Optimal. Leaf size=194 \[ -\frac {a^2 \cot ^{11}(c+d x)}{11 d}-\frac {a^2 \cot ^9(c+d x)}{3 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}+\frac {3 a^2 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}+\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{128 d} \]
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Rubi [A] time = 0.30, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2873, 2607, 14, 2611, 3768, 3770, 270} \[ -\frac {a^2 \cot ^{11}(c+d x)}{11 d}-\frac {a^2 \cot ^9(c+d x)}{3 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}+\frac {3 a^2 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}+\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{128 d} \]
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))^2 \, dx &=\int \left (a^2 \cot ^6(c+d x) \csc ^4(c+d x)+2 a^2 \cot ^6(c+d x) \csc ^5(c+d x)+a^2 \cot ^6(c+d x) \csc ^6(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^6(c+d x) \csc ^4(c+d x) \, dx+a^2 \int \cot ^6(c+d x) \csc ^6(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^6(c+d x) \csc ^5(c+d x) \, dx\\ &=-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}-a^2 \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx+\frac {a^2 \operatorname {Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac {a^2 \operatorname {Subst}\left (\int x^6 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}+\frac {1}{8} \left (3 a^2\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac {a^2 \operatorname {Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac {a^2 \operatorname {Subst}\left (\int \left (x^6+2 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^9(c+d x)}{3 d}-\frac {a^2 \cot ^{11}(c+d x)}{11 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}-\frac {1}{16} a^2 \int \csc ^5(c+d x) \, dx\\ &=-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^9(c+d x)}{3 d}-\frac {a^2 \cot ^{11}(c+d x)}{11 d}+\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}-\frac {1}{64} \left (3 a^2\right ) \int \csc ^3(c+d x) \, dx\\ &=-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^9(c+d x)}{3 d}-\frac {a^2 \cot ^{11}(c+d x)}{11 d}+\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}-\frac {1}{128} \left (3 a^2\right ) \int \csc (c+d x) \, dx\\ &=\frac {3 a^2 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^9(c+d x)}{3 d}-\frac {a^2 \cot ^{11}(c+d x)}{11 d}+\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 3.05, size = 187, normalized size = 0.96 \[ \frac {a^2 (\sin (c+d x)+1)^2 \left (887040 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-\cot (c+d x) \csc ^{10}(c+d x) (1073226 \sin (c+d x)+869484 \sin (3 (c+d x))+727188 \sin (5 (c+d x))+40425 \sin (7 (c+d x))-3465 \sin (9 (c+d x))+1798400 \cos (2 (c+d x))+440320 \cos (4 (c+d x))-81280 \cos (6 (c+d x))-38400 \cos (8 (c+d x))+3200 \cos (10 (c+d x))+1318400)\right )}{37847040 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.94, size = 360, normalized size = 1.86 \[ \frac {12800 \, a^{2} \cos \left (d x + c\right )^{11} - 70400 \, a^{2} \cos \left (d x + c\right )^{9} + 84480 \, a^{2} \cos \left (d x + c\right )^{7} + 3465 \, {\left (a^{2} \cos \left (d x + c\right )^{10} - 5 \, a^{2} \cos \left (d x + c\right )^{8} + 10 \, a^{2} \cos \left (d x + c\right )^{6} - 10 \, a^{2} \cos \left (d x + c\right )^{4} + 5 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 3465 \, {\left (a^{2} \cos \left (d x + c\right )^{10} - 5 \, a^{2} \cos \left (d x + c\right )^{8} + 10 \, a^{2} \cos \left (d x + c\right )^{6} - 10 \, a^{2} \cos \left (d x + c\right )^{4} + 5 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 462 \, {\left (15 \, a^{2} \cos \left (d x + c\right )^{9} - 70 \, a^{2} \cos \left (d x + c\right )^{7} - 128 \, a^{2} \cos \left (d x + c\right )^{5} + 70 \, a^{2} \cos \left (d x + c\right )^{3} - 15 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{295680 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.37, size = 388, normalized size = 2.00 \[ \frac {105 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 462 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 385 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1155 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 2805 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 2310 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1155 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9240 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 16170 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4620 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 55440 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 39270 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {167422 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 39270 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 4620 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 16170 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 9240 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1155 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 2310 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2805 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1155 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 385 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 462 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 105 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11}}}{2365440 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.39, size = 264, normalized size = 1.36 \[ -\frac {5 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{33 d \sin \left (d x +c \right )^{9}}-\frac {10 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{231 d \sin \left (d x +c \right )^{7}}-\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{5 d \sin \left (d x +c \right )^{10}}-\frac {3 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{40 d \sin \left (d x +c \right )^{8}}-\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{80 d \sin \left (d x +c \right )^{6}}+\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{320 d \sin \left (d x +c \right )^{4}}-\frac {3 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{640 d \sin \left (d x +c \right )^{2}}-\frac {3 a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{640 d}-\frac {a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{128 d}-\frac {3 a^{2} \cos \left (d x +c \right )}{128 d}-\frac {3 a^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128 d}-\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{11 d \sin \left (d x +c \right )^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 197, normalized size = 1.02 \[ -\frac {693 \, a^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {14080 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a^{2}}{\tan \left (d x + c\right )^{9}} + \frac {1280 \, {\left (99 \, \tan \left (d x + c\right )^{4} + 154 \, \tan \left (d x + c\right )^{2} + 63\right )} a^{2}}{\tan \left (d x + c\right )^{11}}}{887040 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.97, size = 433, normalized size = 2.23 \[ \frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{1024\,d}-\frac {7\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{1024\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2048\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{512\,d}+\frac {17\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{14336\,d}+\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{6144\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{5120\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{22528\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{512\,d}+\frac {7\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{1024\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2048\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{1024\,d}-\frac {17\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{14336\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{6144\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{5120\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{22528\,d}-\frac {3\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,d}+\frac {17\,a^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{1024\,d}-\frac {17\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{1024\,d} \]
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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