Optimal. Leaf size=125 \[ \frac {\sin ^7(c+d x)}{7 a d}+\frac {\sin ^5(c+d x) \cos ^3(c+d x)}{8 a d}+\frac {5 \sin ^3(c+d x) \cos ^3(c+d x)}{48 a d}+\frac {5 \sin (c+d x) \cos ^3(c+d x)}{64 a d}-\frac {5 \sin (c+d x) \cos (c+d x)}{128 a d}-\frac {5 x}{128 a} \]
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Rubi [A] time = 0.21, 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 = {3872, 2839, 2564, 30, 2568, 2635, 8} \[ \frac {\sin ^7(c+d x)}{7 a d}+\frac {\sin ^5(c+d x) \cos ^3(c+d x)}{8 a d}+\frac {5 \sin ^3(c+d x) \cos ^3(c+d x)}{48 a d}+\frac {5 \sin (c+d x) \cos ^3(c+d x)}{64 a d}-\frac {5 \sin (c+d x) \cos (c+d x)}{128 a d}-\frac {5 x}{128 a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2564
Rule 2568
Rule 2635
Rule 2839
Rule 3872
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 {\operatorname {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 = 1.31, size = 132, normalized size = 1.06 \[ \frac {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \left (1680 \sin (c+d x)+336 \sin (2 (c+d x))-1008 \sin (3 (c+d x))+168 \sin (4 (c+d x))+336 \sin (5 (c+d x))-112 \sin (6 (c+d x))-48 \sin (7 (c+d x))+21 \sin (8 (c+d x))+1176 c-1176 \tan \left (\frac {c}{2}\right )-840 d x\right )}{10752 a d (\sec (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 91, normalized size = 0.73 \[ -\frac {105 \, d x - {\left (336 \, \cos \left (d x + c\right )^{7} - 384 \, \cos \left (d x + c\right )^{6} - 952 \, \cos \left (d x + c\right )^{5} + 1152 \, \cos \left (d x + c\right )^{4} + 826 \, \cos \left (d x + c\right )^{3} - 1152 \, \cos \left (d x + c\right )^{2} - 105 \, \cos \left (d x + c\right ) + 384\right )} \sin \left (d x + c\right )}{2688 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 139, normalized size = 1.11 \[ -\frac {\frac {105 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 805 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 2681 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 44099 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 5053 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 2681 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 805 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{8} a}}{2688 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.44, size = 290, normalized size = 2.32 \[ \frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {115 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {383 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {5053 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1344 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {44099 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1344 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {383 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {115 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {5 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.48, size = 360, normalized size = 2.88 \[ \frac {\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2681 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5053 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {44099 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {2681 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {805 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac {105 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}}}{a + \frac {8 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {28 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {56 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {70 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {56 \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {28 \, a \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {8 \, a \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {a \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}}} - \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{1344 \, d} \]
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 \[ \frac {-\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{64}-\frac {115\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{192}-\frac {383\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{192}+\frac {44099\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{1344}+\frac {5053\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{1344}+\frac {383\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{192}+\frac {115\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192}+\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^8}-\frac {5\,x}{128\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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