Optimal. Leaf size=238 \[ \frac{1}{1024 d \left (a^8-a^8 \sin (c+d x)\right )}-\frac{9}{1024 d \left (a^8 \sin (c+d x)+a^8\right )}-\frac{1}{128 d \left (a^4 \sin (c+d x)+a^4\right )^2}-\frac{3}{256 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac{1}{48 a^2 d (a \sin (c+d x)+a)^6}-\frac{1}{64 a^3 d (a \sin (c+d x)+a)^5}-\frac{7}{768 a^5 d (a \sin (c+d x)+a)^3}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{512 a^8 d}-\frac{a}{36 d (a \sin (c+d x)+a)^9}-\frac{1}{32 d (a \sin (c+d x)+a)^8}-\frac{3}{112 a d (a \sin (c+d x)+a)^7} \]
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Rubi [A] time = 0.171267, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2667, 44, 206} \[ \frac{1}{1024 d \left (a^8-a^8 \sin (c+d x)\right )}-\frac{9}{1024 d \left (a^8 \sin (c+d x)+a^8\right )}-\frac{1}{128 d \left (a^4 \sin (c+d x)+a^4\right )^2}-\frac{3}{256 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac{1}{48 a^2 d (a \sin (c+d x)+a)^6}-\frac{1}{64 a^3 d (a \sin (c+d x)+a)^5}-\frac{7}{768 a^5 d (a \sin (c+d x)+a)^3}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{512 a^8 d}-\frac{a}{36 d (a \sin (c+d x)+a)^9}-\frac{1}{32 d (a \sin (c+d x)+a)^8}-\frac{3}{112 a d (a \sin (c+d x)+a)^7} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 44
Rule 206
Rubi steps
\begin{align*} \int \frac{\sec ^3(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{(a-x)^2 (a+x)^{10}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \left (\frac{1}{1024 a^{10} (a-x)^2}+\frac{1}{4 a^2 (a+x)^{10}}+\frac{1}{4 a^3 (a+x)^9}+\frac{3}{16 a^4 (a+x)^8}+\frac{1}{8 a^5 (a+x)^7}+\frac{5}{64 a^6 (a+x)^6}+\frac{3}{64 a^7 (a+x)^5}+\frac{7}{256 a^8 (a+x)^4}+\frac{1}{64 a^9 (a+x)^3}+\frac{9}{1024 a^{10} (a+x)^2}+\frac{5}{512 a^{10} \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{a}{36 d (a+a \sin (c+d x))^9}-\frac{1}{32 d (a+a \sin (c+d x))^8}-\frac{3}{112 a d (a+a \sin (c+d x))^7}-\frac{1}{48 a^2 d (a+a \sin (c+d x))^6}-\frac{1}{64 a^3 d (a+a \sin (c+d x))^5}-\frac{7}{768 a^5 d (a+a \sin (c+d x))^3}-\frac{3}{256 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac{1}{128 d \left (a^4+a^4 \sin (c+d x)\right )^2}+\frac{1}{1024 d \left (a^8-a^8 \sin (c+d x)\right )}-\frac{9}{1024 d \left (a^8+a^8 \sin (c+d x)\right )}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{512 a^7 d}\\ &=\frac{5 \tanh ^{-1}(\sin (c+d x))}{512 a^8 d}-\frac{a}{36 d (a+a \sin (c+d x))^9}-\frac{1}{32 d (a+a \sin (c+d x))^8}-\frac{3}{112 a d (a+a \sin (c+d x))^7}-\frac{1}{48 a^2 d (a+a \sin (c+d x))^6}-\frac{1}{64 a^3 d (a+a \sin (c+d x))^5}-\frac{7}{768 a^5 d (a+a \sin (c+d x))^3}-\frac{3}{256 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac{1}{128 d \left (a^4+a^4 \sin (c+d x)\right )^2}+\frac{1}{1024 d \left (a^8-a^8 \sin (c+d x)\right )}-\frac{9}{1024 d \left (a^8+a^8 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 1.76929, size = 175, normalized size = 0.74 \[ -\frac{\sec ^2(c+d x) \left (-315 \sin ^9(c+d x)-2520 \sin ^8(c+d x)-8610 \sin ^7(c+d x)-15960 \sin ^6(c+d x)-16128 \sin ^5(c+d x)-5544 \sin ^4(c+d x)+7074 \sin ^3(c+d x)+11736 \sin ^2(c+d x)+9019 \sin (c+d x)-315 \tanh ^{-1}(\sin (c+d x)) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^{18}+5120\right )}{32256 a^8 d (\sin (c+d x)+1)^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.135, size = 216, normalized size = 0.9 \begin{align*} -{\frac{1}{1024\,d{a}^{8} \left ( \sin \left ( dx+c \right ) -1 \right ) }}-{\frac{5\,\ln \left ( \sin \left ( dx+c \right ) -1 \right ) }{1024\,d{a}^{8}}}-{\frac{1}{36\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{1}{32\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{3}{112\,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}{64\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{3}{256\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{7}{768\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{1}{128\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{9}{1024\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) }}+{\frac{5\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{1024\,d{a}^{8}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.980043, size = 335, normalized size = 1.41 \begin{align*} -\frac{\frac{2 \,{\left (315 \, \sin \left (d x + c\right )^{9} + 2520 \, \sin \left (d x + c\right )^{8} + 8610 \, \sin \left (d x + c\right )^{7} + 15960 \, \sin \left (d x + c\right )^{6} + 16128 \, \sin \left (d x + c\right )^{5} + 5544 \, \sin \left (d x + c\right )^{4} - 7074 \, \sin \left (d x + c\right )^{3} - 11736 \, \sin \left (d x + c\right )^{2} - 9019 \, \sin \left (d x + c\right ) - 5120\right )}}{a^{8} \sin \left (d x + c\right )^{10} + 8 \, a^{8} \sin \left (d x + c\right )^{9} + 27 \, a^{8} \sin \left (d x + c\right )^{8} + 48 \, a^{8} \sin \left (d x + c\right )^{7} + 42 \, a^{8} \sin \left (d x + c\right )^{6} - 42 \, a^{8} \sin \left (d x + c\right )^{4} - 48 \, a^{8} \sin \left (d x + c\right )^{3} - 27 \, a^{8} \sin \left (d x + c\right )^{2} - 8 \, a^{8} \sin \left (d x + c\right ) - a^{8}} - \frac{315 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{8}} + \frac{315 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{8}}}{64512 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.16685, size = 1241, normalized size = 5.21 \begin{align*} \frac{5040 \, \cos \left (d x + c\right )^{8} - 52080 \, \cos \left (d x + c\right )^{6} + 137088 \, \cos \left (d x + c\right )^{4} - 114624 \, \cos \left (d x + c\right )^{2} + 315 \,{\left (\cos \left (d x + c\right )^{10} - 32 \, \cos \left (d x + c\right )^{8} + 160 \, \cos \left (d x + c\right )^{6} - 256 \, \cos \left (d x + c\right )^{4} + 128 \, \cos \left (d x + c\right )^{2} - 8 \,{\left (\cos \left (d x + c\right )^{8} - 10 \, \cos \left (d x + c\right )^{6} + 24 \, \cos \left (d x + c\right )^{4} - 16 \, \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 315 \,{\left (\cos \left (d x + c\right )^{10} - 32 \, \cos \left (d x + c\right )^{8} + 160 \, \cos \left (d x + c\right )^{6} - 256 \, \cos \left (d x + c\right )^{4} + 128 \, \cos \left (d x + c\right )^{2} - 8 \,{\left (\cos \left (d x + c\right )^{8} - 10 \, \cos \left (d x + c\right )^{6} + 24 \, \cos \left (d x + c\right )^{4} - 16 \, \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (315 \, \cos \left (d x + c\right )^{8} - 9870 \, \cos \left (d x + c\right )^{6} + 43848 \, \cos \left (d x + c\right )^{4} - 52272 \, \cos \left (d x + c\right )^{2} + 8960\right )} \sin \left (d x + c\right ) + 14336}{64512 \,{\left (a^{8} d \cos \left (d x + c\right )^{10} - 32 \, a^{8} d \cos \left (d x + c\right )^{8} + 160 \, a^{8} d \cos \left (d x + c\right )^{6} - 256 \, a^{8} d \cos \left (d x + c\right )^{4} + 128 \, a^{8} d \cos \left (d x + c\right )^{2} - 8 \,{\left (a^{8} d \cos \left (d x + c\right )^{8} - 10 \, a^{8} d \cos \left (d x + c\right )^{6} + 24 \, a^{8} d \cos \left (d x + c\right )^{4} - 16 \, a^{8} d \cos \left (d x + c\right )^{2}\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 = 1.23705, size = 224, normalized size = 0.94 \begin{align*} \frac{\frac{2520 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{8}} - \frac{2520 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{8}} + \frac{504 \,{\left (5 \, \sin \left (d x + c\right ) - 6\right )}}{a^{8}{\left (\sin \left (d x + c\right ) - 1\right )}} - \frac{7129 \, \sin \left (d x + c\right )^{9} + 68697 \, \sin \left (d x + c\right )^{8} + 296964 \, \sin \left (d x + c\right )^{7} + 758772 \, \sin \left (d x + c\right )^{6} + 1271214 \, \sin \left (d x + c\right )^{5} + 1465758 \, \sin \left (d x + c\right )^{4} + 1191540 \, \sin \left (d x + c\right )^{3} + 693828 \, \sin \left (d x + c\right )^{2} + 295425 \, \sin \left (d x + c\right ) + 89553}{a^{8}{\left (\sin \left (d x + c\right ) + 1\right )}^{9}}}{516096 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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