Optimal. Leaf size=194 \[ -\frac{1}{256 d \left (a^8 \sin (c+d x)+a^8\right )}-\frac{1}{256 d \left (a^4 \sin (c+d x)+a^4\right )^2}-\frac{1}{192 a^5 d (a \sin (c+d x)+a)^3}-\frac{1}{128 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac{1}{80 a^3 d (a \sin (c+d x)+a)^5}-\frac{1}{48 a^2 d (a \sin (c+d x)+a)^6}+\frac{\tanh ^{-1}(\sin (c+d x))}{256 a^8 d}-\frac{1}{28 a d (a \sin (c+d x)+a)^7}-\frac{1}{16 d (a \sin (c+d x)+a)^8} \]
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Rubi [A] time = 0.112255, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2667, 44, 206} \[ -\frac{1}{256 d \left (a^8 \sin (c+d x)+a^8\right )}-\frac{1}{256 d \left (a^4 \sin (c+d x)+a^4\right )^2}-\frac{1}{192 a^5 d (a \sin (c+d x)+a)^3}-\frac{1}{128 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac{1}{80 a^3 d (a \sin (c+d x)+a)^5}-\frac{1}{48 a^2 d (a \sin (c+d x)+a)^6}+\frac{\tanh ^{-1}(\sin (c+d x))}{256 a^8 d}-\frac{1}{28 a d (a \sin (c+d x)+a)^7}-\frac{1}{16 d (a \sin (c+d x)+a)^8} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 44
Rule 206
Rubi steps
\begin{align*} \int \frac{\sec (c+d x)}{(a+a \sin (c+d x))^8} \, dx &=\frac{a \operatorname{Subst}\left (\int \frac{1}{(a-x) (a+x)^9} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a \operatorname{Subst}\left (\int \left (\frac{1}{2 a (a+x)^9}+\frac{1}{4 a^2 (a+x)^8}+\frac{1}{8 a^3 (a+x)^7}+\frac{1}{16 a^4 (a+x)^6}+\frac{1}{32 a^5 (a+x)^5}+\frac{1}{64 a^6 (a+x)^4}+\frac{1}{128 a^7 (a+x)^3}+\frac{1}{256 a^8 (a+x)^2}+\frac{1}{256 a^8 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{1}{16 d (a+a \sin (c+d x))^8}-\frac{1}{28 a d (a+a \sin (c+d x))^7}-\frac{1}{48 a^2 d (a+a \sin (c+d x))^6}-\frac{1}{80 a^3 d (a+a \sin (c+d x))^5}-\frac{1}{192 a^5 d (a+a \sin (c+d x))^3}-\frac{1}{128 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac{1}{256 d \left (a^4+a^4 \sin (c+d x)\right )^2}-\frac{1}{256 d \left (a^8+a^8 \sin (c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{256 a^7 d}\\ &=\frac{\tanh ^{-1}(\sin (c+d x))}{256 a^8 d}-\frac{1}{16 d (a+a \sin (c+d x))^8}-\frac{1}{28 a d (a+a \sin (c+d x))^7}-\frac{1}{48 a^2 d (a+a \sin (c+d x))^6}-\frac{1}{80 a^3 d (a+a \sin (c+d x))^5}-\frac{1}{192 a^5 d (a+a \sin (c+d x))^3}-\frac{1}{128 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac{1}{256 d \left (a^4+a^4 \sin (c+d x)\right )^2}-\frac{1}{256 d \left (a^8+a^8 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.773059, size = 122, normalized size = 0.63 \[ -\frac{105 \sin ^7(c+d x)+840 \sin ^6(c+d x)+2975 \sin ^5(c+d x)+6160 \sin ^4(c+d x)+8351 \sin ^3(c+d x)+8008 \sin ^2(c+d x)+5993 \sin (c+d x)-105 \tanh ^{-1}(\sin (c+d x)) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^{16}+4096}{26880 a^8 d (\sin (c+d x)+1)^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.122, size = 180, normalized size = 0.9 \begin{align*} -{\frac{\ln \left ( \sin \left ( dx+c \right ) -1 \right ) }{512\,d{a}^{8}}}-{\frac{1}{16\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{1}{28\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{1}{48\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{6}}}-{\frac{1}{80\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{1}{128\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{1}{192\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{1}{256\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{1}{256\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) }}+{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{512\,d{a}^{8}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.968938, size = 288, normalized size = 1.48 \begin{align*} -\frac{\frac{2 \,{\left (105 \, \sin \left (d x + c\right )^{7} + 840 \, \sin \left (d x + c\right )^{6} + 2975 \, \sin \left (d x + c\right )^{5} + 6160 \, \sin \left (d x + c\right )^{4} + 8351 \, \sin \left (d x + c\right )^{3} + 8008 \, \sin \left (d x + c\right )^{2} + 5993 \, \sin \left (d x + c\right ) + 4096\right )}}{a^{8} \sin \left (d x + c\right )^{8} + 8 \, a^{8} \sin \left (d x + c\right )^{7} + 28 \, a^{8} \sin \left (d x + c\right )^{6} + 56 \, a^{8} \sin \left (d x + c\right )^{5} + 70 \, a^{8} \sin \left (d x + c\right )^{4} + 56 \, a^{8} \sin \left (d x + c\right )^{3} + 28 \, a^{8} \sin \left (d x + c\right )^{2} + 8 \, a^{8} \sin \left (d x + c\right ) + a^{8}} - \frac{105 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{8}} + \frac{105 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{8}}}{53760 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.96336, size = 1052, normalized size = 5.42 \begin{align*} \frac{1680 \, \cos \left (d x + c\right )^{6} - 17360 \, \cos \left (d x + c\right )^{4} + 45696 \, \cos \left (d x + c\right )^{2} + 105 \,{\left (\cos \left (d x + c\right )^{8} - 32 \, \cos \left (d x + c\right )^{6} + 160 \, \cos \left (d x + c\right )^{4} - 256 \, \cos \left (d x + c\right )^{2} - 8 \,{\left (\cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 24 \, \cos \left (d x + c\right )^{2} - 16\right )} \sin \left (d x + c\right ) + 128\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \,{\left (\cos \left (d x + c\right )^{8} - 32 \, \cos \left (d x + c\right )^{6} + 160 \, \cos \left (d x + c\right )^{4} - 256 \, \cos \left (d x + c\right )^{2} - 8 \,{\left (\cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 24 \, \cos \left (d x + c\right )^{2} - 16\right )} \sin \left (d x + c\right ) + 128\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (105 \, \cos \left (d x + c\right )^{6} - 3290 \, \cos \left (d x + c\right )^{4} + 14616 \, \cos \left (d x + c\right )^{2} - 17424\right )} \sin \left (d x + c\right ) - 38208}{53760 \,{\left (a^{8} d \cos \left (d x + c\right )^{8} - 32 \, a^{8} d \cos \left (d x + c\right )^{6} + 160 \, a^{8} d \cos \left (d x + c\right )^{4} - 256 \, a^{8} d \cos \left (d x + c\right )^{2} + 128 \, a^{8} d - 8 \,{\left (a^{8} d \cos \left (d x + c\right )^{6} - 10 \, a^{8} d \cos \left (d x + c\right )^{4} + 24 \, a^{8} d \cos \left (d x + c\right )^{2} - 16 \, a^{8} d\right )} \sin \left (d x + c\right )\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 [A] time = 1.18536, size = 177, normalized size = 0.91 \begin{align*} \frac{\frac{840 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{8}} - \frac{840 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{8}} - \frac{2283 \, \sin \left (d x + c\right )^{8} + 19944 \, \sin \left (d x + c\right )^{7} + 77364 \, \sin \left (d x + c\right )^{6} + 175448 \, \sin \left (d x + c\right )^{5} + 258370 \, \sin \left (d x + c\right )^{4} + 261464 \, \sin \left (d x + c\right )^{3} + 192052 \, \sin \left (d x + c\right )^{2} + 114152 \, \sin \left (d x + c\right ) + 67819}{a^{8}{\left (\sin \left (d x + c\right ) + 1\right )}^{8}}}{430080 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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