Optimal. Leaf size=66 \[ -\frac {i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{63 d}-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^8}{9 d} \]
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Rubi [A]
time = 0.06, 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 ^9(c+d x) (a+i a \tan (c+d x))^8}{9 d}-\frac {i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{63 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3569
Rule 3578
Rubi steps
\begin {align*} \int \cos ^9(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^8}{9 d}+\frac {1}{9} a \int \cos ^7(c+d x) (a+i a \tan (c+d x))^7 \, dx\\ &=-\frac {i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{63 d}-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^8}{9 d}\\ \end {align*}
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Mathematica [A]
time = 0.40, size = 50, normalized size = 0.76 \begin {gather*} \frac {a^8 (8 \cos (c+d x)-i \sin (c+d x)) (-i \cos (8 (c+d x))+\sin (8 (c+d x)))}{63 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. 446 vs. \(2 (58 ) = 116\).
time = 0.24, size = 447, normalized size = 6.77
method | result | size |
risch | \(-\frac {i a^{8} {\mathrm e}^{9 i \left (d x +c \right )}}{18 d}-\frac {i a^{8} {\mathrm e}^{7 i \left (d x +c \right )}}{14 d}\) | \(38\) |
derivativedivides | \(\frac {\frac {a^{8} \left (\sin ^{9}\left (d x +c \right )\right )}{9}-8 i a^{8} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{6}\left (d x +c \right )\right )}{9}-\frac {2 \left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{21}-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{105}-\frac {16 \left (\cos ^{3}\left (d x +c \right )\right )}{315}\right )-28 a^{8} \left (-\frac {\left (\cos ^{4}\left (d x +c \right )\right ) \left (\sin ^{5}\left (d x +c \right )\right )}{9}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{63}-\frac {\sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{21}+\frac {\left (\cos ^{2}\left (d x +c \right )+2\right ) \sin \left (d x +c \right )}{63}\right )+56 i a^{8} \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 )+70 a^{8} \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 )-56 i a^{8} \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 )-28 a^{8} \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 {8 i a^{8} \left (\cos ^{9}\left (d x +c \right )\right )}{9}+\frac {a^{8} \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}\) | \(447\) |
default | \(\frac {\frac {a^{8} \left (\sin ^{9}\left (d x +c \right )\right )}{9}-8 i a^{8} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{6}\left (d x +c \right )\right )}{9}-\frac {2 \left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{21}-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{105}-\frac {16 \left (\cos ^{3}\left (d x +c \right )\right )}{315}\right )-28 a^{8} \left (-\frac {\left (\cos ^{4}\left (d x +c \right )\right ) \left (\sin ^{5}\left (d x +c \right )\right )}{9}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{63}-\frac {\sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{21}+\frac {\left (\cos ^{2}\left (d x +c \right )+2\right ) \sin \left (d x +c \right )}{63}\right )+56 i a^{8} \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 )+70 a^{8} \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 )-56 i a^{8} \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 )-28 a^{8} \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 {8 i a^{8} \left (\cos ^{9}\left (d x +c \right )\right )}{9}+\frac {a^{8} \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}\) | \(447\) |
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. 302 vs. \(2 (54) = 108\).
time = 0.30, size = 302, normalized size = 4.58 \begin {gather*} -\frac {280 i \, a^{8} \cos \left (d x + c\right )^{9} - 35 \, a^{8} \sin \left (d x + c\right )^{9} + 56 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^{8} + 8 i \, {\left (35 \, \cos \left (d x + c\right )^{9} - 135 \, \cos \left (d x + c\right )^{7} + 189 \, \cos \left (d x + c\right )^{5} - 105 \, \cos \left (d x + c\right )^{3}\right )} a^{8} + 280 i \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{8} - 70 \, {\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^{8} - 28 \, {\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^{8} - {\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^{8} - 140 \, {\left (7 \, \sin \left (d x + c\right )^{9} - 9 \, \sin \left (d x + c\right )^{7}\right )} a^{8}}{315 \, 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 {-7 i \, a^{8} e^{\left (9 i \, d x + 9 i \, c\right )} - 9 i \, a^{8} e^{\left (7 i \, d x + 7 i \, c\right )}}{126 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.55, size = 80, normalized size = 1.21 \begin {gather*} \begin {cases} \frac {- 14 i a^{8} d e^{9 i c} e^{9 i d x} - 18 i a^{8} d e^{7 i c} e^{7 i d x}}{252 d^{2}} & \text {for}\: d^{2} \neq 0 \\x \left (\frac {a^{8} e^{9 i c}}{2} + \frac {a^{8} e^{7 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. 2451 vs. \(2 (54) = 108\).
time = 1.61, size = 2451, normalized size = 37.14 \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 = 3.62, size = 37, normalized size = 0.56 \begin {gather*} -\frac {2\,a^8\,\left (\frac {{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,9{}\mathrm {i}}{4}+\frac {{\mathrm {e}}^{c\,9{}\mathrm {i}+d\,x\,9{}\mathrm {i}}\,7{}\mathrm {i}}{4}\right )}{63\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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