Optimal. Leaf size=152 \[ -\frac {a^2 \cot ^9(c+d x)}{9 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}+\frac {5 a^2 \tanh ^{-1}(\cos (c+d x))}{64 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{4 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{24 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}+\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{64 d} \]
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Rubi [A] time = 0.29, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2873, 2607, 30, 2611, 3768, 3770, 14} \[ -\frac {a^2 \cot ^9(c+d x)}{9 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}+\frac {5 a^2 \tanh ^{-1}(\cos (c+d x))}{64 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{4 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{24 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}+\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{64 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 2607
Rule 2611
Rule 2873
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx &=\int \left (a^2 \cot ^6(c+d x) \csc ^2(c+d x)+2 a^2 \cot ^6(c+d x) \csc ^3(c+d x)+a^2 \cot ^6(c+d x) \csc ^4(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx+a^2 \int \cot ^6(c+d x) \csc ^4(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^6(c+d x) \csc ^3(c+d x) \, dx\\ &=-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{4 d}-\frac {1}{4} \left (5 a^2\right ) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx+\frac {a^2 \operatorname {Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{d}+\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 \cot ^7(c+d x)}{7 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{4 d}+\frac {1}{8} \left (5 a^2\right ) \int \cot ^2(c+d x) \csc ^3(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 {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^9(c+d x)}{9 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{4 d}-\frac {1}{32} \left (5 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)}{9 d}+\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{64 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{4 d}-\frac {1}{64} \left (5 a^2\right ) \int \csc (c+d x) \, dx\\ &=\frac {5 a^2 \tanh ^{-1}(\cos (c+d x))}{64 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^9(c+d x)}{9 d}+\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{64 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{4 d}\\ \end {align*}
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Mathematica [B] time = 1.63, size = 313, normalized size = 2.06 \[ -\frac {a^2 \csc ^9(c+d x) \left (36540 \sin (2 (c+d x))+20916 \sin (4 (c+d x))+16044 \sin (6 (c+d x))+630 \sin (8 (c+d x))+72576 \cos (c+d x)+37632 \cos (3 (c+d x))+6912 \cos (5 (c+d x))-1728 \cos (7 (c+d x))-704 \cos (9 (c+d x))+39690 \sin (c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-26460 \sin (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+11340 \sin (5 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-2835 \sin (7 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+315 \sin (9 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-39690 \sin (c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+26460 \sin (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-11340 \sin (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+2835 \sin (7 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-315 \sin (9 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{1032192 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.77, size = 291, normalized size = 1.91 \[ \frac {1408 \, a^{2} \cos \left (d x + c\right )^{9} - 2304 \, a^{2} \cos \left (d x + c\right )^{7} + 315 \, {\left (a^{2} \cos \left (d x + c\right )^{8} - 4 \, a^{2} \cos \left (d x + c\right )^{6} + 6 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 315 \, {\left (a^{2} \cos \left (d x + c\right )^{8} - 4 \, a^{2} \cos \left (d x + c\right )^{6} + 6 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 42 \, {\left (15 \, a^{2} \cos \left (d x + c\right )^{7} + 73 \, a^{2} \cos \left (d x + c\right )^{5} - 55 \, a^{2} \cos \left (d x + c\right )^{3} + 15 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8064 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.34, size = 324, normalized size = 2.13 \[ \frac {14 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 63 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 18 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 336 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 504 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 504 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1848 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1008 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 5040 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 3276 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {14258 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 3276 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1008 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1848 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 504 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 504 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 336 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 18 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 63 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 14 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{64512 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 216, normalized size = 1.42 \[ -\frac {11 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{63 d \sin \left (d x +c \right )^{7}}-\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{4 d \sin \left (d x +c \right )^{8}}-\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{24 d \sin \left (d x +c \right )^{6}}+\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{96 d \sin \left (d x +c \right )^{4}}-\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{64 d \sin \left (d x +c \right )^{2}}-\frac {a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{64 d}-\frac {5 a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{192 d}-\frac {5 a^{2} \cos \left (d x +c \right )}{64 d}-\frac {5 a^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{64 d}-\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{9 d \sin \left (d x +c \right )^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 155, normalized size = 1.02 \[ -\frac {21 \, a^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {1152 \, a^{2}}{\tan \left (d x + c\right )^{7}} + \frac {128 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a^{2}}{\tan \left (d x + c\right )^{9}}}{8064 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.41, size = 357, normalized size = 2.35 \[ \frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{128\,d}-\frac {11\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}+\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3584\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{1024\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4608\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}+\frac {11\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{128\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3584\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{1024\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4608\,d}-\frac {5\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{64\,d}+\frac {13\,a^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d}-\frac {13\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d} \]
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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