Optimal. Leaf size=110 \[ -\frac{a^8 \sin ^3(c+d x)}{3 d}-\frac{4 a^8 \sin ^2(c+d x)}{d}+\frac{16 a^{10}}{d (a-a \sin (c+d x))^2}-\frac{80 a^9}{d (a-a \sin (c+d x))}-\frac{31 a^8 \sin (c+d x)}{d}-\frac{80 a^8 \log (1-\sin (c+d x))}{d} \]
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Rubi [A] time = 0.091071, antiderivative size = 110, 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} \[ -\frac{a^8 \sin ^3(c+d x)}{3 d}-\frac{4 a^8 \sin ^2(c+d x)}{d}+\frac{16 a^{10}}{d (a-a \sin (c+d x))^2}-\frac{80 a^9}{d (a-a \sin (c+d x))}-\frac{31 a^8 \sin (c+d x)}{d}-\frac{80 a^8 \log (1-\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \sec ^5(c+d x) (a+a \sin (c+d x))^8 \, dx &=\frac{a^5 \operatorname{Subst}\left (\int \frac{(a+x)^5}{(a-x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^5 \operatorname{Subst}\left (\int \left (-31 a^2+\frac{32 a^5}{(a-x)^3}-\frac{80 a^4}{(a-x)^2}+\frac{80 a^3}{a-x}-8 a x-x^2\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{80 a^8 \log (1-\sin (c+d x))}{d}-\frac{31 a^8 \sin (c+d x)}{d}-\frac{4 a^8 \sin ^2(c+d x)}{d}-\frac{a^8 \sin ^3(c+d x)}{3 d}+\frac{16 a^{10}}{d (a-a \sin (c+d x))^2}-\frac{80 a^9}{d (a-a \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.445311, size = 73, normalized size = 0.66 \[ \frac{a^8 \left (-\frac{1}{3} \sin ^3(c+d x)-4 \sin ^2(c+d x)-31 \sin (c+d x)+\frac{16 (5 \sin (c+d x)-4)}{(\sin (c+d x)-1)^2}-80 \log (1-\sin (c+d x))\right )}{d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.116, size = 503, normalized size = 4.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.959797, size = 128, normalized size = 1.16 \begin{align*} -\frac{a^{8} \sin \left (d x + c\right )^{3} + 12 \, a^{8} \sin \left (d x + c\right )^{2} + 240 \, a^{8} \log \left (\sin \left (d x + c\right ) - 1\right ) + 93 \, a^{8} \sin \left (d x + c\right ) - \frac{48 \,{\left (5 \, a^{8} \sin \left (d x + c\right ) - 4 \, a^{8}\right )}}{\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8313, size = 344, normalized size = 3.13 \begin{align*} \frac{10 \, a^{8} \cos \left (d x + c\right )^{4} + 160 \, a^{8} \cos \left (d x + c\right )^{2} + 16 \, a^{8} - 240 \,{\left (a^{8} \cos \left (d x + c\right )^{2} + 2 \, a^{8} \sin \left (d x + c\right ) - 2 \, a^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) +{\left (a^{8} \cos \left (d x + c\right )^{4} - 72 \, a^{8} \cos \left (d x + c\right )^{2} - 64 \, a^{8}\right )} \sin \left (d x + c\right )}{3 \,{\left (d \cos \left (d x + c\right )^{2} + 2 \, d \sin \left (d x + c\right ) - 2 \, 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 = 1.23866, size = 328, normalized size = 2.98 \begin{align*} \frac{2 \,{\left (120 \, a^{8} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) - 240 \, a^{8} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{220 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 93 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 684 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 190 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 684 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 93 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 220 \, a^{8}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{3}} + \frac{4 \,{\left (125 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 536 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 846 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 536 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 125 \, a^{8}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{4}}\right )}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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