Optimal. Leaf size=141 \[ -\frac {i a^5 \cos ^5(c+d x)}{105 d}+\frac {a^5 \sin (c+d x)}{21 d}-\frac {2 a^5 \sin ^3(c+d x)}{63 d}+\frac {a^5 \sin ^5(c+d x)}{105 d}-\frac {2 i a^3 \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{63 d}-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^4}{9 d} \]
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Rubi [A]
time = 0.09, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps
used = 5, 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 {a^5 \sin ^5(c+d x)}{105 d}-\frac {2 a^5 \sin ^3(c+d x)}{63 d}+\frac {a^5 \sin (c+d x)}{21 d}-\frac {i a^5 \cos ^5(c+d x)}{105 d}-\frac {2 i a^3 \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{63 d}-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^4}{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))^5 \, dx &=-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^4}{9 d}+\frac {1}{9} a^2 \int \cos ^7(c+d x) (a+i a \tan (c+d x))^3 \, dx\\ &=-\frac {2 i a^3 \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{63 d}-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^4}{9 d}+\frac {1}{21} a^4 \int \cos ^5(c+d x) (a+i a \tan (c+d x)) \, dx\\ &=-\frac {i a^5 \cos ^5(c+d x)}{105 d}-\frac {2 i a^3 \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{63 d}-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^4}{9 d}+\frac {1}{21} a^5 \int \cos ^5(c+d x) \, dx\\ &=-\frac {i a^5 \cos ^5(c+d x)}{105 d}-\frac {2 i a^3 \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{63 d}-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^4}{9 d}-\frac {a^5 \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{21 d}\\ &=-\frac {i a^5 \cos ^5(c+d x)}{105 d}+\frac {a^5 \sin (c+d x)}{21 d}-\frac {2 a^5 \sin ^3(c+d x)}{63 d}+\frac {a^5 \sin ^5(c+d x)}{105 d}-\frac {2 i a^3 \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{63 d}-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^4}{9 d}\\ \end {align*}
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Mathematica [A]
time = 0.62, size = 94, normalized size = 0.67 \begin {gather*} \frac {a^5 (189+300 \cos (2 (c+d x))+175 \cos (4 (c+d x))-120 i \sin (2 (c+d x))-140 i \sin (4 (c+d x))) (-i \cos (5 (c+2 d x))+\sin (5 (c+2 d x)))}{2520 d (\cos (d x)+i \sin (d x))^5} \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. 286 vs. \(2 (124 ) = 248\).
time = 0.25, size = 287, normalized size = 2.04
method | result | size |
risch | \(-\frac {i a^{5} {\mathrm e}^{9 i \left (d x +c \right )}}{144 d}-\frac {i a^{5} {\mathrm e}^{7 i \left (d x +c \right )}}{28 d}-\frac {3 i a^{5} {\mathrm e}^{5 i \left (d x +c \right )}}{40 d}-\frac {i a^{5} {\mathrm e}^{3 i \left (d x +c \right )}}{12 d}-\frac {i a^{5} {\mathrm e}^{i \left (d x +c \right )}}{16 d}\) | \(92\) |
derivativedivides | \(\frac {i a^{5} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+5 a^{5} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{9}-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{21}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{105}\right )-10 i a^{5} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )-10 a^{5} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{8}\left (d x +c \right )\right )}{9}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{63}\right )-\frac {5 i a^{5} \left (\cos ^{9}\left (d x +c \right )\right )}{9}+\frac {a^{5} \left (\frac {128}{35}+\cos ^{8}\left (d x +c \right )+\frac {8 \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\cos ^{2}\left (d x +c \right )\right )}{35}\right ) \sin \left (d x +c \right )}{9}}{d}\) | \(287\) |
default | \(\frac {i a^{5} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+5 a^{5} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{9}-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{21}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{105}\right )-10 i a^{5} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )-10 a^{5} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{8}\left (d x +c \right )\right )}{9}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{63}\right )-\frac {5 i a^{5} \left (\cos ^{9}\left (d x +c \right )\right )}{9}+\frac {a^{5} \left (\frac {128}{35}+\cos ^{8}\left (d x +c \right )+\frac {8 \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\cos ^{2}\left (d x +c \right )\right )}{35}\right ) \sin \left (d x +c \right )}{9}}{d}\) | \(287\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 217, normalized size = 1.54 \begin {gather*} -\frac {175 i \, a^{5} \cos \left (d x + c\right )^{9} + i \, {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} a^{5} + 50 i \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{5} - 5 \, {\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^{5} - 10 \, {\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^{5} - {\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^{5}}{315 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 76, normalized size = 0.54 \begin {gather*} \frac {-35 i \, a^{5} e^{\left (9 i \, d x + 9 i \, c\right )} - 180 i \, a^{5} e^{\left (7 i \, d x + 7 i \, c\right )} - 378 i \, a^{5} e^{\left (5 i \, d x + 5 i \, c\right )} - 420 i \, a^{5} e^{\left (3 i \, d x + 3 i \, c\right )} - 315 i \, a^{5} e^{\left (i \, d x + i \, c\right )}}{5040 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.44, size = 192, normalized size = 1.36 \begin {gather*} \begin {cases} \frac {- 215040 i a^{5} d^{4} e^{9 i c} e^{9 i d x} - 1105920 i a^{5} d^{4} e^{7 i c} e^{7 i d x} - 2322432 i a^{5} d^{4} e^{5 i c} e^{5 i d x} - 2580480 i a^{5} d^{4} e^{3 i c} e^{3 i d x} - 1935360 i a^{5} d^{4} e^{i c} e^{i d x}}{30965760 d^{5}} & \text {for}\: d^{5} \neq 0 \\x \left (\frac {a^{5} e^{9 i c}}{16} + \frac {a^{5} e^{7 i c}}{4} + \frac {3 a^{5} e^{5 i c}}{8} + \frac {a^{5} e^{3 i c}}{4} + \frac {a^{5} e^{i c}}{16}\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. 1725 vs. \(2 (119) = 238\).
time = 1.01, size = 1725, normalized size = 12.23 \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 = 3.94, size = 79, normalized size = 0.56 \begin {gather*} -\frac {a^5\,\left (\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{16}+\frac {{\mathrm {e}}^{c\,3{}\mathrm {i}+d\,x\,3{}\mathrm {i}}\,1{}\mathrm {i}}{12}+\frac {{\mathrm {e}}^{c\,5{}\mathrm {i}+d\,x\,5{}\mathrm {i}}\,3{}\mathrm {i}}{40}+\frac {{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,1{}\mathrm {i}}{28}+\frac {{\mathrm {e}}^{c\,9{}\mathrm {i}+d\,x\,9{}\mathrm {i}}\,1{}\mathrm {i}}{144}\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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