Optimal. Leaf size=173 \[ -\frac {63 a^8 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {63 i a^8 \sec (c+d x)}{2 d}+\frac {6 i a^3 \cos ^3(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^7}{5 d}-\frac {42 i a^2 \cos (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^3}{5 d}-\frac {21 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{2 d} \]
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Rubi [A]
time = 0.13, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3577, 3579,
3567, 3855} \begin {gather*} -\frac {63 i a^8 \sec (c+d x)}{2 d}-\frac {63 a^8 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {21 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{2 d}+\frac {6 i a^3 \cos ^3(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {42 i a^2 \cos (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^3}{5 d}-\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^7}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3567
Rule 3577
Rule 3579
Rule 3855
Rubi steps
\begin {align*} \int \cos ^5(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^7}{5 d}-\frac {1}{5} \left (9 a^2\right ) \int \cos ^3(c+d x) (a+i a \tan (c+d x))^6 \, dx\\ &=\frac {6 i a^3 \cos ^3(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^7}{5 d}+\frac {1}{5} \left (21 a^4\right ) \int \cos (c+d x) (a+i a \tan (c+d x))^4 \, dx\\ &=-\frac {42 i a^5 \cos (c+d x) (a+i a \tan (c+d x))^3}{5 d}+\frac {6 i a^3 \cos ^3(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^7}{5 d}-\left (21 a^6\right ) \int \sec (c+d x) (a+i a \tan (c+d x))^2 \, dx\\ &=-\frac {42 i a^5 \cos (c+d x) (a+i a \tan (c+d x))^3}{5 d}+\frac {6 i a^3 \cos ^3(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^7}{5 d}-\frac {21 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{2 d}-\frac {1}{2} \left (63 a^7\right ) \int \sec (c+d x) (a+i a \tan (c+d x)) \, dx\\ &=-\frac {63 i a^8 \sec (c+d x)}{2 d}-\frac {42 i a^5 \cos (c+d x) (a+i a \tan (c+d x))^3}{5 d}+\frac {6 i a^3 \cos ^3(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^7}{5 d}-\frac {21 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{2 d}-\frac {1}{2} \left (63 a^8\right ) \int \sec (c+d x) \, dx\\ &=-\frac {63 a^8 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {63 i a^8 \sec (c+d x)}{2 d}-\frac {42 i a^5 \cos (c+d x) (a+i a \tan (c+d x))^3}{5 d}+\frac {6 i a^3 \cos ^3(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^7}{5 d}-\frac {21 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{2 d}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(1162\) vs. \(2(173)=346\).
time = 6.51, size = 1162, normalized size = 6.72 \begin {gather*} \frac {63 \cos (8 c) \cos ^8(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) (a+i a \tan (c+d x))^8}{2 d (\cos (d x)+i \sin (d x))^8}-\frac {63 \cos (8 c) \cos ^8(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) (a+i a \tan (c+d x))^8}{2 d (\cos (d x)+i \sin (d x))^8}+\frac {\cos (5 d x) \cos ^8(c+d x) \left (-\frac {8}{5} i \cos (3 c)-\frac {8}{5} \sin (3 c)\right ) (a+i a \tan (c+d x))^8}{d (\cos (d x)+i \sin (d x))^8}+\frac {\cos (3 d x) \cos ^8(c+d x) (8 i \cos (5 c)+8 \sin (5 c)) (a+i a \tan (c+d x))^8}{d (\cos (d x)+i \sin (d x))^8}+\frac {\cos (d x) \cos ^8(c+d x) (-48 i \cos (7 c)-48 \sin (7 c)) (a+i a \tan (c+d x))^8}{d (\cos (d x)+i \sin (d x))^8}+\frac {\cos ^8(c+d x) \sec (c) (-8 i \cos (8 c)-8 \sin (8 c)) (a+i a \tan (c+d x))^8}{d (\cos (d x)+i \sin (d x))^8}-\frac {63 i \cos ^8(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sin (8 c) (a+i a \tan (c+d x))^8}{2 d (\cos (d x)+i \sin (d x))^8}+\frac {63 i \cos ^8(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sin (8 c) (a+i a \tan (c+d x))^8}{2 d (\cos (d x)+i \sin (d x))^8}+\frac {\cos ^8(c+d x) (48 \cos (7 c)-48 i \sin (7 c)) \sin (d x) (a+i a \tan (c+d x))^8}{d (\cos (d x)+i \sin (d x))^8}+\frac {\cos ^8(c+d x) (-8 \cos (5 c)+8 i \sin (5 c)) \sin (3 d x) (a+i a \tan (c+d x))^8}{d (\cos (d x)+i \sin (d x))^8}+\frac {\cos ^8(c+d x) \left (\frac {8}{5} \cos (3 c)-\frac {8}{5} i \sin (3 c)\right ) \sin (5 d x) (a+i a \tan (c+d x))^8}{d (\cos (d x)+i \sin (d x))^8}+\frac {\cos ^8(c+d x) \left (\frac {1}{4} \cos (8 c)-\frac {1}{4} i \sin (8 c)\right ) (a+i a \tan (c+d x))^8}{d (\cos (d x)+i \sin (d x))^8 \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}-\frac {i \cos ^8(c+d x) (8 \cos (8 c)-8 i \sin (8 c)) \sin \left (\frac {d x}{2}\right ) (a+i a \tan (c+d x))^8}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) (\cos (d x)+i \sin (d x))^8 \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}+\frac {\cos ^8(c+d x) \left (-\frac {1}{4} \cos (8 c)+\frac {1}{4} i \sin (8 c)\right ) (a+i a \tan (c+d x))^8}{d (\cos (d x)+i \sin (d x))^8 \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}+\frac {i \cos ^8(c+d x) (8 \cos (8 c)-8 i \sin (8 c)) \sin \left (\frac {d x}{2}\right ) (a+i a \tan (c+d x))^8}{d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) (\cos (d x)+i \sin (d x))^8 \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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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. 356 vs. \(2 (152 ) = 304\).
time = 0.24, size = 357, normalized size = 2.06
method | result | size |
risch | \(-\frac {8 i a^{8} {\mathrm e}^{5 i \left (d x +c \right )}}{5 d}+\frac {8 i a^{8} {\mathrm e}^{3 i \left (d x +c \right )}}{d}-\frac {48 i a^{8} {\mathrm e}^{i \left (d x +c \right )}}{d}-\frac {i a^{8} \left (17 \,{\mathrm e}^{3 i \left (d x +c \right )}+15 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {63 a^{8} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}-\frac {63 a^{8} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}\) | \(143\) |
derivativedivides | \(\frac {a^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{7}\left (d x +c \right )\right )}{2}+\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{10}+\frac {7 \left (\sin ^{3}\left (d x +c \right )\right )}{6}+\frac {7 \sin \left (d x +c \right )}{2}-\frac {7 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-\frac {56 i a^{8} \left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )}{5}-28 a^{8} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )-\frac {8 i a^{8} \left (\cos ^{5}\left (d x +c \right )\right )}{5}+14 a^{8} \left (\sin ^{5}\left (d x +c \right )\right )-8 i a^{8} \left (\frac {\sin ^{8}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )\right )-28 a^{8} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {\left (\cos ^{2}\left (d x +c \right )+2\right ) \sin \left (d x +c \right )}{15}\right )-56 i a^{8} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )+\frac {a^{8} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}}{d}\) | \(357\) |
default | \(\frac {a^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{7}\left (d x +c \right )\right )}{2}+\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{10}+\frac {7 \left (\sin ^{3}\left (d x +c \right )\right )}{6}+\frac {7 \sin \left (d x +c \right )}{2}-\frac {7 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-\frac {56 i a^{8} \left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )}{5}-28 a^{8} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )-\frac {8 i a^{8} \left (\cos ^{5}\left (d x +c \right )\right )}{5}+14 a^{8} \left (\sin ^{5}\left (d x +c \right )\right )-8 i a^{8} \left (\frac {\sin ^{8}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )\right )-28 a^{8} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {\left (\cos ^{2}\left (d x +c \right )+2\right ) \sin \left (d x +c \right )}{15}\right )-56 i a^{8} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )+\frac {a^{8} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}}{d}\) | \(357\) |
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. 326 vs. \(2 (143) = 286\).
time = 0.29, size = 326, normalized size = 1.88 \begin {gather*} -\frac {96 i \, a^{8} \cos \left (d x + c\right )^{5} - 840 \, a^{8} \sin \left (d x + c\right )^{5} + 224 i \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{8} + 224 i \, {\left (3 \, \cos \left (d x + c\right )^{5} - 10 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} a^{8} + 96 i \, {\left (\cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3} + \frac {5}{\cos \left (d x + c\right )} + 15 \, \cos \left (d x + c\right )\right )} a^{8} - {\left (12 \, \sin \left (d x + c\right )^{5} + 40 \, \sin \left (d x + c\right )^{3} - \frac {30 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 180 \, \sin \left (d x + c\right )\right )} a^{8} - 56 \, {\left (6 \, \sin \left (d x + c\right )^{5} + 10 \, \sin \left (d x + c\right )^{3} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 30 \, \sin \left (d x + c\right )\right )} a^{8} - 112 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 5 \, \sin \left (d x + c\right )^{3}\right )} a^{8} - 4 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a^{8}}{60 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 190, normalized size = 1.10 \begin {gather*} \frac {-16 i \, a^{8} e^{\left (9 i \, d x + 9 i \, c\right )} + 48 i \, a^{8} e^{\left (7 i \, d x + 7 i \, c\right )} - 336 i \, a^{8} e^{\left (5 i \, d x + 5 i \, c\right )} - 1050 i \, a^{8} e^{\left (3 i \, d x + 3 i \, c\right )} - 630 i \, a^{8} e^{\left (i \, d x + i \, c\right )} - 315 \, {\left (a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{8}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) + 315 \, {\left (a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{8}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right )}{10 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.51, size = 235, normalized size = 1.36 \begin {gather*} \frac {63 a^{8} \left (\frac {\log {\left (e^{i d x} - i e^{- i c} \right )}}{2} - \frac {\log {\left (e^{i d x} + i e^{- i c} \right )}}{2}\right )}{d} + \frac {- 17 i a^{8} e^{3 i c} e^{3 i d x} - 15 i a^{8} e^{i c} e^{i d x}}{d e^{4 i c} e^{4 i d x} + 2 d e^{2 i c} e^{2 i d x} + d} + \begin {cases} \frac {- 8 i a^{8} d^{2} e^{5 i c} e^{5 i d x} + 40 i a^{8} d^{2} e^{3 i c} e^{3 i d x} - 240 i a^{8} d^{2} e^{i c} e^{i d x}}{5 d^{3}} & \text {for}\: d^{3} \neq 0 \\x \left (8 a^{8} e^{5 i c} - 24 a^{8} e^{3 i c} + 48 a^{8} e^{i c}\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. 2849 vs. \(2 (143) = 286\).
time = 1.45, size = 2849, normalized size = 16.47 \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 = 7.39, size = 281, normalized size = 1.62 \begin {gather*} -\frac {63\,a^8\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {65\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,309{}\mathrm {i}-761\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,1109{}\mathrm {i}+\frac {7351\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}+a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,1223{}\mathrm {i}-\frac {4407\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}-a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,431{}\mathrm {i}+\frac {496\,a^8}{5}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,5{}\mathrm {i}-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,20{}\mathrm {i}+26\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,26{}\mathrm {i}-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,12{}\mathrm {i}+5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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