Optimal. Leaf size=228 \[ -\frac{2 a^2 \cot ^9(c+d x)}{9 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{13 a^2 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}+\frac{5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{32 d}-\frac{9 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{13 a^2 \cot (c+d x) \csc (c+d x)}{256 d} \]
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Rubi [A] time = 0.39092, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2873, 2611, 3768, 3770, 2607, 14} \[ -\frac{2 a^2 \cot ^9(c+d x)}{9 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{13 a^2 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}+\frac{5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{32 d}-\frac{9 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{13 a^2 \cot (c+d x) \csc (c+d x)}{256 d} \]
Antiderivative was successfully verified.
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Rule 2873
Rule 2611
Rule 3768
Rule 3770
Rule 2607
Rule 14
Rubi steps
\begin{align*} \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx &=\int \left (a^2 \cot ^6(c+d x) \csc ^3(c+d x)+2 a^2 \cot ^6(c+d x) \csc ^4(c+d x)+a^2 \cot ^6(c+d x) \csc ^5(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^6(c+d x) \csc ^3(c+d x) \, dx+a^2 \int \cot ^6(c+d x) \csc ^5(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^6(c+d x) \csc ^4(c+d x) \, dx\\ &=-\frac{a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{1}{2} a^2 \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx-\frac{1}{8} \left (5 a^2\right ) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac{5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac{1}{16} \left (3 a^2\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac{1}{16} \left (5 a^2\right ) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{2 a^2 \cot ^7(c+d x)}{7 d}-\frac{2 a^2 \cot ^9(c+d x)}{9 d}-\frac{5 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{1}{32} a^2 \int \csc ^5(c+d x) \, dx-\frac{1}{64} \left (5 a^2\right ) \int \csc ^3(c+d x) \, dx\\ &=-\frac{2 a^2 \cot ^7(c+d x)}{7 d}-\frac{2 a^2 \cot ^9(c+d x)}{9 d}+\frac{5 a^2 \cot (c+d x) \csc (c+d x)}{128 d}-\frac{9 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{1}{128} \left (3 a^2\right ) \int \csc ^3(c+d x) \, dx-\frac{1}{128} \left (5 a^2\right ) \int \csc (c+d x) \, dx\\ &=\frac{5 a^2 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}-\frac{2 a^2 \cot ^9(c+d x)}{9 d}+\frac{13 a^2 \cot (c+d x) \csc (c+d x)}{256 d}-\frac{9 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{1}{256} \left (3 a^2\right ) \int \csc (c+d x) \, dx\\ &=\frac{13 a^2 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}-\frac{2 a^2 \cot ^9(c+d x)}{9 d}+\frac{13 a^2 \cot (c+d x) \csc (c+d x)}{256 d}-\frac{9 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}\\ \end{align*}
Mathematica [A] time = 1.23422, size = 353, normalized size = 1.55 \[ -\frac{a^2 \csc ^{10}(c+d x) \left (1075200 \sin (2 (c+d x))+1044480 \sin (4 (c+d x))+414720 \sin (6 (c+d x))+51200 \sin (8 (c+d x))-5120 \sin (10 (c+d x))+2732940 \cos (c+d x)+1151640 \cos (3 (c+d x))+388248 \cos (5 (c+d x))-135870 \cos (7 (c+d x))-8190 \cos (9 (c+d x))+515970 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+859950 \cos (2 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-491400 \cos (4 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+184275 \cos (6 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-40950 \cos (8 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+4095 \cos (10 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-515970 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-859950 \cos (2 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+491400 \cos (4 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-184275 \cos (6 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+40950 \cos (8 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-4095 \cos (10 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{41287680 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.086, size = 240, normalized size = 1.1 \begin{align*} -{\frac{13\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{80\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{13\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{480\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{13\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{1920\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{13\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{1280\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{13\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{1280\,d}}-{\frac{13\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{768\,d}}-{\frac{13\,{a}^{2}\cos \left ( dx+c \right ) }{256\,d}}-{\frac{13\,{a}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{256\,d}}-{\frac{2\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{9\,d \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{4\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{63\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{10\,d \left ( \sin \left ( dx+c \right ) \right ) ^{10}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06939, size = 369, normalized size = 1.62 \begin{align*} -\frac{63 \, 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 )} + 210 \, a^{2}{\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{5120 \,{\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a^{2}}{\tan \left (d x + c\right )^{9}}}{161280 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.21268, size = 855, normalized size = 3.75 \begin{align*} -\frac{8190 \, a^{2} \cos \left (d x + c\right )^{9} + 15540 \, a^{2} \cos \left (d x + c\right )^{7} - 69888 \, a^{2} \cos \left (d x + c\right )^{5} + 38220 \, a^{2} \cos \left (d x + c\right )^{3} - 8190 \, a^{2} \cos \left (d x + c\right ) - 4095 \,{\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 ) + 4095 \,{\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 ) + 5120 \,{\left (2 \, a^{2} \cos \left (d x + c\right )^{9} - 9 \, a^{2} \cos \left (d x + c\right )^{7}\right )} \sin \left (d x + c\right )}{161280 \,{\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.38668, size = 437, normalized size = 1.92 \begin{align*} \frac{126 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 560 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 315 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 2160 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 3990 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 7560 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 13440 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 11340 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 65520 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 30240 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{191906 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 30240 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 11340 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 13440 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 7560 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 3990 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 2160 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 315 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 560 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 126 \, a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10}}}{1290240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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