Optimal. Leaf size=120 \[ \frac {5 a^4 \sin (c+d x)}{21 d}-\frac {10 a^4 \sin ^3(c+d x)}{63 d}+\frac {a^4 \sin ^5(c+d x)}{21 d}-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^3}{9 d}-\frac {2 i \cos ^7(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{21 d} \]
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Rubi [A]
time = 0.07, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3577, 2713}
\begin {gather*} \frac {a^4 \sin ^5(c+d x)}{21 d}-\frac {10 a^4 \sin ^3(c+d x)}{63 d}+\frac {5 a^4 \sin (c+d x)}{21 d}-\frac {2 i \cos ^7(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{21 d}-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^3}{9 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2713
Rule 3577
Rubi steps
\begin {align*} \int \cos ^9(c+d x) (a+i a \tan (c+d x))^4 \, dx &=-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^3}{9 d}+\frac {1}{3} a^2 \int \cos ^7(c+d x) (a+i a \tan (c+d x))^2 \, dx\\ &=-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^3}{9 d}-\frac {2 i \cos ^7(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{21 d}+\frac {1}{21} \left (5 a^4\right ) \int \cos ^5(c+d x) \, dx\\ &=-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^3}{9 d}-\frac {2 i \cos ^7(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{21 d}-\frac {\left (5 a^4\right ) \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{21 d}\\ &=\frac {5 a^4 \sin (c+d x)}{21 d}-\frac {10 a^4 \sin ^3(c+d x)}{63 d}+\frac {a^4 \sin ^5(c+d x)}{21 d}-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^3}{9 d}-\frac {2 i \cos ^7(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{21 d}\\ \end {align*}
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Mathematica [A]
time = 0.58, size = 111, normalized size = 0.92 \begin {gather*} \frac {a^4 (-168 i \cos (c+d x)-180 i \cos (3 (c+d x))+28 i \cos (5 (c+d x))-42 \sin (c+d x)-135 \sin (3 (c+d x))+35 \sin (5 (c+d x))) (\cos (4 (c+2 d x))+i \sin (4 (c+2 d x)))}{1008 d (\cos (d x)+i \sin (d x))^4} \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. 232 vs. \(2 (106 ) = 212\).
time = 0.25, size = 233, normalized size = 1.94
method | result | size |
risch | \(-\frac {i a^{4} {\mathrm e}^{9 i \left (d x +c \right )}}{288 d}-\frac {5 i a^{4} {\mathrm e}^{7 i \left (d x +c \right )}}{224 d}-\frac {i a^{4} {\mathrm e}^{5 i \left (d x +c \right )}}{16 d}-\frac {5 i a^{4} {\mathrm e}^{3 i \left (d x +c \right )}}{48 d}-\frac {i a^{4} \cos \left (d x +c \right )}{8 d}+\frac {3 a^{4} \sin \left (d x +c \right )}{16 d}\) | \(103\) |
derivativedivides | \(\frac {a^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{9}-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{21}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{105}\right )-4 i a^{4} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )-6 a^{4} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{8}\left (d x +c \right )\right )}{9}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{63}\right )-\frac {4 i a^{4} \left (\cos ^{9}\left (d x +c \right )\right )}{9}+\frac {a^{4} \left (\frac {128}{35}+\cos ^{8}\left (d x +c \right )+\frac {8 \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\cos ^{2}\left (d x +c \right )\right )}{35}\right ) \sin \left (d x +c \right )}{9}}{d}\) | \(233\) |
default | \(\frac {a^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{9}-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{21}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{105}\right )-4 i a^{4} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )-6 a^{4} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{8}\left (d x +c \right )\right )}{9}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{63}\right )-\frac {4 i a^{4} \left (\cos ^{9}\left (d x +c \right )\right )}{9}+\frac {a^{4} \left (\frac {128}{35}+\cos ^{8}\left (d x +c \right )+\frac {8 \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\cos ^{2}\left (d x +c \right )\right )}{35}\right ) \sin \left (d x +c \right )}{9}}{d}\) | \(233\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 181, normalized size = 1.51 \begin {gather*} -\frac {140 i \, a^{4} \cos \left (d x + c\right )^{9} + 20 i \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{4} - {\left (35 \, \sin \left (d x + c\right )^{9} - 90 \, \sin \left (d x + c\right )^{7} + 63 \, \sin \left (d x + c\right )^{5}\right )} a^{4} - 6 \, {\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^{4} - {\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^{4}}{315 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 90, normalized size = 0.75 \begin {gather*} \frac {{\left (-7 i \, a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} - 45 i \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} - 126 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 210 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 315 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 63 i \, a^{4}\right )} e^{\left (-i \, d x - i \, c\right )}}{2016 \, 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. 228 vs. \(2 (107) = 214\).
time = 0.42, size = 228, normalized size = 1.90 \begin {gather*} \begin {cases} \frac {\left (- 176160768 i a^{4} d^{5} e^{10 i c} e^{9 i d x} - 1132462080 i a^{4} d^{5} e^{8 i c} e^{7 i d x} - 3170893824 i a^{4} d^{5} e^{6 i c} e^{5 i d x} - 5284823040 i a^{4} d^{5} e^{4 i c} e^{3 i d x} - 7927234560 i a^{4} d^{5} e^{2 i c} e^{i d x} + 1585446912 i a^{4} d^{5} e^{- i d x}\right ) e^{- i c}}{50734301184 d^{6}} & \text {for}\: d^{6} e^{i c} \neq 0 \\\frac {x \left (a^{4} e^{10 i c} + 5 a^{4} e^{8 i c} + 10 a^{4} e^{6 i c} + 10 a^{4} e^{4 i c} + 5 a^{4} e^{2 i c} + a^{4}\right ) e^{- i c}}{32} & \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. 1409 vs. \(2 (102) = 204\).
time = 0.92, size = 1409, normalized size = 11.74 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.69, size = 145, normalized size = 1.21 \begin {gather*} \frac {2\,a^4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {89\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}-\frac {55\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{4}+\frac {55\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{4}-\frac {355\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{16}+\frac {35\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{16}-\frac {\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )\,21{}\mathrm {i}}{2}+\frac {\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )\,21{}\mathrm {i}}{2}-\frac {\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )\,87{}\mathrm {i}}{4}+\frac {\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )\,7{}\mathrm {i}}{4}\right )}{63\,d\,\left (\cos \left (4\,c+4\,d\,x\right )-\sin \left (4\,c+4\,d\,x\right )\,1{}\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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