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 = 0.24, antiderivative size = 247, normalized size of antiderivative = 1.00, 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 12
Rule 88
Rule 2836
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*}
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Mathematica [A] time = 6.21, size = 228, normalized size = 0.92 \[ \frac {a^{10} \left (-\frac {193 \log (1-\sin (c+d x))}{512 a^{11}}+\frac {\log (\sin (c+d x))}{a^{11}}-\frac {319 \log (\sin (c+d x)+1)}{512 a^{11}}+\frac {65}{256 a^{10} (a-a \sin (c+d x))}+\frac {1}{2 a^{10} (a \sin (c+d x)+a)}+\frac {37}{512 a^9 (a-a \sin (c+d x))^2}+\frac {93}{512 a^9 (a \sin (c+d x)+a)^2}+\frac {1}{48 a^8 (a-a \sin (c+d x))^3}+\frac {29}{384 a^8 (a \sin (c+d x)+a)^3}+\frac {1}{256 a^7 (a-a \sin (c+d x))^4}+\frac {7}{256 a^7 (a \sin (c+d x)+a)^4}+\frac {1}{160 a^6 (a \sin (c+d x)+a)^5}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 222, normalized size = 0.90 \[ \frac {1890 \, \cos \left (d x + c\right )^{8} + 3210 \, \cos \left (d x + c\right )^{6} + 1668 \, \cos \left (d x + c\right )^{4} + 1136 \, \cos \left (d x + c\right )^{2} + 7680 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 4785 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 2895 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (975 \, \cos \left (d x + c\right )^{6} + 330 \, \cos \left (d x + c\right )^{4} + 136 \, \cos \left (d x + c\right )^{2} + 48\right )} \sin \left (d x + c\right ) + 864}{7680 \, {\left (a d \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 169, normalized size = 0.68 \[ -\frac {\frac {19140 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} + \frac {11580 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} - \frac {30720 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac {5 \, {\left (4825 \, \sin \left (d x + c\right )^{4} - 20860 \, \sin \left (d x + c\right )^{3} + 34074 \, \sin \left (d x + c\right )^{2} - 24996 \, \sin \left (d x + c\right ) + 6981\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac {43703 \, \sin \left (d x + c\right )^{5} + 233875 \, \sin \left (d x + c\right )^{4} + 504050 \, \sin \left (d x + c\right )^{3} + 548250 \, \sin \left (d x + c\right )^{2} + 302175 \, \sin \left (d x + c\right ) + 67995}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{30720 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 212, normalized size = 0.86 \[ \frac {1}{256 a d \left (\sin \left (d x +c \right )-1\right )^{4}}-\frac {1}{48 a d \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {37}{512 a d \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {65}{256 a d \left (\sin \left (d x +c \right )-1\right )}-\frac {193 \ln \left (\sin \left (d x +c \right )-1\right )}{512 a d}+\frac {\ln \left (\sin \left (d x +c \right )\right )}{a d}+\frac {1}{160 a d \left (1+\sin \left (d x +c \right )\right )^{5}}+\frac {7}{256 a d \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {29}{384 a d \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {93}{512 a d \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {1}{2 a d \left (1+\sin \left (d x +c \right )\right )}-\frac {319 \ln \left (1+\sin \left (d x +c \right )\right )}{512 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 226, normalized size = 0.91 \[ \frac {\frac {2 \, {\left (945 \, \sin \left (d x + c\right )^{8} - 975 \, \sin \left (d x + c\right )^{7} - 5385 \, \sin \left (d x + c\right )^{6} + 3255 \, \sin \left (d x + c\right )^{5} + 11319 \, \sin \left (d x + c\right )^{4} - 3721 \, \sin \left (d x + c\right )^{3} - 10831 \, \sin \left (d x + c\right )^{2} + 1489 \, \sin \left (d x + c\right ) + 4384\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} - 4 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} + 6 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} - 4 \, a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) + a} - \frac {4785 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {2895 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a} + \frac {7680 \, \log \left (\sin \left (d x + c\right )\right )}{a}}{7680 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 231, normalized size = 0.94 \[ \frac {\frac {63\,{\sin \left (c+d\,x\right )}^8}{256}-\frac {65\,{\sin \left (c+d\,x\right )}^7}{256}-\frac {359\,{\sin \left (c+d\,x\right )}^6}{256}+\frac {217\,{\sin \left (c+d\,x\right )}^5}{256}+\frac {3773\,{\sin \left (c+d\,x\right )}^4}{1280}-\frac {3721\,{\sin \left (c+d\,x\right )}^3}{3840}-\frac {10831\,{\sin \left (c+d\,x\right )}^2}{3840}+\frac {1489\,\sin \left (c+d\,x\right )}{3840}+\frac {137}{120}}{d\,\left (a\,{\sin \left (c+d\,x\right )}^9+a\,{\sin \left (c+d\,x\right )}^8-4\,a\,{\sin \left (c+d\,x\right )}^7-4\,a\,{\sin \left (c+d\,x\right )}^6+6\,a\,{\sin \left (c+d\,x\right )}^5+6\,a\,{\sin \left (c+d\,x\right )}^4-4\,a\,{\sin \left (c+d\,x\right )}^3-4\,a\,{\sin \left (c+d\,x\right )}^2+a\,\sin \left (c+d\,x\right )+a\right )}-\frac {319\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{512\,a\,d}-\frac {193\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{512\,a\,d}+\frac {\ln \left (\sin \left (c+d\,x\right )\right )}{a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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