Optimal. Leaf size=270 \[ -\frac {2 a^2 \cot ^{11}(c+d x)}{11 d}-\frac {4 a^2 \cot ^9(c+d x)}{9 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}+\frac {17 a^2 \tanh ^{-1}(\cos (c+d x))}{1024 d}-\frac {a^2 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}+\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^2 \cot (c+d x) \csc ^7(c+d x)}{64 d}-\frac {11 a^2 \cot (c+d x) \csc ^5(c+d x)}{384 d}+\frac {17 a^2 \cot (c+d x) \csc ^3(c+d x)}{1536 d}+\frac {17 a^2 \cot (c+d x) \csc (c+d x)}{1024 d} \]
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Rubi [A] time = 0.43, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 18, 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, 270} \[ -\frac {2 a^2 \cot ^{11}(c+d x)}{11 d}-\frac {4 a^2 \cot ^9(c+d x)}{9 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}+\frac {17 a^2 \tanh ^{-1}(\cos (c+d x))}{1024 d}-\frac {a^2 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}+\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^2 \cot (c+d x) \csc ^7(c+d x)}{64 d}-\frac {11 a^2 \cot (c+d x) \csc ^5(c+d x)}{384 d}+\frac {17 a^2 \cot (c+d x) \csc ^3(c+d x)}{1536 d}+\frac {17 a^2 \cot (c+d x) \csc (c+d x)}{1024 d} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2607
Rule 2611
Rule 2873
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx &=\int \left (a^2 \cot ^6(c+d x) \csc ^5(c+d x)+2 a^2 \cot ^6(c+d x) \csc ^6(c+d x)+a^2 \cot ^6(c+d x) \csc ^7(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^6(c+d x) \csc ^5(c+d x) \, dx+a^2 \int \cot ^6(c+d x) \csc ^7(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^6(c+d x) \csc ^6(c+d x) \, dx\\ &=-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {a^2 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac {1}{12} \left (5 a^2\right ) \int \cot ^4(c+d x) \csc ^7(c+d x) \, dx-\frac {1}{2} a^2 \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx+\frac {\left (2 a^2\right ) \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)}{16 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}+\frac {1}{8} a^2 \int \cot ^2(c+d x) \csc ^7(c+d x) \, dx+\frac {1}{16} \left (3 a^2\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac {\left (2 a^2\right ) \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 {4 a^2 \cot ^9(c+d x)}{9 d}-\frac {2 a^2 \cot ^{11}(c+d x)}{11 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 {a^2 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac {a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac {1}{64} a^2 \int \csc ^7(c+d x) \, dx-\frac {1}{32} a^2 \int \csc ^5(c+d x) \, dx\\ &=-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {4 a^2 \cot ^9(c+d x)}{9 d}-\frac {2 a^2 \cot ^{11}(c+d x)}{11 d}+\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac {11 a^2 \cot (c+d x) \csc ^5(c+d x)}{384 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 {a^2 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac {a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac {1}{384} \left (5 a^2\right ) \int \csc ^5(c+d x) \, dx-\frac {1}{128} \left (3 a^2\right ) \int \csc ^3(c+d x) \, dx\\ &=-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {4 a^2 \cot ^9(c+d x)}{9 d}-\frac {2 a^2 \cot ^{11}(c+d x)}{11 d}+\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{256 d}+\frac {17 a^2 \cot (c+d x) \csc ^3(c+d x)}{1536 d}-\frac {11 a^2 \cot (c+d x) \csc ^5(c+d x)}{384 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 {a^2 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac {a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac {1}{512} \left (5 a^2\right ) \int \csc ^3(c+d x) \, dx-\frac {1}{256} \left (3 a^2\right ) \int \csc (c+d x) \, dx\\ &=\frac {3 a^2 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {4 a^2 \cot ^9(c+d x)}{9 d}-\frac {2 a^2 \cot ^{11}(c+d x)}{11 d}+\frac {17 a^2 \cot (c+d x) \csc (c+d x)}{1024 d}+\frac {17 a^2 \cot (c+d x) \csc ^3(c+d x)}{1536 d}-\frac {11 a^2 \cot (c+d x) \csc ^5(c+d x)}{384 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 {a^2 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac {a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac {\left (5 a^2\right ) \int \csc (c+d x) \, dx}{1024}\\ &=\frac {17 a^2 \tanh ^{-1}(\cos (c+d x))}{1024 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {4 a^2 \cot ^9(c+d x)}{9 d}-\frac {2 a^2 \cot ^{11}(c+d x)}{11 d}+\frac {17 a^2 \cot (c+d x) \csc (c+d x)}{1024 d}+\frac {17 a^2 \cot (c+d x) \csc ^3(c+d x)}{1536 d}-\frac {11 a^2 \cot (c+d x) \csc ^5(c+d x)}{384 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 {a^2 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac {a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}\\ \end {align*}
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Mathematica [A] time = 4.72, size = 197, normalized size = 0.73 \[ \frac {a^2 (\sin (c+d x)+1)^2 \left (30159360 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-\cot (c+d x) \csc ^{11}(c+d x) (29655040 \sin (c+d x)+51445760 \sin (3 (c+d x))+25600000 \sin (5 (c+d x))+3235840 \sin (7 (c+d x))-532480 \sin (9 (c+d x))+40960 \sin (11 (c+d x))+67499586 \cos (2 (c+d x))+25966248 \cos (4 (c+d x))-6944091 \cos (6 (c+d x))-746130 \cos (8 (c+d x))+58905 \cos (10 (c+d x))+65553642)\right )}{1816657920 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 384, normalized size = 1.42 \[ -\frac {117810 \, a^{2} \cos \left (d x + c\right )^{11} - 667590 \, a^{2} \cos \left (d x + c\right )^{9} + 135828 \, a^{2} \cos \left (d x + c\right )^{7} + 1555092 \, a^{2} \cos \left (d x + c\right )^{5} - 667590 \, a^{2} \cos \left (d x + c\right )^{3} + 117810 \, a^{2} \cos \left (d x + c\right ) - 58905 \, {\left (a^{2} \cos \left (d x + c\right )^{12} - 6 \, a^{2} \cos \left (d x + c\right )^{10} + 15 \, a^{2} \cos \left (d x + c\right )^{8} - 20 \, a^{2} \cos \left (d x + c\right )^{6} + 15 \, a^{2} \cos \left (d x + c\right )^{4} - 6 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 58905 \, {\left (a^{2} \cos \left (d x + c\right )^{12} - 6 \, a^{2} \cos \left (d x + c\right )^{10} + 15 \, a^{2} \cos \left (d x + c\right )^{8} - 20 \, a^{2} \cos \left (d x + c\right )^{6} + 15 \, a^{2} \cos \left (d x + c\right )^{4} - 6 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 20480 \, {\left (8 \, a^{2} \cos \left (d x + c\right )^{11} - 44 \, a^{2} \cos \left (d x + c\right )^{9} + 99 \, a^{2} \cos \left (d x + c\right )^{7}\right )} \sin \left (d x + c\right )}{7096320 \, {\left (d \cos \left (d x + c\right )^{12} - 6 \, d \cos \left (d x + c\right )^{10} + 15 \, d \cos \left (d x + c\right )^{8} - 20 \, d \cos \left (d x + c\right )^{6} + 15 \, d \cos \left (d x + c\right )^{4} - 6 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 420, normalized size = 1.56 \[ \frac {1155 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 5040 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 5544 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 6160 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 24255 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 39600 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 27720 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 55440 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 162855 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 184800 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 55440 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 942480 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 554400 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {2924714 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 554400 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 55440 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 184800 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 162855 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 55440 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 27720 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 39600 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 24255 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 6160 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5544 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 5040 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1155 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12}}}{56770560 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 288, normalized size = 1.07 \[ -\frac {17 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{120 d \sin \left (d x +c \right )^{10}}-\frac {17 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{320 d \sin \left (d x +c \right )^{8}}-\frac {17 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{1920 d \sin \left (d x +c \right )^{6}}+\frac {17 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{7680 d \sin \left (d x +c \right )^{4}}-\frac {17 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{5120 d \sin \left (d x +c \right )^{2}}-\frac {17 a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{5120 d}-\frac {17 a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{3072 d}-\frac {17 a^{2} \cos \left (d x +c \right )}{1024 d}-\frac {17 a^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{1024 d}-\frac {2 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{11 d \sin \left (d x +c \right )^{11}}-\frac {8 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{99 d \sin \left (d x +c \right )^{9}}-\frac {16 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{693 d \sin \left (d x +c \right )^{7}}-\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{12 d \sin \left (d x +c \right )^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 323, normalized size = 1.20 \[ -\frac {1155 \, a^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{11} - 85 \, \cos \left (d x + c\right )^{9} + 198 \, \cos \left (d x + c\right )^{7} + 198 \, \cos \left (d x + c\right )^{5} - 85 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{12} - 6 \, \cos \left (d x + c\right )^{10} + 15 \, \cos \left (d x + c\right )^{8} - 20 \, \cos \left (d x + c\right )^{6} + 15 \, \cos \left (d x + c\right )^{4} - 6 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 2772 \, a^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {20480 \, {\left (99 \, \tan \left (d x + c\right )^{4} + 154 \, \tan \left (d x + c\right )^{2} + 63\right )} a^{2}}{\tan \left (d x + c\right )^{11}}}{7096320 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.36, size = 471, normalized size = 1.74 \[ \frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}-\frac {5\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{1536\,d}-\frac {47\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{16384\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{1024\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {5\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{7168\,d}+\frac {7\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{16384\,d}+\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{9216\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{11264\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{49152\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {5\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{1536\,d}+\frac {47\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{16384\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{1024\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}-\frac {5\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{7168\,d}-\frac {7\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{16384\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{9216\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{11264\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{49152\,d}-\frac {17\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{1024\,d}+\frac {5\,a^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{512\,d}-\frac {5\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{512\,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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