Optimal. Leaf size=152 \[ \frac {a^8 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}-\frac {2 i a^2 \cos ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^3}{3 d}+\frac {2 i \cos (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{d} \]
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Rubi [A]
time = 0.12, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3577, 3855}
\begin {gather*} \frac {a^8 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {2 i \cos (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{d}+\frac {2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {2 i a^2 \cos ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^3}{3 d}-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3577
Rule 3855
Rubi steps
\begin {align*} \int \cos ^7(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}-a^2 \int \cos ^5(c+d x) (a+i a \tan (c+d x))^6 \, dx\\ &=\frac {2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}+a^4 \int \cos ^3(c+d x) (a+i a \tan (c+d x))^4 \, dx\\ &=-\frac {2 i a^5 \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}+\frac {2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}-a^6 \int \cos (c+d x) (a+i a \tan (c+d x))^2 \, dx\\ &=-\frac {2 i a^5 \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}+\frac {2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}+\frac {2 i \cos (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{d}+a^8 \int \sec (c+d x) \, dx\\ &=\frac {a^8 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {2 i a^5 \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}+\frac {2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}+\frac {2 i \cos (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{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. \(305\) vs. \(2(152)=304\).
time = 2.05, size = 305, normalized size = 2.01 \begin {gather*} \frac {a^8 \left (-70 i \cos \left (\frac {1}{2} (c+d x)\right )+42 i \cos \left (\frac {3}{2} (c+d x)\right )+210 i \cos \left (\frac {5}{2} (c+d x)\right )-30 i \cos \left (\frac {7}{2} (c+d x)\right )-105 \cos \left (\frac {7}{2} (c+d x)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+105 \cos \left (\frac {7}{2} (c+d x)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-70 \sin \left (\frac {1}{2} (c+d x)\right )-42 \sin \left (\frac {3}{2} (c+d x)\right )+210 \sin \left (\frac {5}{2} (c+d x)\right )+30 \sin \left (\frac {7}{2} (c+d x)\right )+105 i \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin \left (\frac {7}{2} (c+d x)\right )-105 i \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin \left (\frac {7}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (7 c+23 d x)\right )+i \sin \left (\frac {1}{2} (7 c+23 d x)\right )\right )}{105 d (\cos (d x)+i \sin (d x))^8} \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. 368 vs. \(2 (138 ) = 276\).
time = 0.23, size = 369, normalized size = 2.43
method | result | size |
risch | \(-\frac {2 i a^{8} {\mathrm e}^{7 i \left (d x +c \right )}}{7 d}+\frac {2 i a^{8} {\mathrm e}^{5 i \left (d x +c \right )}}{5 d}-\frac {2 i a^{8} {\mathrm e}^{3 i \left (d x +c \right )}}{3 d}+\frac {2 i a^{8} {\mathrm e}^{i \left (d x +c \right )}}{d}+\frac {a^{8} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {a^{8} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}\) | \(115\) |
derivativedivides | \(\frac {a^{8} \left (-\frac {\left (\sin ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )-\frac {8 i a^{8} \left (\cos ^{7}\left (d x +c \right )\right )}{7}-4 a^{8} \left (\sin ^{7}\left (d x +c \right )\right )-56 i a^{8} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+70 a^{8} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{7}-\frac {3 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {\left (\cos ^{2}\left (d x +c \right )+2\right ) \sin \left (d x +c \right )}{35}\right )+\frac {8 i a^{8} \left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )}{7}-28 a^{8} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )+56 i a^{8} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{7}-\frac {4 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{35}-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right )}{105}\right )+\frac {a^{8} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d}\) | \(369\) |
default | \(\frac {a^{8} \left (-\frac {\left (\sin ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )-\frac {8 i a^{8} \left (\cos ^{7}\left (d x +c \right )\right )}{7}-4 a^{8} \left (\sin ^{7}\left (d x +c \right )\right )-56 i a^{8} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+70 a^{8} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{7}-\frac {3 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {\left (\cos ^{2}\left (d x +c \right )+2\right ) \sin \left (d x +c \right )}{35}\right )+\frac {8 i a^{8} \left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )}{7}-28 a^{8} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )+56 i a^{8} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{7}-\frac {4 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{35}-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right )}{105}\right )+\frac {a^{8} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d}\) | \(369\) |
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. 309 vs. \(2 (130) = 260\).
time = 0.29, size = 309, normalized size = 2.03 \begin {gather*} -\frac {240 i \, a^{8} \cos \left (d x + c\right )^{7} + 840 \, a^{8} \sin \left (d x + c\right )^{7} + 112 i \, {\left (15 \, \cos \left (d x + c\right )^{7} - 42 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3}\right )} a^{8} + 336 i \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{8} + 48 i \, {\left (5 \, \cos \left (d x + c\right )^{7} - 21 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3} - 35 \, \cos \left (d x + c\right )\right )} a^{8} + {\left (30 \, \sin \left (d x + c\right )^{7} + 42 \, \sin \left (d x + c\right )^{5} + 70 \, \sin \left (d x + c\right )^{3} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 210 \, \sin \left (d x + c\right )\right )} a^{8} + 56 \, {\left (15 \, \sin \left (d x + c\right )^{7} - 42 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3}\right )} a^{8} + 420 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 7 \, \sin \left (d x + c\right )^{5}\right )} a^{8} + 6 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} a^{8}}{210 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 96, normalized size = 0.63 \begin {gather*} \frac {-30 i \, a^{8} e^{\left (7 i \, d x + 7 i \, c\right )} + 42 i \, a^{8} e^{\left (5 i \, d x + 5 i \, c\right )} - 70 i \, a^{8} e^{\left (3 i \, d x + 3 i \, c\right )} + 210 i \, a^{8} e^{\left (i \, d x + i \, c\right )} + 105 \, a^{8} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 105 \, a^{8} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right )}{105 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.57, size = 187, normalized size = 1.23 \begin {gather*} \frac {a^{8} \left (- \log {\left (e^{i d x} - i e^{- i c} \right )} + \log {\left (e^{i d x} + i e^{- i c} \right )}\right )}{d} + \begin {cases} \frac {- 30 i a^{8} d^{3} e^{7 i c} e^{7 i d x} + 42 i a^{8} d^{3} e^{5 i c} e^{5 i d x} - 70 i a^{8} d^{3} e^{3 i c} e^{3 i d x} + 210 i a^{8} d^{3} e^{i c} e^{i d x}}{105 d^{4}} & \text {for}\: d^{4} \neq 0 \\x \left (2 a^{8} e^{7 i c} - 2 a^{8} e^{5 i c} + 2 a^{8} e^{3 i c} - 2 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. 2863 vs. \(2 (130) = 260\).
time = 1.55, size = 2863, normalized size = 18.84 \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 = 6.69, size = 207, normalized size = 1.36 \begin {gather*} \frac {2\,a^8\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,16{}\mathrm {i}-\frac {80\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}-\frac {a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,224{}\mathrm {i}}{3}+\frac {224\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,304{}\mathrm {i}}{15}-\frac {304\,a^8}{105}}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,7{}\mathrm {i}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,35{}\mathrm {i}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,21{}\mathrm {i}+7\,\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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