Optimal. Leaf size=121 \[ \frac{a^8 \sin ^5(c+d x)}{5 d}+\frac{2 a^8 \sin ^4(c+d x)}{d}+\frac{10 a^8 \sin ^3(c+d x)}{d}+\frac{36 a^8 \sin ^2(c+d x)}{d}+\frac{64 a^9}{d (a-a \sin (c+d x))}+\frac{129 a^8 \sin (c+d x)}{d}+\frac{192 a^8 \log (1-\sin (c+d x))}{d} \]
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Rubi [A] time = 0.0940941, antiderivative size = 121, normalized size of antiderivative = 1., 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 2667
Rule 43
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*}
Mathematica [A] time = 0.259012, 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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Maple [B] time = 0.112, size = 345, normalized size = 2.9 \begin{align*} 192\,{\frac{{a}^{8}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}-192\,{\frac{{a}^{8}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+68\,{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d}}+{\frac{385\,{a}^{8}\sin \left ( dx+c \right ) }{2\,d}}+{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{2\,d}}+4\,{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{d}}+{\frac{147\,{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{10\,d}}+34\,{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d}}+{\frac{119\,{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{2\,d}}+28\,{\frac{{a}^{8} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{d}}+4\,{\frac{{a}^{8}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+4\,{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+14\,{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+28\,{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+35\,{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+14\,{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{8}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.959338, size = 131, normalized size = 1.08 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87999, size = 328, normalized size = 2.71 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.26611, size = 371, normalized size = 3.07 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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