Optimal. Leaf size=125 \[ -\frac {5 x}{128 a}-\frac {5 \cos (c+d x) \sin (c+d x)}{128 a d}+\frac {5 \cos ^3(c+d x) \sin (c+d x)}{64 a d}+\frac {5 \cos ^3(c+d x) \sin ^3(c+d x)}{48 a d}+\frac {\cos ^3(c+d x) \sin ^5(c+d x)}{8 a d}+\frac {\sin ^7(c+d x)}{7 a d} \]
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Rubi [A]
time = 0.16, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3957, 2918,
2644, 30, 2648, 2715, 8} \begin {gather*} \frac {\sin ^7(c+d x)}{7 a d}+\frac {\sin ^5(c+d x) \cos ^3(c+d x)}{8 a d}+\frac {5 \sin ^3(c+d x) \cos ^3(c+d x)}{48 a d}+\frac {5 \sin (c+d x) \cos ^3(c+d x)}{64 a d}-\frac {5 \sin (c+d x) \cos (c+d x)}{128 a d}-\frac {5 x}{128 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2644
Rule 2648
Rule 2715
Rule 2918
Rule 3957
Rubi steps
\begin {align*} \int \frac {\sin ^8(c+d x)}{a+a \sec (c+d x)} \, dx &=-\int \frac {\cos (c+d x) \sin ^8(c+d x)}{-a-a \cos (c+d x)} \, dx\\ &=\frac {\int \cos (c+d x) \sin ^6(c+d x) \, dx}{a}-\frac {\int \cos ^2(c+d x) \sin ^6(c+d x) \, dx}{a}\\ &=\frac {\cos ^3(c+d x) \sin ^5(c+d x)}{8 a d}-\frac {5 \int \cos ^2(c+d x) \sin ^4(c+d x) \, dx}{8 a}+\frac {\text {Subst}\left (\int x^6 \, dx,x,\sin (c+d x)\right )}{a d}\\ &=\frac {5 \cos ^3(c+d x) \sin ^3(c+d x)}{48 a d}+\frac {\cos ^3(c+d x) \sin ^5(c+d x)}{8 a d}+\frac {\sin ^7(c+d x)}{7 a d}-\frac {5 \int \cos ^2(c+d x) \sin ^2(c+d x) \, dx}{16 a}\\ &=\frac {5 \cos ^3(c+d x) \sin (c+d x)}{64 a d}+\frac {5 \cos ^3(c+d x) \sin ^3(c+d x)}{48 a d}+\frac {\cos ^3(c+d x) \sin ^5(c+d x)}{8 a d}+\frac {\sin ^7(c+d x)}{7 a d}-\frac {5 \int \cos ^2(c+d x) \, dx}{64 a}\\ &=-\frac {5 \cos (c+d x) \sin (c+d x)}{128 a d}+\frac {5 \cos ^3(c+d x) \sin (c+d x)}{64 a d}+\frac {5 \cos ^3(c+d x) \sin ^3(c+d x)}{48 a d}+\frac {\cos ^3(c+d x) \sin ^5(c+d x)}{8 a d}+\frac {\sin ^7(c+d x)}{7 a d}-\frac {5 \int 1 \, dx}{128 a}\\ &=-\frac {5 x}{128 a}-\frac {5 \cos (c+d x) \sin (c+d x)}{128 a d}+\frac {5 \cos ^3(c+d x) \sin (c+d x)}{64 a d}+\frac {5 \cos ^3(c+d x) \sin ^3(c+d x)}{48 a d}+\frac {\cos ^3(c+d x) \sin ^5(c+d x)}{8 a d}+\frac {\sin ^7(c+d x)}{7 a d}\\ \end {align*}
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Mathematica [A]
time = 0.82, size = 132, normalized size = 1.06 \begin {gather*} \frac {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \left (1176 c-840 d x+1680 \sin (c+d x)+336 \sin (2 (c+d x))-1008 \sin (3 (c+d x))+168 \sin (4 (c+d x))+336 \sin (5 (c+d x))-112 \sin (6 (c+d x))-48 \sin (7 (c+d x))+21 \sin (8 (c+d x))-1176 \tan \left (\frac {c}{2}\right )\right )}{10752 a d (1+\sec (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 142, normalized size = 1.14
method | result | size |
risch | \(-\frac {5 x}{128 a}+\frac {5 \sin \left (d x +c \right )}{64 a d}+\frac {\sin \left (8 d x +8 c \right )}{1024 a d}-\frac {\sin \left (7 d x +7 c \right )}{448 a d}-\frac {\sin \left (6 d x +6 c \right )}{192 a d}+\frac {\sin \left (5 d x +5 c \right )}{64 a d}+\frac {\sin \left (4 d x +4 c \right )}{128 a d}-\frac {3 \sin \left (3 d x +3 c \right )}{64 a d}+\frac {\sin \left (2 d x +2 c \right )}{64 a d}\) | \(141\) |
derivativedivides | \(\frac {-\frac {256 \left (-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16384}-\frac {115 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{49152}-\frac {383 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{49152}-\frac {5053 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{344064}-\frac {44099 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{344064}+\frac {383 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{49152}+\frac {115 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{49152}+\frac {5 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16384}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{a d}\) | \(142\) |
default | \(\frac {-\frac {256 \left (-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16384}-\frac {115 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{49152}-\frac {383 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{49152}-\frac {5053 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{344064}-\frac {44099 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{344064}+\frac {383 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{49152}+\frac {115 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{49152}+\frac {5 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16384}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{a d}\) | \(142\) |
norman | \(\frac {-\frac {5 x}{128 a}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 a d}+\frac {115 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 a d}+\frac {383 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 a d}+\frac {5053 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1344 a d}+\frac {44099 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1344 a d}-\frac {383 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 a d}-\frac {115 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 a d}-\frac {5 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a d}-\frac {5 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}-\frac {35 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 a}-\frac {35 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}-\frac {175 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a}-\frac {35 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}-\frac {35 x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 a}-\frac {5 x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}-\frac {5 x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}\) | \(310\) |
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. 360 vs.
\(2 (113) = 226\).
time = 0.49, size = 360, normalized size = 2.88 \begin {gather*} \frac {\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2681 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5053 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {44099 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {2681 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {805 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac {105 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}}}{a + \frac {8 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {28 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {56 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {70 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {56 \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {28 \, a \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {8 \, a \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {a \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}}} - \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{1344 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.40, size = 91, normalized size = 0.73 \begin {gather*} -\frac {105 \, d x - {\left (336 \, \cos \left (d x + c\right )^{7} - 384 \, \cos \left (d x + c\right )^{6} - 952 \, \cos \left (d x + c\right )^{5} + 1152 \, \cos \left (d x + c\right )^{4} + 826 \, \cos \left (d x + c\right )^{3} - 1152 \, \cos \left (d x + c\right )^{2} - 105 \, \cos \left (d x + c\right ) + 384\right )} \sin \left (d x + c\right )}{2688 \, a 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 = 0.45, size = 139, normalized size = 1.11 \begin {gather*} -\frac {\frac {105 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 805 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 2681 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 44099 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 5053 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 2681 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 805 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 105 \, \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}}{2688 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.90, size = 132, normalized size = 1.06 \begin {gather*} \frac {-\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{64}-\frac {115\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{192}-\frac {383\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{192}+\frac {44099\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{1344}+\frac {5053\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{1344}+\frac {383\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{192}+\frac {115\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192}+\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^8}-\frac {5\,x}{128\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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