Optimal. Leaf size=102 \[ -\frac {5 a \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {\tan ^5(c+d x) (5 a \sec (c+d x)+6 a)}{30 d}-\frac {\tan ^3(c+d x) (5 a \sec (c+d x)+8 a)}{24 d}+\frac {\tan (c+d x) (5 a \sec (c+d x)+16 a)}{16 d}-a x \]
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Rubi [A] time = 0.09, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3881, 3770} \[ -\frac {5 a \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {\tan ^5(c+d x) (5 a \sec (c+d x)+6 a)}{30 d}-\frac {\tan ^3(c+d x) (5 a \sec (c+d x)+8 a)}{24 d}+\frac {\tan (c+d x) (5 a \sec (c+d x)+16 a)}{16 d}-a x \]
Antiderivative was successfully verified.
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Rule 3770
Rule 3881
Rubi steps
\begin {align*} \int (a+a \sec (c+d x)) \tan ^6(c+d x) \, dx &=\frac {(6 a+5 a \sec (c+d x)) \tan ^5(c+d x)}{30 d}-\frac {1}{6} \int (6 a+5 a \sec (c+d x)) \tan ^4(c+d x) \, dx\\ &=-\frac {(8 a+5 a \sec (c+d x)) \tan ^3(c+d x)}{24 d}+\frac {(6 a+5 a \sec (c+d x)) \tan ^5(c+d x)}{30 d}+\frac {1}{24} \int (24 a+15 a \sec (c+d x)) \tan ^2(c+d x) \, dx\\ &=\frac {(16 a+5 a \sec (c+d x)) \tan (c+d x)}{16 d}-\frac {(8 a+5 a \sec (c+d x)) \tan ^3(c+d x)}{24 d}+\frac {(6 a+5 a \sec (c+d x)) \tan ^5(c+d x)}{30 d}-\frac {1}{48} \int (48 a+15 a \sec (c+d x)) \, dx\\ &=-a x+\frac {(16 a+5 a \sec (c+d x)) \tan (c+d x)}{16 d}-\frac {(8 a+5 a \sec (c+d x)) \tan ^3(c+d x)}{24 d}+\frac {(6 a+5 a \sec (c+d x)) \tan ^5(c+d x)}{30 d}-\frac {1}{16} (5 a) \int \sec (c+d x) \, dx\\ &=-a x-\frac {5 a \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {(16 a+5 a \sec (c+d x)) \tan (c+d x)}{16 d}-\frac {(8 a+5 a \sec (c+d x)) \tan ^3(c+d x)}{24 d}+\frac {(6 a+5 a \sec (c+d x)) \tan ^5(c+d x)}{30 d}\\ \end {align*}
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Mathematica [A] time = 1.25, size = 95, normalized size = 0.93 \[ -\frac {a \left (240 \tan ^{-1}(\tan (c+d x))+75 \tanh ^{-1}(\sin (c+d x))-\frac {1}{8} (1168 \cos (c+d x)+140 \cos (2 (c+d x))+568 \cos (3 (c+d x))+165 \cos (4 (c+d x))+184 \cos (5 (c+d x))+295) \tan (c+d x) \sec ^5(c+d x)\right )}{240 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 134, normalized size = 1.31 \[ -\frac {480 \, a d x \cos \left (d x + c\right )^{6} + 75 \, a \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 75 \, a \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (368 \, a \cos \left (d x + c\right )^{5} + 165 \, a \cos \left (d x + c\right )^{4} - 176 \, a \cos \left (d x + c\right )^{3} - 130 \, a \cos \left (d x + c\right )^{2} + 48 \, a \cos \left (d x + c\right ) + 40 \, a\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 8.79, size = 146, normalized size = 1.43 \[ -\frac {240 \, {\left (d x + c\right )} a + 75 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 75 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (165 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 1095 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 3138 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 5118 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1945 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 315 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{6}}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.51, size = 178, normalized size = 1.75 \[ \frac {a \left (\tan ^{5}\left (d x +c \right )\right )}{5 d}-\frac {a \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}+\frac {a \tan \left (d x +c \right )}{d}-a x -\frac {c a}{d}+\frac {a \left (\sin ^{7}\left (d x +c \right )\right )}{6 d \cos \left (d x +c \right )^{6}}-\frac {a \left (\sin ^{7}\left (d x +c \right )\right )}{24 d \cos \left (d x +c \right )^{4}}+\frac {a \left (\sin ^{7}\left (d x +c \right )\right )}{16 d \cos \left (d x +c \right )^{2}}+\frac {a \left (\sin ^{5}\left (d x +c \right )\right )}{16 d}+\frac {5 a \left (\sin ^{3}\left (d x +c \right )\right )}{48 d}+\frac {5 a \sin \left (d x +c \right )}{16 d}-\frac {5 a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.82, size = 134, normalized size = 1.31 \[ \frac {32 \, {\left (3 \, \tan \left (d x + c\right )^{5} - 5 \, \tan \left (d x + c\right )^{3} - 15 \, d x - 15 \, c + 15 \, \tan \left (d x + c\right )\right )} a - 5 \, a {\left (\frac {2 \, {\left (33 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.38, size = 188, normalized size = 1.84 \[ \frac {-\frac {11\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+\frac {73\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{8}-\frac {523\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{20}+\frac {853\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{20}-\frac {389\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {21\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-a\,x-\frac {5\,a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,d} \]
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 \[ a \left (\int \tan ^{6}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \tan ^{6}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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