Optimal. Leaf size=228 \[ -\frac{a^3 \cot ^9(c+d x)}{3 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{33 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}-\frac{29 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{33 a^3 \cot (c+d x) \csc (c+d x)}{256 d} \]
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Rubi [A] time = 0.427153, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 18, 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} \[ -\frac{a^3 \cot ^9(c+d x)}{3 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{33 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}-\frac{29 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{33 a^3 \cot (c+d x) \csc (c+d x)}{256 d} \]
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 ^5(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (a^3 \cot ^6(c+d x) \csc ^2(c+d x)+3 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)+a^3 \cot ^6(c+d x) \csc ^5(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx+a^3 \int \cot ^6(c+d x) \csc ^5(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^4(c+d x) \, dx\\ &=-\frac{3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{1}{2} a^3 \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx-\frac{1}{8} \left (15 a^3\right ) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx+\frac{a^3 \operatorname{Subst}\left (\int x^6 \, 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{a^3 \cot ^7(c+d x)}{7 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac{1}{16} \left (3 a^3\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac{1}{16} \left (15 a^3\right ) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\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{a^3 \cot ^9(c+d x)}{3 d}-\frac{15 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)}{16 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{1}{32} a^3 \int \csc ^5(c+d x) \, dx-\frac{1}{64} \left (15 a^3\right ) \int \csc ^3(c+d x) \, dx\\ &=-\frac{4 a^3 \cot ^7(c+d x)}{7 d}-\frac{a^3 \cot ^9(c+d x)}{3 d}+\frac{15 a^3 \cot (c+d x) \csc (c+d x)}{128 d}-\frac{29 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)}{16 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{1}{128} \left (3 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac{1}{128} \left (15 a^3\right ) \int \csc (c+d x) \, dx\\ &=\frac{15 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}-\frac{a^3 \cot ^9(c+d x)}{3 d}+\frac{33 a^3 \cot (c+d x) \csc (c+d x)}{256 d}-\frac{29 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)}{16 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{1}{256} \left (3 a^3\right ) \int \csc (c+d x) \, dx\\ &=\frac{33 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}-\frac{a^3 \cot ^9(c+d x)}{3 d}+\frac{33 a^3 \cot (c+d x) \csc (c+d x)}{256 d}-\frac{29 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)}{16 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}\\ \end{align*}
Mathematica [A] time = 2.04623, size = 365, normalized size = 1.6 \[ \frac{a^3 (\sin (c+d x)+1)^3 \left (-51200 \tan \left (\frac{1}{2} (c+d x)\right )+51200 \cot \left (\frac{1}{2} (c+d x)\right )+13860 \csc ^2\left (\frac{1}{2} (c+d x)\right )+42 \sec ^{10}\left (\frac{1}{2} (c+d x)\right )+315 \sec ^8\left (\frac{1}{2} (c+d x)\right )-5250 \sec ^6\left (\frac{1}{2} (c+d x)\right )+19320 \sec ^4\left (\frac{1}{2} (c+d x)\right )-13860 \sec ^2\left (\frac{1}{2} (c+d x)\right )-55440 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+55440 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+3840 \sin ^6\left (\frac{1}{2} (c+d x)\right ) \csc ^5(c+d x)+164800 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-14 (10 \sin (c+d x)+3) \csc ^{10}\left (\frac{1}{2} (c+d x)\right )+5 (172 \sin (c+d x)-63) \csc ^8\left (\frac{1}{2} (c+d x)\right )+(5250-60 \sin (c+d x)) \csc ^6\left (\frac{1}{2} (c+d x)\right )-20 (515 \sin (c+d x)+966) \csc ^4\left (\frac{1}{2} (c+d x)\right )+280 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^8\left (\frac{1}{2} (c+d x)\right )-1720 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^6\left (\frac{1}{2} (c+d x)\right )\right )}{430080 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.094, size = 240, normalized size = 1.1 \begin{align*} -{\frac{5\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{21\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{33\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{80\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{11\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{160\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{11\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{640\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{33\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{1280\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{33\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{1280\,d}}-{\frac{11\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{256\,d}}-{\frac{33\,{a}^{3}\cos \left ( dx+c \right ) }{256\,d}}-{\frac{33\,{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{256\,d}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{{a}^{3} \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 = 1.05929, size = 386, normalized size = 1.69 \begin{align*} -\frac{21 \, 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 )} + 210 \, 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{7680 \, a^{3}}{\tan \left (d x + c\right )^{7}} + \frac{2560 \,{\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a^{3}}{\tan \left (d x + c\right )^{9}}}{53760 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.26405, size = 855, normalized size = 3.75 \begin{align*} -\frac{6930 \, a^{3} \cos \left (d x + c\right )^{9} + 21420 \, a^{3} \cos \left (d x + c\right )^{7} - 59136 \, a^{3} \cos \left (d x + c\right )^{5} + 32340 \, a^{3} \cos \left (d x + c\right )^{3} - 6930 \, a^{3} \cos \left (d x + c\right ) - 3465 \,{\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 ) + 3465 \,{\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 ) + 2560 \,{\left (5 \, a^{3} \cos \left (d x + c\right )^{9} - 12 \, a^{3} \cos \left (d x + c\right )^{7}\right )} \sin \left (d x + c\right )}{53760 \,{\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 = 1.4621, size = 481, normalized size = 2.11 \begin{align*} \frac{42 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 280 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 525 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 600 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 3570 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 3360 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 5880 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 16800 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 10500 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 55440 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 31920 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{162382 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 31920 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 10500 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 16800 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 5880 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 3360 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3570 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 600 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 525 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 280 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 42 \, a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10}}}{430080 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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