Optimal. Leaf size=279 \[ \frac {\csc (c+d x)}{a d}-\frac {\csc ^2(c+d x)}{2 a d}-\frac {843 \log (1-\sin (c+d x))}{512 a d}+\frac {6 \log (\sin (c+d x))}{a d}-\frac {2229 \log (1+\sin (c+d x))}{512 a d}+\frac {a^3}{256 d (a-a \sin (c+d x))^4}+\frac {a^2}{32 d (a-a \sin (c+d x))^3}+\frac {81 a}{512 d (a-a \sin (c+d x))^2}+\frac {203}{256 d (a-a \sin (c+d x))}+\frac {a^4}{160 d (a+a \sin (c+d x))^5}+\frac {11 a^3}{256 d (a+a \sin (c+d x))^4}+\frac {23 a^2}{128 d (a+a \sin (c+d x))^3}+\frac {325 a}{512 d (a+a \sin (c+d x))^2}+\frac {5}{2 d (a+a \sin (c+d x))} \]
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Rubi [A]
time = 0.21, antiderivative size = 279, 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 = {2915, 12, 90}
\begin {gather*} \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 {11 a^3}{256 d (a \sin (c+d x)+a)^4}+\frac {a^2}{32 d (a-a \sin (c+d x))^3}+\frac {23 a^2}{128 d (a \sin (c+d x)+a)^3}+\frac {81 a}{512 d (a-a \sin (c+d x))^2}+\frac {325 a}{512 d (a \sin (c+d x)+a)^2}+\frac {203}{256 d (a-a \sin (c+d x))}+\frac {5}{2 d (a \sin (c+d x)+a)}-\frac {\csc ^2(c+d x)}{2 a d}+\frac {\csc (c+d x)}{a d}-\frac {843 \log (1-\sin (c+d x))}{512 a d}+\frac {6 \log (\sin (c+d x))}{a d}-\frac {2229 \log (\sin (c+d x)+1)}{512 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 90
Rule 2915
Rubi steps
\begin {align*} \int \frac {\csc ^3(c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {a^9 \text {Subst}\left (\int \frac {a^3}{(a-x)^5 x^3 (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^{12} \text {Subst}\left (\int \frac {1}{(a-x)^5 x^3 (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^{12} \text {Subst}\left (\int \left (\frac {1}{64 a^9 (a-x)^5}+\frac {3}{32 a^{10} (a-x)^4}+\frac {81}{256 a^{11} (a-x)^3}+\frac {203}{256 a^{12} (a-x)^2}+\frac {843}{512 a^{13} (a-x)}+\frac {1}{a^{11} x^3}-\frac {1}{a^{12} x^2}+\frac {6}{a^{13} x}-\frac {1}{32 a^8 (a+x)^6}-\frac {11}{64 a^9 (a+x)^5}-\frac {69}{128 a^{10} (a+x)^4}-\frac {325}{256 a^{11} (a+x)^3}-\frac {5}{2 a^{12} (a+x)^2}-\frac {2229}{512 a^{13} (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\csc (c+d x)}{a d}-\frac {\csc ^2(c+d x)}{2 a d}-\frac {843 \log (1-\sin (c+d x))}{512 a d}+\frac {6 \log (\sin (c+d x))}{a d}-\frac {2229 \log (1+\sin (c+d x))}{512 a d}+\frac {a^3}{256 d (a-a \sin (c+d x))^4}+\frac {a^2}{32 d (a-a \sin (c+d x))^3}+\frac {81 a}{512 d (a-a \sin (c+d x))^2}+\frac {203}{256 d (a-a \sin (c+d x))}+\frac {a^4}{160 d (a+a \sin (c+d x))^5}+\frac {11 a^3}{256 d (a+a \sin (c+d x))^4}+\frac {23 a^2}{128 d (a+a \sin (c+d x))^3}+\frac {325 a}{512 d (a+a \sin (c+d x))^2}+\frac {5}{2 d (a+a \sin (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 6.19, size = 254, normalized size = 0.91 \begin {gather*} \frac {a^{12} \left (\frac {\csc (c+d x)}{a^{13}}-\frac {\csc ^2(c+d x)}{2 a^{13}}-\frac {843 \log (1-\sin (c+d x))}{512 a^{13}}+\frac {6 \log (\sin (c+d x))}{a^{13}}-\frac {2229 \log (1+\sin (c+d x))}{512 a^{13}}+\frac {1}{256 a^9 (a-a \sin (c+d x))^4}+\frac {1}{32 a^{10} (a-a \sin (c+d x))^3}+\frac {81}{512 a^{11} (a-a \sin (c+d x))^2}+\frac {203}{256 a^{12} (a-a \sin (c+d x))}+\frac {1}{160 a^8 (a+a \sin (c+d x))^5}+\frac {11}{256 a^9 (a+a \sin (c+d x))^4}+\frac {23}{128 a^{10} (a+a \sin (c+d x))^3}+\frac {325}{512 a^{11} (a+a \sin (c+d x))^2}+\frac {5}{2 a^{12} (a+a \sin (c+d x))}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.63, size = 166, normalized size = 0.59
method | result | size |
derivativedivides | \(\frac {-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\frac {1}{\sin \left (d x +c \right )}+6 \ln \left (\sin \left (d x +c \right )\right )+\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}+\frac {11}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {23}{128 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {325}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {5}{2 \left (1+\sin \left (d x +c \right )\right )}-\frac {2229 \ln \left (1+\sin \left (d x +c \right )\right )}{512}+\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}-\frac {1}{32 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {81}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {203}{256 \left (\sin \left (d x +c \right )-1\right )}-\frac {843 \ln \left (\sin \left (d x +c \right )-1\right )}{512}}{d a}\) | \(166\) |
default | \(\frac {-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\frac {1}{\sin \left (d x +c \right )}+6 \ln \left (\sin \left (d x +c \right )\right )+\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}+\frac {11}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {23}{128 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {325}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {5}{2 \left (1+\sin \left (d x +c \right )\right )}-\frac {2229 \ln \left (1+\sin \left (d x +c \right )\right )}{512}+\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}-\frac {1}{32 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {81}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {203}{256 \left (\sin \left (d x +c \right )-1\right )}-\frac {843 \ln \left (\sin \left (d x +c \right )-1\right )}{512}}{d a}\) | \(166\) |
risch | \(\frac {i \left (-870 i {\mathrm e}^{4 i \left (d x +c \right )}-58336 i {\mathrm e}^{8 i \left (d x +c \right )}+3465 \,{\mathrm e}^{i \left (d x +c \right )}+85906 \,{\mathrm e}^{9 i \left (d x +c \right )}-750 i {\mathrm e}^{20 i \left (d x +c \right )}+182360 \,{\mathrm e}^{15 i \left (d x +c \right )}+870 i {\mathrm e}^{18 i \left (d x +c \right )}-21296 i {\mathrm e}^{6 i \left (d x +c \right )}+111333 \,{\mathrm e}^{17 i \left (d x +c \right )}+31530 \,{\mathrm e}^{3 i \left (d x +c \right )}+3465 \,{\mathrm e}^{21 i \left (d x +c \right )}+46852 i {\mathrm e}^{12 i \left (d x +c \right )}+85906 \,{\mathrm e}^{13 i \left (d x +c \right )}+750 i {\mathrm e}^{2 i \left (d x +c \right )}+111333 \,{\mathrm e}^{5 i \left (d x +c \right )}+182360 \,{\mathrm e}^{7 i \left (d x +c \right )}+31530 \,{\mathrm e}^{19 i \left (d x +c \right )}-173828 \,{\mathrm e}^{11 i \left (d x +c \right )}+21296 i {\mathrm e}^{16 i \left (d x +c \right )}+58336 i {\mathrm e}^{14 i \left (d x +c \right )}-46852 i {\mathrm e}^{10 i \left (d x +c \right )}\right )}{640 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{10} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{8} a d}-\frac {843 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{256 a d}-\frac {2229 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{256 a d}+\frac {6 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a d}\) | \(356\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 257, normalized size = 0.92 \begin {gather*} \frac {\frac {2 \, {\left (3465 \, \sin \left (d x + c\right )^{10} - 375 \, \sin \left (d x + c\right )^{9} - 16545 \, \sin \left (d x + c\right )^{8} + 735 \, \sin \left (d x + c\right )^{7} + 30303 \, \sin \left (d x + c\right )^{6} + 223 \, \sin \left (d x + c\right )^{5} - 25847 \, \sin \left (d x + c\right )^{4} - 1207 \, \sin \left (d x + c\right )^{3} + 9408 \, \sin \left (d x + c\right )^{2} + 640 \, \sin \left (d x + c\right ) - 640\right )}}{a \sin \left (d x + c\right )^{11} + a \sin \left (d x + c\right )^{10} - 4 \, a \sin \left (d x + c\right )^{9} - 4 \, a \sin \left (d x + c\right )^{8} + 6 \, a \sin \left (d x + c\right )^{7} + 6 \, a \sin \left (d x + c\right )^{6} - 4 \, a \sin \left (d x + c\right )^{5} - 4 \, a \sin \left (d x + c\right )^{4} + a \sin \left (d x + c\right )^{3} + a \sin \left (d x + c\right )^{2}} - \frac {11145 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {4215 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a} + \frac {15360 \, \log \left (\sin \left (d x + c\right )\right )}{a}}{2560 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 331, normalized size = 1.19 \begin {gather*} \frac {6930 \, \cos \left (d x + c\right )^{10} - 1560 \, \cos \left (d x + c\right )^{8} - 2454 \, \cos \left (d x + c\right )^{6} - 884 \, \cos \left (d x + c\right )^{4} - 464 \, \cos \left (d x + c\right )^{2} + 15360 \, {\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8} + {\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 11145 \, {\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8} + {\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 4215 \, {\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8} + {\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (375 \, \cos \left (d x + c\right )^{8} - 765 \, \cos \left (d x + c\right )^{6} - 178 \, \cos \left (d x + c\right )^{4} - 56 \, \cos \left (d x + c\right )^{2} - 16\right )} \sin \left (d x + c\right ) - 288}{2560 \, {\left (a d \cos \left (d x + c\right )^{10} - a d \cos \left (d x + c\right )^{8} + {\left (a d \cos \left (d x + c\right )^{10} - a d \cos \left (d x + c\right )^{8}\right )} \sin \left (d x + c\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.60, size = 202, normalized size = 0.72 \begin {gather*} -\frac {\frac {44580 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} + \frac {16860 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} - \frac {61440 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} + \frac {5120 \, {\left (18 \, \sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right )}}{a \sin \left (d x + c\right )^{2}} - \frac {5 \, {\left (7025 \, \sin \left (d x + c\right )^{4} - 29724 \, \sin \left (d x + c\right )^{3} + 47346 \, \sin \left (d x + c\right )^{2} - 33684 \, \sin \left (d x + c\right ) + 9045\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac {101791 \, \sin \left (d x + c\right )^{5} + 534555 \, \sin \left (d x + c\right )^{4} + 1126810 \, \sin \left (d x + c\right )^{3} + 1192850 \, \sin \left (d x + c\right )^{2} + 634975 \, \sin \left (d x + c\right ) + 136235}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{10240 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.64, size = 263, normalized size = 0.94 \begin {gather*} \frac {\frac {693\,{\sin \left (c+d\,x\right )}^{10}}{256}-\frac {75\,{\sin \left (c+d\,x\right )}^9}{256}-\frac {3309\,{\sin \left (c+d\,x\right )}^8}{256}+\frac {147\,{\sin \left (c+d\,x\right )}^7}{256}+\frac {30303\,{\sin \left (c+d\,x\right )}^6}{1280}+\frac {223\,{\sin \left (c+d\,x\right )}^5}{1280}-\frac {25847\,{\sin \left (c+d\,x\right )}^4}{1280}-\frac {1207\,{\sin \left (c+d\,x\right )}^3}{1280}+\frac {147\,{\sin \left (c+d\,x\right )}^2}{20}+\frac {\sin \left (c+d\,x\right )}{2}-\frac {1}{2}}{d\,\left (a\,{\sin \left (c+d\,x\right )}^{11}+a\,{\sin \left (c+d\,x\right )}^{10}-4\,a\,{\sin \left (c+d\,x\right )}^9-4\,a\,{\sin \left (c+d\,x\right )}^8+6\,a\,{\sin \left (c+d\,x\right )}^7+6\,a\,{\sin \left (c+d\,x\right )}^6-4\,a\,{\sin \left (c+d\,x\right )}^5-4\,a\,{\sin \left (c+d\,x\right )}^4+a\,{\sin \left (c+d\,x\right )}^3+a\,{\sin \left (c+d\,x\right )}^2\right )}-\frac {2229\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{512\,a\,d}-\frac {843\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{512\,a\,d}+\frac {6\,\ln \left (\sin \left (c+d\,x\right )\right )}{a\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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