Optimal. Leaf size=246 \[ -\frac {a^3 \cot ^{11}(c+d x)}{11 d}-\frac {5 a^3 \cot ^9(c+d x)}{9 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}+\frac {19 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}-\frac {7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {19 a^3 \cot (c+d x) \csc (c+d x)}{256 d} \]
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Rubi [A] time = 0.44, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2873, 2611, 3768, 3770, 2607, 14, 270} \[ -\frac {a^3 \cot ^{11}(c+d x)}{11 d}-\frac {5 a^3 \cot ^9(c+d x)}{9 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}+\frac {19 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}-\frac {7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {19 a^3 \cot (c+d x) \csc (c+d x)}{256 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 270
Rule 2607
Rule 2611
Rule 2873
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (a^3 \cot ^6(c+d x) \csc ^3(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^4(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^5(c+d x)+a^3 \cot ^6(c+d x) \csc ^6(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^6(c+d x) \csc ^3(c+d x) \, dx+a^3 \int \cot ^6(c+d x) \csc ^6(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^4(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^5(c+d x) \, dx\\ &=-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{8} \left (5 a^3\right ) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx-\frac {1}{2} \left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx+\frac {a^3 \operatorname {Subst}\left (\int x^6 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{d}+\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {1}{16} \left (5 a^3\right ) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\frac {1}{16} \left (9 a^3\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac {a^3 \operatorname {Subst}\left (\int \left (x^6+2 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {5 a^3 \cot ^9(c+d x)}{9 d}-\frac {a^3 \cot ^{11}(c+d x)}{11 d}-\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{64} \left (5 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac {1}{32} \left (3 a^3\right ) \int \csc ^5(c+d x) \, dx\\ &=-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {5 a^3 \cot ^9(c+d x)}{9 d}-\frac {a^3 \cot ^{11}(c+d x)}{11 d}+\frac {5 a^3 \cot (c+d x) \csc (c+d x)}{128 d}-\frac {7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{128} \left (5 a^3\right ) \int \csc (c+d x) \, dx-\frac {1}{128} \left (9 a^3\right ) \int \csc ^3(c+d x) \, dx\\ &=\frac {5 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {5 a^3 \cot ^9(c+d x)}{9 d}-\frac {a^3 \cot ^{11}(c+d x)}{11 d}+\frac {19 a^3 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{256} \left (9 a^3\right ) \int \csc (c+d x) \, dx\\ &=\frac {19 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {5 a^3 \cot ^9(c+d x)}{9 d}-\frac {a^3 \cot ^{11}(c+d x)}{11 d}+\frac {19 a^3 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}\\ \end {align*}
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Mathematica [A] time = 3.58, size = 187, normalized size = 0.76 \[ \frac {a^3 (\sin (c+d x)+1)^3 \left (16853760 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-\cot (c+d x) \csc ^{10}(c+d x) (14477694 \sin (c+d x)+5875716 \sin (3 (c+d x))+7902972 \sin (5 (c+d x))-414645 \sin (7 (c+d x))-65835 \sin (9 (c+d x))+12423680 \cos (2 (c+d x))+839680 \cos (4 (c+d x))-2149120 \cos (6 (c+d x))-568320 \cos (8 (c+d x))+47360 \cos (10 (c+d x))+10050560)\right )}{227082240 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 360, normalized size = 1.46 \[ \frac {189440 \, a^{3} \cos \left (d x + c\right )^{11} - 1041920 \, a^{3} \cos \left (d x + c\right )^{9} + 1013760 \, a^{3} \cos \left (d x + c\right )^{7} + 65835 \, {\left (a^{3} \cos \left (d x + c\right )^{10} - 5 \, a^{3} \cos \left (d x + c\right )^{8} + 10 \, a^{3} \cos \left (d x + c\right )^{6} - 10 \, a^{3} \cos \left (d x + c\right )^{4} + 5 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 65835 \, {\left (a^{3} \cos \left (d x + c\right )^{10} - 5 \, a^{3} \cos \left (d x + c\right )^{8} + 10 \, a^{3} \cos \left (d x + c\right )^{6} - 10 \, a^{3} \cos \left (d x + c\right )^{4} + 5 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 462 \, {\left (285 \, a^{3} \cos \left (d x + c\right )^{9} - 50 \, a^{3} \cos \left (d x + c\right )^{7} - 2432 \, a^{3} \cos \left (d x + c\right )^{5} + 1330 \, a^{3} \cos \left (d x + c\right )^{3} - 285 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1774080 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 388, normalized size = 1.58 \[ \frac {630 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 4158 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 8470 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 3465 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 40590 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 57750 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 6930 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 138600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 244860 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 152460 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1053360 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 568260 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {3181018 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 568260 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 152460 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 244860 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 138600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 6930 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 57750 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40590 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 3465 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 8470 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4158 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 630 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11}}}{14192640 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 264, normalized size = 1.07 \[ -\frac {19 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{80 d \sin \left (d x +c \right )^{8}}-\frac {19 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{480 d \sin \left (d x +c \right )^{6}}+\frac {19 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{1920 d \sin \left (d x +c \right )^{4}}-\frac {19 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{1280 d \sin \left (d x +c \right )^{2}}-\frac {19 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{1280 d}-\frac {19 a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{768 d}-\frac {19 a^{3} \cos \left (d x +c \right )}{256 d}-\frac {19 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256 d}-\frac {37 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{99 d \sin \left (d x +c \right )^{9}}-\frac {74 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{693 d \sin \left (d x +c \right )^{7}}-\frac {3 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{10 d \sin \left (d x +c \right )^{10}}-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{11 d \sin \left (d x +c \right )^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 308, normalized size = 1.25 \[ -\frac {2079 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 2310 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {84480 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a^{3}}{\tan \left (d x + c\right )^{9}} + \frac {2560 \, {\left (99 \, \tan \left (d x + c\right )^{4} + 154 \, \tan \left (d x + c\right )^{2} + 63\right )} a^{3}}{\tan \left (d x + c\right )^{11}}}{1774080 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.98, size = 433, normalized size = 1.76 \[ \frac {25\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{6144\,d}-\frac {53\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3072\,d}-\frac {5\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2048\,d}-\frac {11\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {41\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{14336\,d}+\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}-\frac {11\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{18432\,d}-\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{22528\,d}+\frac {11\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {53\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3072\,d}+\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2048\,d}-\frac {25\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{6144\,d}-\frac {41\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{14336\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}+\frac {11\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{18432\,d}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{22528\,d}-\frac {19\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{256\,d}+\frac {41\,a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{1024\,d}-\frac {41\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{1024\,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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