Optimal. Leaf size=246 \[ -\frac{a^3 \cot ^{11}(c+d x)}{11 d}-\frac{5 a^3 \cot ^9(c+d x)}{9 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{19 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{3 a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}-\frac{7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{19 a^3 \cot (c+d x) \csc (c+d x)}{256 d} \]
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Rubi [A] time = 0.435741, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2873, 2611, 3768, 3770, 2607, 14, 270} \[ -\frac{a^3 \cot ^{11}(c+d x)}{11 d}-\frac{5 a^3 \cot ^9(c+d x)}{9 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{19 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{3 a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}-\frac{7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{19 a^3 \cot (c+d x) \csc (c+d x)}{256 d} \]
Antiderivative was successfully verified.
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Rule 2873
Rule 2611
Rule 3768
Rule 3770
Rule 2607
Rule 14
Rule 270
Rubi steps
\begin{align*} \int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (a^3 \cot ^6(c+d x) \csc ^3(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^4(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^5(c+d x)+a^3 \cot ^6(c+d x) \csc ^6(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^6(c+d x) \csc ^3(c+d x) \, dx+a^3 \int \cot ^6(c+d x) \csc ^6(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^4(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^5(c+d x) \, dx\\ &=-\frac{a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{1}{8} \left (5 a^3\right ) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx-\frac{1}{2} \left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx+\frac{a^3 \operatorname{Subst}\left (\int x^6 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{d}+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac{1}{16} \left (5 a^3\right ) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\frac{1}{16} \left (9 a^3\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac{a^3 \operatorname{Subst}\left (\int \left (x^6+2 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{4 a^3 \cot ^7(c+d x)}{7 d}-\frac{5 a^3 \cot ^9(c+d x)}{9 d}-\frac{a^3 \cot ^{11}(c+d x)}{11 d}-\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{3 a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{1}{64} \left (5 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac{1}{32} \left (3 a^3\right ) \int \csc ^5(c+d x) \, dx\\ &=-\frac{4 a^3 \cot ^7(c+d x)}{7 d}-\frac{5 a^3 \cot ^9(c+d x)}{9 d}-\frac{a^3 \cot ^{11}(c+d x)}{11 d}+\frac{5 a^3 \cot (c+d x) \csc (c+d x)}{128 d}-\frac{7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{3 a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{1}{128} \left (5 a^3\right ) \int \csc (c+d x) \, dx-\frac{1}{128} \left (9 a^3\right ) \int \csc ^3(c+d x) \, dx\\ &=\frac{5 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}-\frac{5 a^3 \cot ^9(c+d x)}{9 d}-\frac{a^3 \cot ^{11}(c+d x)}{11 d}+\frac{19 a^3 \cot (c+d x) \csc (c+d x)}{256 d}-\frac{7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{3 a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{1}{256} \left (9 a^3\right ) \int \csc (c+d x) \, dx\\ &=\frac{19 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}-\frac{5 a^3 \cot ^9(c+d x)}{9 d}-\frac{a^3 \cot ^{11}(c+d x)}{11 d}+\frac{19 a^3 \cot (c+d x) \csc (c+d x)}{256 d}-\frac{7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{3 a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}\\ \end{align*}
Mathematica [A] time = 3.44735, size = 187, normalized size = 0.76 \[ \frac{a^3 (\sin (c+d x)+1)^3 \left (16853760 \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )-\cot (c+d x) \csc ^{10}(c+d x) (14477694 \sin (c+d x)+5875716 \sin (3 (c+d x))+7902972 \sin (5 (c+d x))-414645 \sin (7 (c+d x))-65835 \sin (9 (c+d x))+12423680 \cos (2 (c+d x))+839680 \cos (4 (c+d x))-2149120 \cos (6 (c+d x))-568320 \cos (8 (c+d x))+47360 \cos (10 (c+d x))+10050560)\right )}{227082240 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.098, size = 264, normalized size = 1.1 \begin{align*} -{\frac{19\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{80\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{19\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{480\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{19\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{1920\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{19\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{1280\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{19\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{1280\,d}}-{\frac{19\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{768\,d}}-{\frac{19\,{a}^{3}\cos \left ( dx+c \right ) }{256\,d}}-{\frac{19\,{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{256\,d}}-{\frac{37\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{99\,d \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{74\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{693\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{3\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{10\,d \left ( \sin \left ( dx+c \right ) \right ) ^{10}}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{11\,d \left ( \sin \left ( dx+c \right ) \right ) ^{11}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08224, size = 416, normalized size = 1.69 \begin{align*} -\frac{2079 \, a^{3}{\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 )} + 2310 \, a^{3}{\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{84480 \,{\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a^{3}}{\tan \left (d x + c\right )^{9}} + \frac{2560 \,{\left (99 \, \tan \left (d x + c\right )^{4} + 154 \, \tan \left (d x + c\right )^{2} + 63\right )} a^{3}}{\tan \left (d x + c\right )^{11}}}{1774080 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33549, size = 957, normalized size = 3.89 \begin{align*} \frac{189440 \, a^{3} \cos \left (d x + c\right )^{11} - 1041920 \, a^{3} \cos \left (d x + c\right )^{9} + 1013760 \, a^{3} \cos \left (d x + c\right )^{7} + 65835 \,{\left (a^{3} \cos \left (d x + c\right )^{10} - 5 \, a^{3} \cos \left (d x + c\right )^{8} + 10 \, a^{3} \cos \left (d x + c\right )^{6} - 10 \, a^{3} \cos \left (d x + c\right )^{4} + 5 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 65835 \,{\left (a^{3} \cos \left (d x + c\right )^{10} - 5 \, a^{3} \cos \left (d x + c\right )^{8} + 10 \, a^{3} \cos \left (d x + c\right )^{6} - 10 \, a^{3} \cos \left (d x + c\right )^{4} + 5 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 462 \,{\left (285 \, a^{3} \cos \left (d x + c\right )^{9} - 50 \, a^{3} \cos \left (d x + c\right )^{7} - 2432 \, a^{3} \cos \left (d x + c\right )^{5} + 1330 \, a^{3} \cos \left (d x + c\right )^{3} - 285 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1774080 \,{\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 )} \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 = 1.3471, size = 524, normalized size = 2.13 \begin{align*} \frac{630 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 4158 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 8470 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 3465 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 40590 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 57750 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 6930 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 138600 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 244860 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 152460 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1053360 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 568260 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{3181018 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 568260 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} - 152460 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 244860 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 138600 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 6930 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 57750 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 40590 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 3465 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 8470 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 4158 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 630 \, a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11}}}{14192640 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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