Optimal. Leaf size=217 \[ -\frac {a^3}{64 d (a \sin (c+d x)+a)^4}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}-\frac {a^2}{12 d (a \sin (c+d x)+a)^3}+\frac {9 a}{128 d (a-a \sin (c+d x))^2}-\frac {19 a}{64 d (a \sin (c+d x)+a)^2}+\frac {47}{128 d (a-a \sin (c+d x))}-\frac {35}{32 d (a \sin (c+d x)+a)}-\frac {\csc (c+d x)}{a d}-\frac {187 \log (1-\sin (c+d x))}{256 a d}-\frac {\log (\sin (c+d x))}{a d}+\frac {443 \log (\sin (c+d x)+1)}{256 a d} \]
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Rubi [A] time = 0.24, antiderivative size = 217, 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} \[ -\frac {a^3}{64 d (a \sin (c+d x)+a)^4}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}-\frac {a^2}{12 d (a \sin (c+d x)+a)^3}+\frac {9 a}{128 d (a-a \sin (c+d x))^2}-\frac {19 a}{64 d (a \sin (c+d x)+a)^2}+\frac {47}{128 d (a-a \sin (c+d x))}-\frac {35}{32 d (a \sin (c+d x)+a)}-\frac {\csc (c+d x)}{a d}-\frac {187 \log (1-\sin (c+d x))}{256 a d}-\frac {\log (\sin (c+d x))}{a d}+\frac {443 \log (\sin (c+d x)+1)}{256 a d} \]
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 ^7(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {a^7 \operatorname {Subst}\left (\int \frac {a^2}{(a-x)^4 x^2 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^9 \operatorname {Subst}\left (\int \frac {1}{(a-x)^4 x^2 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^9 \operatorname {Subst}\left (\int \left (\frac {1}{32 a^7 (a-x)^4}+\frac {9}{64 a^8 (a-x)^3}+\frac {47}{128 a^9 (a-x)^2}+\frac {187}{256 a^{10} (a-x)}+\frac {1}{a^9 x^2}-\frac {1}{a^{10} x}+\frac {1}{16 a^6 (a+x)^5}+\frac {1}{4 a^7 (a+x)^4}+\frac {19}{32 a^8 (a+x)^3}+\frac {35}{32 a^9 (a+x)^2}+\frac {443}{256 a^{10} (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {\csc (c+d x)}{a d}-\frac {187 \log (1-\sin (c+d x))}{256 a d}-\frac {\log (\sin (c+d x))}{a d}+\frac {443 \log (1+\sin (c+d x))}{256 a d}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}+\frac {9 a}{128 d (a-a \sin (c+d x))^2}+\frac {47}{128 d (a-a \sin (c+d x))}-\frac {a^3}{64 d (a+a \sin (c+d x))^4}-\frac {a^2}{12 d (a+a \sin (c+d x))^3}-\frac {19 a}{64 d (a+a \sin (c+d x))^2}-\frac {35}{32 d (a+a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 6.14, size = 201, normalized size = 0.93 \[ \frac {a^9 \left (-\frac {\csc (c+d x)}{a^{10}}-\frac {187 \log (1-\sin (c+d x))}{256 a^{10}}-\frac {\log (\sin (c+d x))}{a^{10}}+\frac {443 \log (\sin (c+d x)+1)}{256 a^{10}}+\frac {47}{128 a^9 (a-a \sin (c+d x))}-\frac {35}{32 a^9 (a \sin (c+d x)+a)}+\frac {9}{128 a^8 (a-a \sin (c+d x))^2}-\frac {19}{64 a^8 (a \sin (c+d x)+a)^2}+\frac {1}{96 a^7 (a-a \sin (c+d x))^3}-\frac {1}{12 a^7 (a \sin (c+d x)+a)^3}-\frac {1}{64 a^6 (a \sin (c+d x)+a)^4}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 258, normalized size = 1.19 \[ \frac {1506 \, \cos \left (d x + c\right )^{6} - 438 \, \cos \left (d x + c\right )^{4} - 188 \, \cos \left (d x + c\right )^{2} - 768 \, {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) - \cos \left (d x + c\right )^{6}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 1329 \, {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) - \cos \left (d x + c\right )^{6}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 561 \, {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) - \cos \left (d x + c\right )^{6}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (945 \, \cos \left (d x + c\right )^{6} - 123 \, \cos \left (d x + c\right )^{4} - 30 \, \cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right ) - 112}{768 \, {\left (a d \cos \left (d x + c\right )^{8} - a d \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) - a d \cos \left (d x + c\right )^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 170, normalized size = 0.78 \[ \frac {\frac {5316 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {2244 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} - \frac {3072 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} + \frac {3072 \, {\left (\sin \left (d x + c\right ) - 1\right )}}{a \sin \left (d x + c\right )} + \frac {2 \, {\left (2057 \, \sin \left (d x + c\right )^{3} - 6735 \, \sin \left (d x + c\right )^{2} + 7407 \, \sin \left (d x + c\right ) - 2745\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac {11075 \, \sin \left (d x + c\right )^{4} + 47660 \, \sin \left (d x + c\right )^{3} + 77442 \, \sin \left (d x + c\right )^{2} + 56460 \, \sin \left (d x + c\right ) + 15651}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{3072 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.48, size = 193, normalized size = 0.89 \[ -\frac {1}{96 a d \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {9}{128 a d \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {47}{128 a d \left (\sin \left (d x +c \right )-1\right )}-\frac {187 \ln \left (\sin \left (d x +c \right )-1\right )}{256 a d}-\frac {1}{d a \sin \left (d x +c \right )}-\frac {\ln \left (\sin \left (d x +c \right )\right )}{a d}-\frac {1}{64 a d \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {1}{12 a d \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {19}{64 a d \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {35}{32 a d \left (1+\sin \left (d x +c \right )\right )}+\frac {443 \ln \left (1+\sin \left (d x +c \right )\right )}{256 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 205, normalized size = 0.94 \[ -\frac {\frac {2 \, {\left (945 \, \sin \left (d x + c\right )^{7} + 753 \, \sin \left (d x + c\right )^{6} - 2712 \, \sin \left (d x + c\right )^{5} - 2040 \, \sin \left (d x + c\right )^{4} + 2559 \, \sin \left (d x + c\right )^{3} + 1727 \, \sin \left (d x + c\right )^{2} - 784 \, \sin \left (d x + c\right ) - 384\right )}}{a \sin \left (d x + c\right )^{8} + a \sin \left (d x + c\right )^{7} - 3 \, a \sin \left (d x + c\right )^{6} - 3 \, a \sin \left (d x + c\right )^{5} + 3 \, a \sin \left (d x + c\right )^{4} + 3 \, a \sin \left (d x + c\right )^{3} - a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right )} - \frac {1329 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {561 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a} + \frac {768 \, \log \left (\sin \left (d x + c\right )\right )}{a}}{768 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.33, size = 212, normalized size = 0.98 \[ \frac {443\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{256\,a\,d}-\frac {187\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{256\,a\,d}-\frac {-\frac {315\,{\sin \left (c+d\,x\right )}^7}{128}-\frac {251\,{\sin \left (c+d\,x\right )}^6}{128}+\frac {113\,{\sin \left (c+d\,x\right )}^5}{16}+\frac {85\,{\sin \left (c+d\,x\right )}^4}{16}-\frac {853\,{\sin \left (c+d\,x\right )}^3}{128}-\frac {1727\,{\sin \left (c+d\,x\right )}^2}{384}+\frac {49\,\sin \left (c+d\,x\right )}{24}+1}{d\,\left (-a\,{\sin \left (c+d\,x\right )}^8-a\,{\sin \left (c+d\,x\right )}^7+3\,a\,{\sin \left (c+d\,x\right )}^6+3\,a\,{\sin \left (c+d\,x\right )}^5-3\,a\,{\sin \left (c+d\,x\right )}^4-3\,a\,{\sin \left (c+d\,x\right )}^3+a\,{\sin \left (c+d\,x\right )}^2+a\,\sin \left (c+d\,x\right )\right )}-\frac {\ln \left (\sin \left (c+d\,x\right )\right )}{a\,d} \]
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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