Optimal. Leaf size=157 \[ -\frac {29 x}{128 a^3}-\frac {29 \cos (c+d x) \sin (c+d x)}{128 a^3 d}-\frac {29 \cos ^3(c+d x) \sin (c+d x)}{192 a^3 d}+\frac {23 \cos ^5(c+d x) \sin (c+d x)}{48 a^3 d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{8 a^3 d}+\frac {4 \sin ^3(c+d x)}{3 a^3 d}-\frac {7 \sin ^5(c+d x)}{5 a^3 d}+\frac {3 \sin ^7(c+d x)}{7 a^3 d} \]
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Rubi [A]
time = 0.33, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3957, 2954,
2952, 2644, 14, 2648, 2715, 8, 276} \begin {gather*} \frac {3 \sin ^7(c+d x)}{7 a^3 d}-\frac {7 \sin ^5(c+d x)}{5 a^3 d}+\frac {4 \sin ^3(c+d x)}{3 a^3 d}+\frac {\sin (c+d x) \cos ^7(c+d x)}{8 a^3 d}+\frac {23 \sin (c+d x) \cos ^5(c+d x)}{48 a^3 d}-\frac {29 \sin (c+d x) \cos ^3(c+d x)}{192 a^3 d}-\frac {29 \sin (c+d x) \cos (c+d x)}{128 a^3 d}-\frac {29 x}{128 a^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 14
Rule 276
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))^3} \, dx &=-\int \frac {\cos ^3(c+d x) \sin ^8(c+d x)}{(-a-a \cos (c+d x))^3} \, dx\\ &=-\frac {\int \cos ^3(c+d x) (-a+a \cos (c+d x))^3 \sin ^2(c+d x) \, dx}{a^6}\\ &=-\frac {\int \left (-a^3 \cos ^3(c+d x) \sin ^2(c+d x)+3 a^3 \cos ^4(c+d x) \sin ^2(c+d x)-3 a^3 \cos ^5(c+d x) \sin ^2(c+d x)+a^3 \cos ^6(c+d x) \sin ^2(c+d x)\right ) \, dx}{a^6}\\ &=\frac {\int \cos ^3(c+d x) \sin ^2(c+d x) \, dx}{a^3}-\frac {\int \cos ^6(c+d x) \sin ^2(c+d x) \, dx}{a^3}-\frac {3 \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx}{a^3}+\frac {3 \int \cos ^5(c+d x) \sin ^2(c+d x) \, dx}{a^3}\\ &=\frac {\cos ^5(c+d x) \sin (c+d x)}{2 a^3 d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{8 a^3 d}-\frac {\int \cos ^6(c+d x) \, dx}{8 a^3}-\frac {\int \cos ^4(c+d x) \, dx}{2 a^3}+\frac {\text {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\sin (c+d x)\right )}{a^3 d}+\frac {3 \text {Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\sin (c+d x)\right )}{a^3 d}\\ &=-\frac {\cos ^3(c+d x) \sin (c+d x)}{8 a^3 d}+\frac {23 \cos ^5(c+d x) \sin (c+d x)}{48 a^3 d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{8 a^3 d}-\frac {5 \int \cos ^4(c+d x) \, dx}{48 a^3}-\frac {3 \int \cos ^2(c+d x) \, dx}{8 a^3}+\frac {\text {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\sin (c+d x)\right )}{a^3 d}+\frac {3 \text {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\sin (c+d x)\right )}{a^3 d}\\ &=-\frac {3 \cos (c+d x) \sin (c+d x)}{16 a^3 d}-\frac {29 \cos ^3(c+d x) \sin (c+d x)}{192 a^3 d}+\frac {23 \cos ^5(c+d x) \sin (c+d x)}{48 a^3 d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{8 a^3 d}+\frac {4 \sin ^3(c+d x)}{3 a^3 d}-\frac {7 \sin ^5(c+d x)}{5 a^3 d}+\frac {3 \sin ^7(c+d x)}{7 a^3 d}-\frac {5 \int \cos ^2(c+d x) \, dx}{64 a^3}-\frac {3 \int 1 \, dx}{16 a^3}\\ &=-\frac {3 x}{16 a^3}-\frac {29 \cos (c+d x) \sin (c+d x)}{128 a^3 d}-\frac {29 \cos ^3(c+d x) \sin (c+d x)}{192 a^3 d}+\frac {23 \cos ^5(c+d x) \sin (c+d x)}{48 a^3 d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{8 a^3 d}+\frac {4 \sin ^3(c+d x)}{3 a^3 d}-\frac {7 \sin ^5(c+d x)}{5 a^3 d}+\frac {3 \sin ^7(c+d x)}{7 a^3 d}-\frac {5 \int 1 \, dx}{128 a^3}\\ &=-\frac {29 x}{128 a^3}-\frac {29 \cos (c+d x) \sin (c+d x)}{128 a^3 d}-\frac {29 \cos ^3(c+d x) \sin (c+d x)}{192 a^3 d}+\frac {23 \cos ^5(c+d x) \sin (c+d x)}{48 a^3 d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{8 a^3 d}+\frac {4 \sin ^3(c+d x)}{3 a^3 d}-\frac {7 \sin ^5(c+d x)}{5 a^3 d}+\frac {3 \sin ^7(c+d x)}{7 a^3 d}\\ \end {align*}
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Mathematica [A]
time = 3.07, size = 131, normalized size = 0.83 \begin {gather*} \frac {\cos ^6\left (\frac {1}{2} (c+d x)\right ) \sec ^3(c+d x) \left (-24360 d x+38640 \sin (c+d x)-6720 \sin (2 (c+d x))-3920 \sin (3 (c+d x))+5880 \sin (4 (c+d x))-4368 \sin (5 (c+d x))+2240 \sin (6 (c+d x))-720 \sin (7 (c+d x))+105 \sin (8 (c+d x))+294 \tan \left (\frac {c}{2}\right )\right )}{13440 a^3 d (1+\sec (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 142, normalized size = 0.90
method | result | size |
risch | \(-\frac {29 x}{128 a^{3}}+\frac {23 \sin \left (d x +c \right )}{64 a^{3} d}+\frac {\sin \left (8 d x +8 c \right )}{1024 a^{3} d}-\frac {3 \sin \left (7 d x +7 c \right )}{448 a^{3} d}+\frac {\sin \left (6 d x +6 c \right )}{48 a^{3} d}-\frac {13 \sin \left (5 d x +5 c \right )}{320 a^{3} d}+\frac {7 \sin \left (4 d x +4 c \right )}{128 a^{3} d}-\frac {7 \sin \left (3 d x +3 c \right )}{192 a^{3} d}-\frac {\sin \left (2 d x +2 c \right )}{16 a^{3} d}\) | \(141\) |
derivativedivides | \(\frac {-\frac {64 \left (-\frac {29 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4096}-\frac {667 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12288}-\frac {11107 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{61440}-\frac {146537 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{430080}-\frac {72669 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{143360}-\frac {1759 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20480}-\frac {1143 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4096}+\frac {29 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4096}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {29 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{a^{3} d}\) | \(142\) |
default | \(\frac {-\frac {64 \left (-\frac {29 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4096}-\frac {667 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12288}-\frac {11107 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{61440}-\frac {146537 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{430080}-\frac {72669 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{143360}-\frac {1759 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20480}-\frac {1143 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4096}+\frac {29 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4096}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {29 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{a^{3} d}\) | \(142\) |
norman | \(\frac {-\frac {29 x}{128 a}+\frac {29 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 a d}+\frac {667 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 a d}+\frac {11107 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{960 a d}+\frac {146537 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6720 a d}+\frac {72669 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2240 a d}+\frac {1759 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{320 a d}+\frac {1143 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a d}-\frac {29 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a d}-\frac {29 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}-\frac {203 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 a}-\frac {203 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}-\frac {1015 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a}-\frac {203 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}-\frac {203 x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 a}-\frac {29 x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}-\frac {29 x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8} a^{2}}\) | \(313\) |
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.
\(2 (141) = 282\).
time = 0.49, size = 378, normalized size = 2.41 \begin {gather*} \frac {\frac {\frac {3045 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {23345 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {77749 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {146537 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {218007 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {36939 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + \frac {120015 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac {3045 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}}}{a^{3} + \frac {8 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {28 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {56 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {70 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {56 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {28 \, a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {8 \, a^{3} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {a^{3} \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}}} - \frac {3045 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{6720 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.95, size = 91, normalized size = 0.58 \begin {gather*} -\frac {3045 \, d x - {\left (1680 \, \cos \left (d x + c\right )^{7} - 5760 \, \cos \left (d x + c\right )^{6} + 6440 \, \cos \left (d x + c\right )^{5} - 1536 \, \cos \left (d x + c\right )^{4} - 2030 \, \cos \left (d x + c\right )^{3} + 2432 \, \cos \left (d x + c\right )^{2} - 3045 \, \cos \left (d x + c\right ) + 4864\right )} \sin \left (d x + c\right )}{13440 \, a^{3} 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 = 0.54, size = 139, normalized size = 0.89 \begin {gather*} -\frac {\frac {3045 \, {\left (d x + c\right )}}{a^{3}} + \frac {2 \, {\left (3045 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} - 120015 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 36939 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 218007 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 146537 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 77749 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 23345 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3045 \, \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^{3}}}{13440 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.87, size = 132, normalized size = 0.84 \begin {gather*} \frac {-\frac {29\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{64}+\frac {1143\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{64}+\frac {1759\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{320}+\frac {72669\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{2240}+\frac {146537\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{6720}+\frac {11107\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{960}+\frac {667\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192}+\frac {29\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{a^3\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^8}-\frac {29\,x}{128\,a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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