Optimal. Leaf size=247 \[ \frac{a^4}{160 d (a \sin (c+d x)+a)^5}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}+\frac{7 a^3}{256 d (a \sin (c+d x)+a)^4}+\frac{a^2}{48 d (a-a \sin (c+d x))^3}+\frac{29 a^2}{384 d (a \sin (c+d x)+a)^3}+\frac{37 a}{512 d (a-a \sin (c+d x))^2}+\frac{93 a}{512 d (a \sin (c+d x)+a)^2}+\frac{65}{256 d (a-a \sin (c+d x))}+\frac{1}{2 d (a \sin (c+d x)+a)}-\frac{193 \log (1-\sin (c+d x))}{512 a d}+\frac{\log (\sin (c+d x))}{a d}-\frac{319 \log (\sin (c+d x)+1)}{512 a d} \]
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Rubi [A] time = 0.244415, antiderivative size = 247, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2836, 12, 88} \[ \frac{a^4}{160 d (a \sin (c+d x)+a)^5}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}+\frac{7 a^3}{256 d (a \sin (c+d x)+a)^4}+\frac{a^2}{48 d (a-a \sin (c+d x))^3}+\frac{29 a^2}{384 d (a \sin (c+d x)+a)^3}+\frac{37 a}{512 d (a-a \sin (c+d x))^2}+\frac{93 a}{512 d (a \sin (c+d x)+a)^2}+\frac{65}{256 d (a-a \sin (c+d x))}+\frac{1}{2 d (a \sin (c+d x)+a)}-\frac{193 \log (1-\sin (c+d x))}{512 a d}+\frac{\log (\sin (c+d x))}{a d}-\frac{319 \log (\sin (c+d x)+1)}{512 a d} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 12
Rule 88
Rubi steps
\begin{align*} \int \frac{\csc (c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{a^9 \operatorname{Subst}\left (\int \frac{a}{(a-x)^5 x (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^{10} \operatorname{Subst}\left (\int \frac{1}{(a-x)^5 x (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^{10} \operatorname{Subst}\left (\int \left (\frac{1}{64 a^7 (a-x)^5}+\frac{1}{16 a^8 (a-x)^4}+\frac{37}{256 a^9 (a-x)^3}+\frac{65}{256 a^{10} (a-x)^2}+\frac{193}{512 a^{11} (a-x)}+\frac{1}{a^{11} x}-\frac{1}{32 a^6 (a+x)^6}-\frac{7}{64 a^7 (a+x)^5}-\frac{29}{128 a^8 (a+x)^4}-\frac{93}{256 a^9 (a+x)^3}-\frac{1}{2 a^{10} (a+x)^2}-\frac{319}{512 a^{11} (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{193 \log (1-\sin (c+d x))}{512 a d}+\frac{\log (\sin (c+d x))}{a d}-\frac{319 \log (1+\sin (c+d x))}{512 a d}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}+\frac{a^2}{48 d (a-a \sin (c+d x))^3}+\frac{37 a}{512 d (a-a \sin (c+d x))^2}+\frac{65}{256 d (a-a \sin (c+d x))}+\frac{a^4}{160 d (a+a \sin (c+d x))^5}+\frac{7 a^3}{256 d (a+a \sin (c+d x))^4}+\frac{29 a^2}{384 d (a+a \sin (c+d x))^3}+\frac{93 a}{512 d (a+a \sin (c+d x))^2}+\frac{1}{2 d (a+a \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 6.19715, size = 228, normalized size = 0.92 \[ \frac{a^{10} \left (\frac{65}{256 a^{10} (a-a \sin (c+d x))}+\frac{1}{2 a^{10} (a \sin (c+d x)+a)}+\frac{37}{512 a^9 (a-a \sin (c+d x))^2}+\frac{93}{512 a^9 (a \sin (c+d x)+a)^2}+\frac{1}{48 a^8 (a-a \sin (c+d x))^3}+\frac{29}{384 a^8 (a \sin (c+d x)+a)^3}+\frac{1}{256 a^7 (a-a \sin (c+d x))^4}+\frac{7}{256 a^7 (a \sin (c+d x)+a)^4}+\frac{1}{160 a^6 (a \sin (c+d x)+a)^5}-\frac{193 \log (1-\sin (c+d x))}{512 a^{11}}+\frac{\log (\sin (c+d x))}{a^{11}}-\frac{319 \log (\sin (c+d x)+1)}{512 a^{11}}\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.1, size = 212, normalized size = 0.9 \begin{align*}{\frac{1}{256\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{4}}}-{\frac{1}{48\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{3}}}+{\frac{37}{512\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}}}-{\frac{65}{256\,da \left ( \sin \left ( dx+c \right ) -1 \right ) }}-{\frac{193\,\ln \left ( \sin \left ( dx+c \right ) -1 \right ) }{512\,da}}+{\frac{1}{160\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{7}{256\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{29}{384\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{93}{512\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{1}{2\,da \left ( 1+\sin \left ( dx+c \right ) \right ) }}-{\frac{319\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{512\,da}}+{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02366, size = 305, normalized size = 1.23 \begin{align*} \frac{\frac{2 \,{\left (945 \, \sin \left (d x + c\right )^{8} - 975 \, \sin \left (d x + c\right )^{7} - 5385 \, \sin \left (d x + c\right )^{6} + 3255 \, \sin \left (d x + c\right )^{5} + 11319 \, \sin \left (d x + c\right )^{4} - 3721 \, \sin \left (d x + c\right )^{3} - 10831 \, \sin \left (d x + c\right )^{2} + 1489 \, \sin \left (d x + c\right ) + 4384\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} - 4 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} + 6 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} - 4 \, a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) + a} - \frac{4785 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac{2895 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a} + \frac{7680 \, \log \left (\sin \left (d x + c\right )\right )}{a}}{7680 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.22443, size = 636, normalized size = 2.57 \begin{align*} \frac{1890 \, \cos \left (d x + c\right )^{8} + 3210 \, \cos \left (d x + c\right )^{6} + 1668 \, \cos \left (d x + c\right )^{4} + 1136 \, \cos \left (d x + c\right )^{2} + 7680 \,{\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) - 4785 \,{\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 2895 \,{\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (975 \, \cos \left (d x + c\right )^{6} + 330 \, \cos \left (d x + c\right )^{4} + 136 \, \cos \left (d x + c\right )^{2} + 48\right )} \sin \left (d x + c\right ) + 864}{7680 \,{\left (a d \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{8}\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 = 1.33199, size = 228, normalized size = 0.92 \begin{align*} -\frac{\frac{19140 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} + \frac{11580 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} - \frac{30720 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac{5 \,{\left (4825 \, \sin \left (d x + c\right )^{4} - 20860 \, \sin \left (d x + c\right )^{3} + 34074 \, \sin \left (d x + c\right )^{2} - 24996 \, \sin \left (d x + c\right ) + 6981\right )}}{a{\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac{43703 \, \sin \left (d x + c\right )^{5} + 233875 \, \sin \left (d x + c\right )^{4} + 504050 \, \sin \left (d x + c\right )^{3} + 548250 \, \sin \left (d x + c\right )^{2} + 302175 \, \sin \left (d x + c\right ) + 67995}{a{\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{30720 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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