Optimal. Leaf size=121 \[ \frac {64 a^9}{d (a-a \sin (c+d x))}+\frac {a^8 \sin ^5(c+d x)}{5 d}+\frac {2 a^8 \sin ^4(c+d x)}{d}+\frac {10 a^8 \sin ^3(c+d x)}{d}+\frac {36 a^8 \sin ^2(c+d x)}{d}+\frac {129 a^8 \sin (c+d x)}{d}+\frac {192 a^8 \log (1-\sin (c+d x))}{d} \]
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Rubi [A] time = 0.09, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2667, 43} \[ \frac {a^8 \sin ^5(c+d x)}{5 d}+\frac {2 a^8 \sin ^4(c+d x)}{d}+\frac {10 a^8 \sin ^3(c+d x)}{d}+\frac {36 a^8 \sin ^2(c+d x)}{d}+\frac {64 a^9}{d (a-a \sin (c+d x))}+\frac {129 a^8 \sin (c+d x)}{d}+\frac {192 a^8 \log (1-\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+a \sin (c+d x))^8 \, dx &=\frac {a^3 \operatorname {Subst}\left (\int \frac {(a+x)^6}{(a-x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^3 \operatorname {Subst}\left (\int \left (129 a^4+\frac {64 a^6}{(a-x)^2}-\frac {192 a^5}{a-x}+72 a^3 x+30 a^2 x^2+8 a x^3+x^4\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {192 a^8 \log (1-\sin (c+d x))}{d}+\frac {129 a^8 \sin (c+d x)}{d}+\frac {36 a^8 \sin ^2(c+d x)}{d}+\frac {10 a^8 \sin ^3(c+d x)}{d}+\frac {2 a^8 \sin ^4(c+d x)}{d}+\frac {a^8 \sin ^5(c+d x)}{5 d}+\frac {64 a^9}{d (a-a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 111, normalized size = 0.92 \[ \frac {a^8 (1-\sin (c+d x)) (\sin (c+d x)+1) \sec ^2(c+d x) \left (\frac {1}{5} \sin ^5(c+d x)+2 \sin ^4(c+d x)+10 \sin ^3(c+d x)+36 \sin ^2(c+d x)+129 \sin (c+d x)+\frac {64}{1-\sin (c+d x)}+192 \log (1-\sin (c+d x))\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 130, normalized size = 1.07 \[ -\frac {4 \, a^{8} \cos \left (d x + c\right )^{6} - 172 \, a^{8} \cos \left (d x + c\right )^{4} + 2192 \, a^{8} \cos \left (d x + c\right )^{2} - 1119 \, a^{8} - 3840 \, {\left (a^{8} \sin \left (d x + c\right ) - a^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - {\left (36 \, a^{8} \cos \left (d x + c\right )^{4} - 592 \, a^{8} \cos \left (d x + c\right )^{2} - 2399 \, a^{8}\right )} \sin \left (d x + c\right )}{20 \, {\left (d \sin \left (d x + c\right ) - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.94, size = 275, normalized size = 2.27 \[ -\frac {2 \, {\left (480 \, a^{8} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 960 \, a^{8} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {160 \, {\left (9 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 20 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, a^{8}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{2}} - \frac {1096 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 645 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 5840 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 2780 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12120 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 4286 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12120 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2780 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5840 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 645 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1096 \, a^{8}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}\right )}}{5 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.28, size = 345, normalized size = 2.85 \[ \frac {a^{8} \left (\sin ^{7}\left (d x +c \right )\right )}{2 d}+\frac {4 a^{8} \left (\sin ^{6}\left (d x +c \right )\right )}{d}+\frac {147 a^{8} \left (\sin ^{5}\left (d x +c \right )\right )}{10 d}+\frac {34 a^{8} \left (\sin ^{4}\left (d x +c \right )\right )}{d}+\frac {119 a^{8} \left (\sin ^{3}\left (d x +c \right )\right )}{2 d}+\frac {68 a^{8} \left (\sin ^{2}\left (d x +c \right )\right )}{d}+\frac {385 a^{8} \sin \left (d x +c \right )}{2 d}+\frac {a^{8} \left (\sin ^{9}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}+\frac {4 a^{8} \left (\sin ^{8}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{2}}+\frac {14 a^{8} \left (\sin ^{7}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{2}}+\frac {192 a^{8} \ln \left (\cos \left (d x +c \right )\right )}{d}-\frac {192 a^{8} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {28 a^{8} \left (\sin ^{6}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{2}}+\frac {35 a^{8} \left (\sin ^{5}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{2}}+\frac {14 a^{8} \left (\sin ^{3}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{2}}+\frac {a^{8} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {28 a^{8} \left (\tan ^{2}\left (d x +c \right )\right )}{d}+\frac {4 a^{8}}{d \cos \left (d x +c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 97, normalized size = 0.80 \[ \frac {a^{8} \sin \left (d x + c\right )^{5} + 10 \, a^{8} \sin \left (d x + c\right )^{4} + 50 \, a^{8} \sin \left (d x + c\right )^{3} + 180 \, a^{8} \sin \left (d x + c\right )^{2} + 960 \, a^{8} \log \left (\sin \left (d x + c\right ) - 1\right ) + 645 \, a^{8} \sin \left (d x + c\right ) - \frac {320 \, a^{8}}{\sin \left (d x + c\right ) - 1}}{5 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.62, size = 97, normalized size = 0.80 \[ \frac {192\,a^8\,\ln \left (\sin \left (c+d\,x\right )-1\right )-\frac {64\,a^8}{\sin \left (c+d\,x\right )-1}+129\,a^8\,\sin \left (c+d\,x\right )+36\,a^8\,{\sin \left (c+d\,x\right )}^2+10\,a^8\,{\sin \left (c+d\,x\right )}^3+2\,a^8\,{\sin \left (c+d\,x\right )}^4+\frac {a^8\,{\sin \left (c+d\,x\right )}^5}{5}}{d} \]
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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