Optimal. Leaf size=162 \[ -\frac{64 a^8 \sin (c+d x)}{d}-\frac{16 a^5 (a \sin (c+d x)+a)^3}{3 d}-\frac{4 a^3 (a \sin (c+d x)+a)^5}{5 d}-\frac{a^2 (a \sin (c+d x)+a)^6}{3 d}-\frac{2 \left (a^2 \sin (c+d x)+a^2\right )^4}{d}-\frac{16 \left (a^4 \sin (c+d x)+a^4\right )^2}{d}-\frac{128 a^8 \log (1-\sin (c+d x))}{d}-\frac{a (a \sin (c+d x)+a)^7}{7 d} \]
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Rubi [A] time = 0.0768911, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2667, 43} \[ -\frac{64 a^8 \sin (c+d x)}{d}-\frac{16 a^5 (a \sin (c+d x)+a)^3}{3 d}-\frac{4 a^3 (a \sin (c+d x)+a)^5}{5 d}-\frac{a^2 (a \sin (c+d x)+a)^6}{3 d}-\frac{2 \left (a^2 \sin (c+d x)+a^2\right )^4}{d}-\frac{16 \left (a^4 \sin (c+d x)+a^4\right )^2}{d}-\frac{128 a^8 \log (1-\sin (c+d x))}{d}-\frac{a (a \sin (c+d x)+a)^7}{7 d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \sec (c+d x) (a+a \sin (c+d x))^8 \, dx &=\frac{a \operatorname{Subst}\left (\int \frac{(a+x)^7}{a-x} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a \operatorname{Subst}\left (\int \left (-64 a^6+\frac{128 a^7}{a-x}-32 a^5 (a+x)-16 a^4 (a+x)^2-8 a^3 (a+x)^3-4 a^2 (a+x)^4-2 a (a+x)^5-(a+x)^6\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{128 a^8 \log (1-\sin (c+d x))}{d}-\frac{64 a^8 \sin (c+d x)}{d}-\frac{16 a^5 (a+a \sin (c+d x))^3}{3 d}-\frac{4 a^3 (a+a \sin (c+d x))^5}{5 d}-\frac{a^2 (a+a \sin (c+d x))^6}{3 d}-\frac{a (a+a \sin (c+d x))^7}{7 d}-\frac{2 \left (a^2+a^2 \sin (c+d x)\right )^4}{d}-\frac{16 \left (a^4+a^4 \sin (c+d x)\right )^2}{d}\\ \end{align*}
Mathematica [A] time = 0.176694, size = 95, normalized size = 0.59 \[ \frac{a^8 \left (-\frac{1}{7} \sin ^7(c+d x)-\frac{4}{3} \sin ^6(c+d x)-\frac{29}{5} \sin ^5(c+d x)-16 \sin ^4(c+d x)-33 \sin ^3(c+d x)-60 \sin ^2(c+d x)-127 \sin (c+d x)-128 \log (1-\sin (c+d x))\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.076, size = 149, normalized size = 0.9 \begin{align*} -128\,{\frac{{a}^{8}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+128\,{\frac{{a}^{8}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{7\,d}}-{\frac{4\,{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{3\,d}}-{\frac{29\,{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-16\,{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d}}-33\,{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{d}}-60\,{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d}}-127\,{\frac{{a}^{8}\sin \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.950359, size = 147, normalized size = 0.91 \begin{align*} -\frac{15 \, a^{8} \sin \left (d x + c\right )^{7} + 140 \, a^{8} \sin \left (d x + c\right )^{6} + 609 \, a^{8} \sin \left (d x + c\right )^{5} + 1680 \, a^{8} \sin \left (d x + c\right )^{4} + 3465 \, a^{8} \sin \left (d x + c\right )^{3} + 6300 \, a^{8} \sin \left (d x + c\right )^{2} + 13440 \, a^{8} \log \left (\sin \left (d x + c\right ) - 1\right ) + 13335 \, a^{8} \sin \left (d x + c\right )}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83921, size = 302, normalized size = 1.86 \begin{align*} \frac{140 \, a^{8} \cos \left (d x + c\right )^{6} - 2100 \, a^{8} \cos \left (d x + c\right )^{4} + 10080 \, a^{8} \cos \left (d x + c\right )^{2} - 13440 \, a^{8} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 3 \,{\left (5 \, a^{8} \cos \left (d x + c\right )^{6} - 218 \, a^{8} \cos \left (d x + c\right )^{4} + 1576 \, a^{8} \cos \left (d x + c\right )^{2} - 5808 \, a^{8}\right )} \sin \left (d x + c\right )}{105 \, d} \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 [A] time = 1.27936, size = 389, normalized size = 2.4 \begin{align*} \frac{2 \,{\left (6720 \, a^{8} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) - 13440 \, a^{8} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{17424 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{14} + 13335 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 134568 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} + 93870 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 442344 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 265209 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 780640 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 370308 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 780640 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 265209 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 442344 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 93870 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 134568 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 13335 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 17424 \, a^{8}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{7}}\right )}}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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