Optimal. Leaf size=167 \[ \frac {11 x}{128 a^2}+\frac {11 \cos (c+d x) \sin (c+d x)}{128 a^2 d}-\frac {7 \cos ^3(c+d x) \sin (c+d x)}{64 a^2 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{16 a^2 d}-\frac {\cos ^3(c+d x) \sin ^3(c+d x)}{6 a^2 d}-\frac {\cos ^5(c+d x) \sin ^3(c+d x)}{8 a^2 d}-\frac {2 \sin ^5(c+d x)}{5 a^2 d}+\frac {2 \sin ^7(c+d x)}{7 a^2 d} \]
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Rubi [A]
time = 0.32, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3957, 2954,
2952, 2648, 2715, 8, 2644, 14} \begin {gather*} \frac {2 \sin ^7(c+d x)}{7 a^2 d}-\frac {2 \sin ^5(c+d x)}{5 a^2 d}-\frac {\sin ^3(c+d x) \cos ^5(c+d x)}{8 a^2 d}-\frac {\sin (c+d x) \cos ^5(c+d x)}{16 a^2 d}-\frac {\sin ^3(c+d x) \cos ^3(c+d x)}{6 a^2 d}-\frac {7 \sin (c+d x) \cos ^3(c+d x)}{64 a^2 d}+\frac {11 \sin (c+d x) \cos (c+d x)}{128 a^2 d}+\frac {11 x}{128 a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 14
Rule 2644
Rule 2648
Rule 2715
Rule 2952
Rule 2954
Rule 3957
Rubi steps
\begin {align*} \int \frac {\sin ^8(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=\int \frac {\cos ^2(c+d x) \sin ^8(c+d x)}{(-a-a \cos (c+d x))^2} \, dx\\ &=\frac {\int \cos ^2(c+d x) (-a+a \cos (c+d x))^2 \sin ^4(c+d x) \, dx}{a^4}\\ &=\frac {\int \left (a^2 \cos ^2(c+d x) \sin ^4(c+d x)-2 a^2 \cos ^3(c+d x) \sin ^4(c+d x)+a^2 \cos ^4(c+d x) \sin ^4(c+d x)\right ) \, dx}{a^4}\\ &=\frac {\int \cos ^2(c+d x) \sin ^4(c+d x) \, dx}{a^2}+\frac {\int \cos ^4(c+d x) \sin ^4(c+d x) \, dx}{a^2}-\frac {2 \int \cos ^3(c+d x) \sin ^4(c+d x) \, dx}{a^2}\\ &=-\frac {\cos ^3(c+d x) \sin ^3(c+d x)}{6 a^2 d}-\frac {\cos ^5(c+d x) \sin ^3(c+d x)}{8 a^2 d}+\frac {3 \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx}{8 a^2}+\frac {\int \cos ^2(c+d x) \sin ^2(c+d x) \, dx}{2 a^2}-\frac {2 \text {Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\sin (c+d x)\right )}{a^2 d}\\ &=-\frac {\cos ^3(c+d x) \sin (c+d x)}{8 a^2 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{16 a^2 d}-\frac {\cos ^3(c+d x) \sin ^3(c+d x)}{6 a^2 d}-\frac {\cos ^5(c+d x) \sin ^3(c+d x)}{8 a^2 d}+\frac {\int \cos ^4(c+d x) \, dx}{16 a^2}+\frac {\int \cos ^2(c+d x) \, dx}{8 a^2}-\frac {2 \text {Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\sin (c+d x)\right )}{a^2 d}\\ &=\frac {\cos (c+d x) \sin (c+d x)}{16 a^2 d}-\frac {7 \cos ^3(c+d x) \sin (c+d x)}{64 a^2 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{16 a^2 d}-\frac {\cos ^3(c+d x) \sin ^3(c+d x)}{6 a^2 d}-\frac {\cos ^5(c+d x) \sin ^3(c+d x)}{8 a^2 d}-\frac {2 \sin ^5(c+d x)}{5 a^2 d}+\frac {2 \sin ^7(c+d x)}{7 a^2 d}+\frac {3 \int \cos ^2(c+d x) \, dx}{64 a^2}+\frac {\int 1 \, dx}{16 a^2}\\ &=\frac {x}{16 a^2}+\frac {11 \cos (c+d x) \sin (c+d x)}{128 a^2 d}-\frac {7 \cos ^3(c+d x) \sin (c+d x)}{64 a^2 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{16 a^2 d}-\frac {\cos ^3(c+d x) \sin ^3(c+d x)}{6 a^2 d}-\frac {\cos ^5(c+d x) \sin ^3(c+d x)}{8 a^2 d}-\frac {2 \sin ^5(c+d x)}{5 a^2 d}+\frac {2 \sin ^7(c+d x)}{7 a^2 d}+\frac {3 \int 1 \, dx}{128 a^2}\\ &=\frac {11 x}{128 a^2}+\frac {11 \cos (c+d x) \sin (c+d x)}{128 a^2 d}-\frac {7 \cos ^3(c+d x) \sin (c+d x)}{64 a^2 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{16 a^2 d}-\frac {\cos ^3(c+d x) \sin ^3(c+d x)}{6 a^2 d}-\frac {\cos ^5(c+d x) \sin ^3(c+d x)}{8 a^2 d}-\frac {2 \sin ^5(c+d x)}{5 a^2 d}+\frac {2 \sin ^7(c+d x)}{7 a^2 d}\\ \end {align*}
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Mathematica [A]
time = 1.77, size = 131, normalized size = 0.78 \begin {gather*} \frac {\cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^2(c+d x) \left (9240 d x-10080 \sin (c+d x)-1680 \sin (2 (c+d x))+3360 \sin (3 (c+d x))-2520 \sin (4 (c+d x))+672 \sin (5 (c+d x))+560 \sin (6 (c+d x))-480 \sin (7 (c+d x))+105 \sin (8 (c+d x))+980 \tan \left (\frac {c}{2}\right )\right )}{26880 a^2 d (1+\sec (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 141, normalized size = 0.84
method | result | size |
derivativedivides | \(\frac {\frac {128 \left (-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8192}-\frac {253 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24576}-\frac {4213 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{122880}-\frac {55583 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{860160}+\frac {31007 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{860160}-\frac {20363 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{122880}+\frac {253 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24576}+\frac {11 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8192}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {11 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{a^{2} d}\) | \(141\) |
default | \(\frac {\frac {128 \left (-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8192}-\frac {253 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24576}-\frac {4213 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{122880}-\frac {55583 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{860160}+\frac {31007 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{860160}-\frac {20363 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{122880}+\frac {253 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24576}+\frac {11 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8192}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {11 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{a^{2} d}\) | \(141\) |
risch | \(\frac {11 x}{128 a^{2}}-\frac {3 \sin \left (d x +c \right )}{32 a^{2} d}+\frac {\sin \left (8 d x +8 c \right )}{1024 a^{2} d}-\frac {\sin \left (7 d x +7 c \right )}{224 a^{2} d}+\frac {\sin \left (6 d x +6 c \right )}{192 a^{2} d}+\frac {\sin \left (5 d x +5 c \right )}{160 a^{2} d}-\frac {3 \sin \left (4 d x +4 c \right )}{128 a^{2} d}+\frac {\sin \left (3 d x +3 c \right )}{32 a^{2} d}-\frac {\sin \left (2 d x +2 c \right )}{64 a^{2} d}\) | \(141\) |
norman | \(\frac {\frac {11 x}{128 a}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 a d}-\frac {253 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 a d}-\frac {4213 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{960 a d}-\frac {55583 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6720 a d}+\frac {31007 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6720 a d}-\frac {20363 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{960 a d}+\frac {253 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 a d}+\frac {11 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a d}+\frac {11 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}+\frac {77 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 a}+\frac {77 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}+\frac {385 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a}+\frac {77 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}+\frac {77 x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 a}+\frac {11 x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}+\frac {11 x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8} a}\) | \(313\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 378 vs.
\(2 (151) = 302\).
time = 0.49, size = 378, normalized size = 2.26 \begin {gather*} -\frac {\frac {\frac {1155 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {8855 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {29491 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {55583 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {31007 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {142541 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {8855 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac {1155 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}}}{a^{2} + \frac {8 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {28 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {56 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {70 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {56 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {28 \, a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {8 \, a^{2} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {a^{2} \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}}} - \frac {1155 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{6720 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.55, size = 90, normalized size = 0.54 \begin {gather*} \frac {1155 \, d x + {\left (1680 \, \cos \left (d x + c\right )^{7} - 3840 \, \cos \left (d x + c\right )^{6} - 280 \, \cos \left (d x + c\right )^{5} + 6144 \, \cos \left (d x + c\right )^{4} - 3710 \, \cos \left (d x + c\right )^{3} - 768 \, \cos \left (d x + c\right )^{2} + 1155 \, \cos \left (d x + c\right ) - 1536\right )} \sin \left (d x + c\right )}{13440 \, a^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 139, normalized size = 0.83 \begin {gather*} \frac {\frac {1155 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (1155 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 8855 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 142541 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 31007 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 55583 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 29491 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 8855 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1155 \, \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} a^{2}}}{13440 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.93, size = 133, normalized size = 0.80 \begin {gather*} \frac {11\,x}{128\,a^2}-\frac {-\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{64}-\frac {253\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{192}+\frac {20363\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{960}-\frac {31007\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{6720}+\frac {55583\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{6720}+\frac {4213\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{960}+\frac {253\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192}+\frac {11\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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