Optimal. Leaf size=129 \[ \frac {35 a \tanh ^{-1}(\sin (c+d x))}{128 d}+\frac {\tan ^7(c+d x) (7 a \sec (c+d x)+8 a)}{56 d}-\frac {\tan ^5(c+d x) (35 a \sec (c+d x)+48 a)}{240 d}+\frac {\tan ^3(c+d x) (35 a \sec (c+d x)+64 a)}{192 d}-\frac {\tan (c+d x) (35 a \sec (c+d x)+128 a)}{128 d}+a x \]
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Rubi [A] time = 0.13, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3881, 3770} \[ \frac {35 a \tanh ^{-1}(\sin (c+d x))}{128 d}+\frac {\tan ^7(c+d x) (7 a \sec (c+d x)+8 a)}{56 d}-\frac {\tan ^5(c+d x) (35 a \sec (c+d x)+48 a)}{240 d}+\frac {\tan ^3(c+d x) (35 a \sec (c+d x)+64 a)}{192 d}-\frac {\tan (c+d x) (35 a \sec (c+d x)+128 a)}{128 d}+a x \]
Antiderivative was successfully verified.
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Rule 3770
Rule 3881
Rubi steps
\begin {align*} \int (a+a \sec (c+d x)) \tan ^8(c+d x) \, dx &=\frac {(8 a+7 a \sec (c+d x)) \tan ^7(c+d x)}{56 d}-\frac {1}{8} \int (8 a+7 a \sec (c+d x)) \tan ^6(c+d x) \, dx\\ &=-\frac {(48 a+35 a \sec (c+d x)) \tan ^5(c+d x)}{240 d}+\frac {(8 a+7 a \sec (c+d x)) \tan ^7(c+d x)}{56 d}+\frac {1}{48} \int (48 a+35 a \sec (c+d x)) \tan ^4(c+d x) \, dx\\ &=\frac {(64 a+35 a \sec (c+d x)) \tan ^3(c+d x)}{192 d}-\frac {(48 a+35 a \sec (c+d x)) \tan ^5(c+d x)}{240 d}+\frac {(8 a+7 a \sec (c+d x)) \tan ^7(c+d x)}{56 d}-\frac {1}{192} \int (192 a+105 a \sec (c+d x)) \tan ^2(c+d x) \, dx\\ &=-\frac {(128 a+35 a \sec (c+d x)) \tan (c+d x)}{128 d}+\frac {(64 a+35 a \sec (c+d x)) \tan ^3(c+d x)}{192 d}-\frac {(48 a+35 a \sec (c+d x)) \tan ^5(c+d x)}{240 d}+\frac {(8 a+7 a \sec (c+d x)) \tan ^7(c+d x)}{56 d}+\frac {1}{384} \int (384 a+105 a \sec (c+d x)) \, dx\\ &=a x-\frac {(128 a+35 a \sec (c+d x)) \tan (c+d x)}{128 d}+\frac {(64 a+35 a \sec (c+d x)) \tan ^3(c+d x)}{192 d}-\frac {(48 a+35 a \sec (c+d x)) \tan ^5(c+d x)}{240 d}+\frac {(8 a+7 a \sec (c+d x)) \tan ^7(c+d x)}{56 d}+\frac {1}{128} (35 a) \int \sec (c+d x) \, dx\\ &=a x+\frac {35 a \tanh ^{-1}(\sin (c+d x))}{128 d}-\frac {(128 a+35 a \sec (c+d x)) \tan (c+d x)}{128 d}+\frac {(64 a+35 a \sec (c+d x)) \tan ^3(c+d x)}{192 d}-\frac {(48 a+35 a \sec (c+d x)) \tan ^5(c+d x)}{240 d}+\frac {(8 a+7 a \sec (c+d x)) \tan ^7(c+d x)}{56 d}\\ \end {align*}
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Mathematica [A] time = 1.89, size = 115, normalized size = 0.89 \[ \frac {a \left (13440 \tan ^{-1}(\tan (c+d x))+3675 \tanh ^{-1}(\sin (c+d x))-\frac {1}{32} (223232 \cos (c+d x)+75915 \cos (2 (c+d x))+147968 \cos (3 (c+d x))+12950 \cos (4 (c+d x))+47616 \cos (5 (c+d x))+9765 \cos (6 (c+d x))+11264 \cos (7 (c+d x))+18970) \tan (c+d x) \sec ^7(c+d x)\right )}{13440 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 156, normalized size = 1.21 \[ \frac {26880 \, a d x \cos \left (d x + c\right )^{8} + 3675 \, a \cos \left (d x + c\right )^{8} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3675 \, a \cos \left (d x + c\right )^{8} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (22528 \, a \cos \left (d x + c\right )^{7} + 9765 \, a \cos \left (d x + c\right )^{6} - 15616 \, a \cos \left (d x + c\right )^{5} - 11410 \, a \cos \left (d x + c\right )^{4} + 8448 \, a \cos \left (d x + c\right )^{3} + 7000 \, a \cos \left (d x + c\right )^{2} - 1920 \, a \cos \left (d x + c\right ) - 1680 \, a\right )} \sin \left (d x + c\right )}{26880 \, d \cos \left (d x + c\right )^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 12.22, size = 174, normalized size = 1.35 \[ \frac {13440 \, {\left (d x + c\right )} a + 3675 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3675 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (9765 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} - 83825 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 321013 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 724649 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1078359 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 508683 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 140175 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 17115 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{8}}}{13440 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.54, size = 227, normalized size = 1.76 \[ \frac {a \left (\tan ^{7}\left (d x +c \right )\right )}{7 d}-\frac {a \left (\tan ^{5}\left (d x +c \right )\right )}{5 d}+\frac {a \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {a \tan \left (d x +c \right )}{d}+a x +\frac {c a}{d}+\frac {a \left (\sin ^{9}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{8}}-\frac {a \left (\sin ^{9}\left (d x +c \right )\right )}{48 d \cos \left (d x +c \right )^{6}}+\frac {a \left (\sin ^{9}\left (d x +c \right )\right )}{64 d \cos \left (d x +c \right )^{4}}-\frac {5 a \left (\sin ^{9}\left (d x +c \right )\right )}{128 d \cos \left (d x +c \right )^{2}}-\frac {5 a \left (\sin ^{7}\left (d x +c \right )\right )}{128 d}-\frac {7 a \left (\sin ^{5}\left (d x +c \right )\right )}{128 d}-\frac {35 a \left (\sin ^{3}\left (d x +c \right )\right )}{384 d}-\frac {35 a \sin \left (d x +c \right )}{128 d}+\frac {35 a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 164, normalized size = 1.27 \[ \frac {256 \, {\left (15 \, \tan \left (d x + c\right )^{7} - 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 105 \, d x + 105 \, c - 105 \, \tan \left (d x + c\right )\right )} a + 35 \, a {\left (\frac {2 \, {\left (279 \, \sin \left (d x + c\right )^{7} - 511 \, \sin \left (d x + c\right )^{5} + 385 \, \sin \left (d x + c\right )^{3} - 105 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} + 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{26880 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.47, size = 242, normalized size = 1.88 \[ a\,x-\frac {-\frac {93\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{64}+\frac {2395\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{192}-\frac {45859\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{960}+\frac {724649\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{6720}-\frac {359453\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2240}+\frac {24223\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{320}-\frac {1335\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{64}+\frac {163\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {35\,a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{64\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a \left (\int \tan ^{8}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \tan ^{8}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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