37548 \, a^{3} e^{\left (9 i \, d x + 3 i \, c\right )} \log \left (i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 56322 \, a^{3} e^{\left (7 i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 37548 \, a^{3} e^{\left (5 i \, d x - i \, c\right )} \log \left (i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 9387 \, a^{3} e^{\left (3 i \, d x - 3 i \, c\right )} \log \left (i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) - 9387 \, a^{3} e^{\left (11 i \, d x + 5 i \, c\right )} \log \left (-i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) - 37548 \, a^{3} e^{\left (9 i \, d x + 3 i \, c\right )} \log \left (-i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) - 56322 \, a^{3} e^{\left (7 i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) - 37548 \, a^{3} e^{\left (5 i \, d x - i \, c\right )} \log \left (-i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) - 9387 \, a^{3} e^{\left (3 i \, d x - 3 i \, c\right )} \log \left (-i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) - 896 i \, a^{3} e^{\left (20 i \, d x + 14 i \, c\right )} - 10496 i \, a^{3} e^{\left (18 i \, d x + 12 i \, c\right )} - 57216 i \, a^{3} e^{\left (16 i \, d x + 10 i \, c\right )} - 195584 i \, a^{3} e^{\left (14 i \, d x + 8 i \, c\right )} - 509696 i \, a^{3} e^{\left (12 i \, d x + 6 i \, c\right )} - 861696 i \, a^{3} e^{\left (10 i \, d x + 4 i \, c\right )} - 768768 i \, a^{3} e^{\left (8 i \, d x + 2 i \, c\right )} + 88704 i \, a^{3} e^{\left (4 i \, d x - 2 i \, c\right )} + 59136 i \, a^{3} e^{\left (2 i \, d x - 4 i \, c\right )} - 236544 i \, a^{3} e^{\left (6 i \, d x\right )} + 2688 i \, a^{3} e^{\left (-6 i \, c\right )}}{516096 \, {\left (d e^{\left (11 i \, d x + 5 i \, c\right )} + 4 \, d e^{\left (9 i \, d x + 3 i \, c\right )} + 6 \, d e^{\left (7 i \, d x + i \, c\right )} + 4 \, d e^{\left (5 i \, d x - i \, c\right )} + d e^{\left (3 i \, d x - 3 i \, c\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.68, size = 330, normalized size = 2.66 \begin {gather*} \frac {2\,a^3\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-3{}\mathrm {i}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {2048\,a^3\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\mathrm {i}\right )}{9\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^9}-\frac {1024\,a^3\,\left (8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-9{}\mathrm {i}\right )}{9\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^8}-\frac {4\,a^3\,\left (14\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-39{}\mathrm {i}\right )}{3\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2}+\frac {8\,a^3\,\left (43\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-97{}\mathrm {i}\right )}{3\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^3}-\frac {16\,a^3\,\left (188\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-357{}\mathrm {i}\right )}{7\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^4}+\frac {128\,a^3\,\left (263\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-333{}\mathrm {i}\right )}{21\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7}-\frac {64\,a^3\,\left (1598\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-2289{}\mathrm {i}\right )}{63\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6}+\frac {32\,a^3\,\left (2041\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-3339{}\mathrm {i}\right )}{63\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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