Optimal. Leaf size=177 \[ \frac {\tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac {\sec ^4(c+d x) \tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {13 \sec ^3(c+d x) \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {34 \sec ^2(c+d x) \tan (c+d x)}{105 a^2 d (a+a \sec (c+d x))^3}+\frac {173 \tan (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}-\frac {661 \tan (c+d x)}{315 d \left (a^5+a^5 \sec (c+d x)\right )} \]
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Rubi [A]
time = 0.29, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3901, 4104,
4093, 4083, 3855, 3879} \begin {gather*} \frac {\tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac {661 \tan (c+d x)}{315 d \left (a^5 \sec (c+d x)+a^5\right )}+\frac {173 \tan (c+d x)}{315 a^3 d (a \sec (c+d x)+a)^2}-\frac {34 \tan (c+d x) \sec ^2(c+d x)}{105 a^2 d (a \sec (c+d x)+a)^3}-\frac {\tan (c+d x) \sec ^4(c+d x)}{9 d (a \sec (c+d x)+a)^5}-\frac {13 \tan (c+d x) \sec ^3(c+d x)}{63 a d (a \sec (c+d x)+a)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 3855
Rule 3879
Rule 3901
Rule 4083
Rule 4093
Rule 4104
Rubi steps
\begin {align*} \int \frac {\sec ^6(c+d x)}{(a+a \sec (c+d x))^5} \, dx &=-\frac {\sec ^4(c+d x) \tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {\int \frac {\sec ^4(c+d x) (4 a-9 a \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx}{9 a^2}\\ &=-\frac {\sec ^4(c+d x) \tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {13 \sec ^3(c+d x) \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {\int \frac {\sec ^3(c+d x) \left (39 a^2-63 a^2 \sec (c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx}{63 a^4}\\ &=-\frac {\sec ^4(c+d x) \tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {13 \sec ^3(c+d x) \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {34 \sec ^2(c+d x) \tan (c+d x)}{105 a^2 d (a+a \sec (c+d x))^3}-\frac {\int \frac {\sec ^2(c+d x) \left (204 a^3-315 a^3 \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{315 a^6}\\ &=-\frac {\sec ^4(c+d x) \tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {13 \sec ^3(c+d x) \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {34 \sec ^2(c+d x) \tan (c+d x)}{105 a^2 d (a+a \sec (c+d x))^3}+\frac {173 \tan (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}+\frac {\int \frac {\sec (c+d x) \left (-1038 a^4+945 a^4 \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{945 a^8}\\ &=-\frac {\sec ^4(c+d x) \tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {13 \sec ^3(c+d x) \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {34 \sec ^2(c+d x) \tan (c+d x)}{105 a^2 d (a+a \sec (c+d x))^3}+\frac {173 \tan (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}+\frac {\int \sec (c+d x) \, dx}{a^5}-\frac {661 \int \frac {\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{315 a^4}\\ &=\frac {\tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac {\sec ^4(c+d x) \tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {13 \sec ^3(c+d x) \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {34 \sec ^2(c+d x) \tan (c+d x)}{105 a^2 d (a+a \sec (c+d x))^3}+\frac {173 \tan (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}-\frac {661 \tan (c+d x)}{315 d \left (a^5+a^5 \sec (c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 1.86, size = 219, normalized size = 1.24 \begin {gather*} -\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec ^5(c+d x) \left (80640 \cos ^9\left (\frac {1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\sec \left (\frac {c}{2}\right ) \left (35973 \sin \left (\frac {d x}{2}\right )-25515 \sin \left (c+\frac {d x}{2}\right )+29757 \sin \left (c+\frac {3 d x}{2}\right )-11235 \sin \left (2 c+\frac {3 d x}{2}\right )+14733 \sin \left (2 c+\frac {5 d x}{2}\right )-2835 \sin \left (3 c+\frac {5 d x}{2}\right )+4077 \sin \left (3 c+\frac {7 d x}{2}\right )-315 \sin \left (4 c+\frac {7 d x}{2}\right )+488 \sin \left (4 c+\frac {9 d x}{2}\right )\right )\right )}{2520 a^5 d (1+\sec (c+d x))^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 101, normalized size = 0.57
method | result | size |
derivativedivides | \(\frac {-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {6 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {16 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {26 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-16 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+16 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d \,a^{5}}\) | \(101\) |
default | \(\frac {-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {6 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {16 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {26 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-16 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+16 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d \,a^{5}}\) | \(101\) |
risch | \(-\frac {2 i \left (315 \,{\mathrm e}^{8 i \left (d x +c \right )}+2835 \,{\mathrm e}^{7 i \left (d x +c \right )}+11235 \,{\mathrm e}^{6 i \left (d x +c \right )}+25515 \,{\mathrm e}^{5 i \left (d x +c \right )}+35973 \,{\mathrm e}^{4 i \left (d x +c \right )}+29757 \,{\mathrm e}^{3 i \left (d x +c \right )}+14733 \,{\mathrm e}^{2 i \left (d x +c \right )}+4077 \,{\mathrm e}^{i \left (d x +c \right )}+488\right )}{315 d \,a^{5} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{9}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{a^{5} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a^{5} d}\) | \(155\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 159, normalized size = 0.90 \begin {gather*} -\frac {\frac {\frac {9765 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {2730 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1008 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {270 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{5}} - \frac {5040 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{5}} + \frac {5040 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{5}}}{5040 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 7.02, size = 246, normalized size = 1.39 \begin {gather*} \frac {315 \, {\left (\cos \left (d x + c\right )^{5} + 5 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3} + 10 \, \cos \left (d x + c\right )^{2} + 5 \, \cos \left (d x + c\right ) + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 315 \, {\left (\cos \left (d x + c\right )^{5} + 5 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3} + 10 \, \cos \left (d x + c\right )^{2} + 5 \, \cos \left (d x + c\right ) + 1\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (488 \, \cos \left (d x + c\right )^{4} + 2125 \, \cos \left (d x + c\right )^{3} + 3549 \, \cos \left (d x + c\right )^{2} + 2740 \, \cos \left (d x + c\right ) + 863\right )} \sin \left (d x + c\right )}{630 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \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*} \frac {\int \frac {\sec ^{6}{\left (c + d x \right )}}{\sec ^{5}{\left (c + d x \right )} + 5 \sec ^{4}{\left (c + d x \right )} + 10 \sec ^{3}{\left (c + d x \right )} + 10 \sec ^{2}{\left (c + d x \right )} + 5 \sec {\left (c + d x \right )} + 1}\, dx}{a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.52, size = 126, normalized size = 0.71 \begin {gather*} \frac {\frac {5040 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{5}} - \frac {5040 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{5}} - \frac {35 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 270 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1008 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2730 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9765 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{45}}}{5040 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.69, size = 99, normalized size = 0.56 \begin {gather*} -\frac {\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a^5}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{5\,a^5}+\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{56\,a^5}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{144\,a^5}-\frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^5}+\frac {31\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,a^5}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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