Optimal. Leaf size=102 \[ -a x-\frac {5 a \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {(16 a+5 a \sec (c+d x)) \tan (c+d x)}{16 d}-\frac {(8 a+5 a \sec (c+d x)) \tan ^3(c+d x)}{24 d}+\frac {(6 a+5 a \sec (c+d x)) \tan ^5(c+d x)}{30 d} \]
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Rubi [A]
time = 0.07, 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 = {3966, 3855}
\begin {gather*} -\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 \end {gather*}
Antiderivative was successfully verified.
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Rule 3855
Rule 3966
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 = 1.30, size = 95, normalized size = 0.93 \begin {gather*} -\frac {a \left (240 \text {ArcTan}(\tan (c+d x))+75 \tanh ^{-1}(\sin (c+d x))-\frac {1}{8} (295+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))) \sec ^5(c+d x) \tan (c+d x)\right )}{240 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 143, normalized size = 1.40
method | result | size |
derivativedivides | \(\frac {a \left (\frac {\sin ^{7}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}-\frac {\sin ^{7}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{7}\left (d x +c \right )}{16 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{16}+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{48}+\frac {5 \sin \left (d x +c \right )}{16}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+a \left (\frac {\left (\tan ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}+\tan \left (d x +c \right )-d x -c \right )}{d}\) | \(143\) |
default | \(\frac {a \left (\frac {\sin ^{7}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}-\frac {\sin ^{7}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{7}\left (d x +c \right )}{16 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{16}+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{48}+\frac {5 \sin \left (d x +c \right )}{16}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+a \left (\frac {\left (\tan ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}+\tan \left (d x +c \right )-d x -c \right )}{d}\) | \(143\) |
risch | \(-a x -\frac {i a \left (165 \,{\mathrm e}^{11 i \left (d x +c \right )}-720 \,{\mathrm e}^{10 i \left (d x +c \right )}-25 \,{\mathrm e}^{9 i \left (d x +c \right )}-2160 \,{\mathrm e}^{8 i \left (d x +c \right )}+450 \,{\mathrm e}^{7 i \left (d x +c \right )}-3680 \,{\mathrm e}^{6 i \left (d x +c \right )}-450 \,{\mathrm e}^{5 i \left (d x +c \right )}-3360 \,{\mathrm e}^{4 i \left (d x +c \right )}+25 \,{\mathrm e}^{3 i \left (d x +c \right )}-1488 \,{\mathrm e}^{2 i \left (d x +c \right )}-165 \,{\mathrm e}^{i \left (d x +c \right )}-368\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}-\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{16 d}+\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{16 d}\) | \(187\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 134, normalized size = 1.31 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.84, size = 134, normalized size = 1.31 \begin {gather*} -\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}} \end {gather*}
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 \begin {gather*} 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.36, size = 146, normalized size = 1.43 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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