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 = 0.285175, antiderivative size = 152, normalized size of antiderivative = 1., 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 2873
Rule 2607
Rule 30
Rule 2611
Rule 3768
Rule 3770
Rule 14
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*}
Mathematica [B] time = 1.42304, 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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Maple [A] time = 0.09, size = 216, normalized size = 1.4 \begin{align*} -{\frac{11\,{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}}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{24\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{96\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{64\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{64\,d}}-{\frac{5\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{192\,d}}-{\frac{5\,{a}^{2}\cos \left ( dx+c \right ) }{64\,d}}-{\frac{5\,{a}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{64\,d}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{9\,d \left ( \sin \left ( dx+c \right ) \right ) ^{9}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02776, size = 209, normalized size = 1.38 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.21343, size = 756, normalized size = 4.97 \begin{align*} \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 )} \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 [B] time = 1.24185, size = 437, normalized size = 2.88 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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