Optimal. Leaf size=262 \[ -\frac {a^4}{160 d (a \sin (c+d x)+a)^5}+\frac {a^3}{256 d (a-a \sin (c+d x))^4}-\frac {9 a^3}{256 d (a \sin (c+d x)+a)^4}+\frac {5 a^2}{192 d (a-a \sin (c+d x))^3}-\frac {47 a^2}{384 d (a \sin (c+d x)+a)^3}+\frac {57 a}{512 d (a-a \sin (c+d x))^2}-\frac {187 a}{512 d (a \sin (c+d x)+a)^2}+\frac {61}{128 d (a-a \sin (c+d x))}-\frac {315}{256 d (a \sin (c+d x)+a)}-\frac {\csc (c+d x)}{a d}-\frac {437 \log (1-\sin (c+d x))}{512 a d}-\frac {\log (\sin (c+d x))}{a d}+\frac {949 \log (\sin (c+d x)+1)}{512 a d} \]
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Rubi [A] time = 0.29, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2836, 12, 88} \[ -\frac {a^4}{160 d (a \sin (c+d x)+a)^5}+\frac {a^3}{256 d (a-a \sin (c+d x))^4}-\frac {9 a^3}{256 d (a \sin (c+d x)+a)^4}+\frac {5 a^2}{192 d (a-a \sin (c+d x))^3}-\frac {47 a^2}{384 d (a \sin (c+d x)+a)^3}+\frac {57 a}{512 d (a-a \sin (c+d x))^2}-\frac {187 a}{512 d (a \sin (c+d x)+a)^2}+\frac {61}{128 d (a-a \sin (c+d x))}-\frac {315}{256 d (a \sin (c+d x)+a)}-\frac {\csc (c+d x)}{a d}-\frac {437 \log (1-\sin (c+d x))}{512 a d}-\frac {\log (\sin (c+d x))}{a d}+\frac {949 \log (\sin (c+d x)+1)}{512 a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \frac {\csc ^2(c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {a^9 \operatorname {Subst}\left (\int \frac {a^2}{(a-x)^5 x^2 (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^{11} \operatorname {Subst}\left (\int \frac {1}{(a-x)^5 x^2 (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^{11} \operatorname {Subst}\left (\int \left (\frac {1}{64 a^8 (a-x)^5}+\frac {5}{64 a^9 (a-x)^4}+\frac {57}{256 a^{10} (a-x)^3}+\frac {61}{128 a^{11} (a-x)^2}+\frac {437}{512 a^{12} (a-x)}+\frac {1}{a^{11} x^2}-\frac {1}{a^{12} x}+\frac {1}{32 a^7 (a+x)^6}+\frac {9}{64 a^8 (a+x)^5}+\frac {47}{128 a^9 (a+x)^4}+\frac {187}{256 a^{10} (a+x)^3}+\frac {315}{256 a^{11} (a+x)^2}+\frac {949}{512 a^{12} (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {\csc (c+d x)}{a d}-\frac {437 \log (1-\sin (c+d x))}{512 a d}-\frac {\log (\sin (c+d x))}{a d}+\frac {949 \log (1+\sin (c+d x))}{512 a d}+\frac {a^3}{256 d (a-a \sin (c+d x))^4}+\frac {5 a^2}{192 d (a-a \sin (c+d x))^3}+\frac {57 a}{512 d (a-a \sin (c+d x))^2}+\frac {61}{128 d (a-a \sin (c+d x))}-\frac {a^4}{160 d (a+a \sin (c+d x))^5}-\frac {9 a^3}{256 d (a+a \sin (c+d x))^4}-\frac {47 a^2}{384 d (a+a \sin (c+d x))^3}-\frac {187 a}{512 d (a+a \sin (c+d x))^2}-\frac {315}{256 d (a+a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 6.21, size = 240, normalized size = 0.92 \[ \frac {a^{11} \left (-\frac {\csc (c+d x)}{a^{12}}-\frac {437 \log (1-\sin (c+d x))}{512 a^{12}}-\frac {\log (\sin (c+d x))}{a^{12}}+\frac {949 \log (\sin (c+d x)+1)}{512 a^{12}}+\frac {61}{128 a^{11} (a-a \sin (c+d x))}-\frac {315}{256 a^{11} (a \sin (c+d x)+a)}+\frac {57}{512 a^{10} (a-a \sin (c+d x))^2}-\frac {187}{512 a^{10} (a \sin (c+d x)+a)^2}+\frac {5}{192 a^9 (a-a \sin (c+d x))^3}-\frac {47}{384 a^9 (a \sin (c+d x)+a)^3}+\frac {1}{256 a^8 (a-a \sin (c+d x))^4}-\frac {9}{256 a^8 (a \sin (c+d x)+a)^4}-\frac {1}{160 a^7 (a \sin (c+d x)+a)^5}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 278, normalized size = 1.06 \[ \frac {16950 \, \cos \left (d x + c\right )^{8} - 5010 \, \cos \left (d x + c\right )^{6} - 2132 \, \cos \left (d x + c\right )^{4} - 1264 \, \cos \left (d x + c\right )^{2} - 7680 \, {\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) - \cos \left (d x + c\right )^{8}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 14235 \, {\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) - \cos \left (d x + c\right )^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 6555 \, {\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) - \cos \left (d x + c\right )^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (10395 \, \cos \left (d x + c\right )^{8} - 1545 \, \cos \left (d x + c\right )^{6} - 426 \, \cos \left (d x + c\right )^{4} - 152 \, \cos \left (d x + c\right )^{2} - 48\right )} \sin \left (d x + c\right ) - 864}{7680 \, {\left (a d \cos \left (d x + c\right )^{10} - a d \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) - a d \cos \left (d x + c\right )^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 190, normalized size = 0.73 \[ \frac {\frac {56940 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {26220 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} - \frac {30720 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} + \frac {30720 \, {\left (\sin \left (d x + c\right ) - 1\right )}}{a \sin \left (d x + c\right )} + \frac {5 \, {\left (10925 \, \sin \left (d x + c\right )^{4} - 46628 \, \sin \left (d x + c\right )^{3} + 75018 \, \sin \left (d x + c\right )^{2} - 54012 \, \sin \left (d x + c\right ) + 14721\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac {130013 \, \sin \left (d x + c\right )^{5} + 687865 \, \sin \left (d x + c\right )^{4} + 1462550 \, \sin \left (d x + c\right )^{3} + 1564350 \, \sin \left (d x + c\right )^{2} + 843525 \, \sin \left (d x + c\right ) + 184065}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{30720 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.47, size = 229, normalized size = 0.87 \[ \frac {1}{256 a d \left (\sin \left (d x +c \right )-1\right )^{4}}-\frac {5}{192 a d \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {57}{512 a d \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {61}{128 a d \left (\sin \left (d x +c \right )-1\right )}-\frac {437 \ln \left (\sin \left (d x +c \right )-1\right )}{512 a d}-\frac {1}{d a \sin \left (d x +c \right )}-\frac {\ln \left (\sin \left (d x +c \right )\right )}{a d}-\frac {1}{160 a d \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {9}{256 a d \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {47}{384 a d \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {187}{512 a d \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {315}{256 a d \left (1+\sin \left (d x +c \right )\right )}+\frac {949 \ln \left (1+\sin \left (d x +c \right )\right )}{512 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 245, normalized size = 0.94 \[ -\frac {\frac {2 \, {\left (10395 \, \sin \left (d x + c\right )^{9} + 8475 \, \sin \left (d x + c\right )^{8} - 40035 \, \sin \left (d x + c\right )^{7} - 31395 \, \sin \left (d x + c\right )^{6} + 57309 \, \sin \left (d x + c\right )^{5} + 42269 \, \sin \left (d x + c\right )^{4} - 35941 \, \sin \left (d x + c\right )^{3} - 23621 \, \sin \left (d x + c\right )^{2} + 8224 \, \sin \left (d x + c\right ) + 3840\right )}}{a \sin \left (d x + c\right )^{10} + a \sin \left (d x + c\right )^{9} - 4 \, a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} + 6 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} - 4 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} + a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right )} - \frac {14235 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {6555 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a} + \frac {7680 \, \log \left (\sin \left (d x + c\right )\right )}{a}}{7680 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.46, size = 252, normalized size = 0.96 \[ \frac {949\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{512\,a\,d}-\frac {437\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{512\,a\,d}-\frac {\ln \left (\sin \left (c+d\,x\right )\right )}{a\,d}-\frac {\frac {693\,{\sin \left (c+d\,x\right )}^9}{256}+\frac {565\,{\sin \left (c+d\,x\right )}^8}{256}-\frac {2669\,{\sin \left (c+d\,x\right )}^7}{256}-\frac {2093\,{\sin \left (c+d\,x\right )}^6}{256}+\frac {19103\,{\sin \left (c+d\,x\right )}^5}{1280}+\frac {42269\,{\sin \left (c+d\,x\right )}^4}{3840}-\frac {35941\,{\sin \left (c+d\,x\right )}^3}{3840}-\frac {23621\,{\sin \left (c+d\,x\right )}^2}{3840}+\frac {257\,\sin \left (c+d\,x\right )}{120}+1}{d\,\left (a\,{\sin \left (c+d\,x\right )}^{10}+a\,{\sin \left (c+d\,x\right )}^9-4\,a\,{\sin \left (c+d\,x\right )}^8-4\,a\,{\sin \left (c+d\,x\right )}^7+6\,a\,{\sin \left (c+d\,x\right )}^6+6\,a\,{\sin \left (c+d\,x\right )}^5-4\,a\,{\sin \left (c+d\,x\right )}^4-4\,a\,{\sin \left (c+d\,x\right )}^3+a\,{\sin \left (c+d\,x\right )}^2+a\,\sin \left (c+d\,x\right )\right )} \]
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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