Optimal. Leaf size=270 \[ -\frac{3 a^3 \cot ^{11}(c+d x)}{11 d}-\frac{7 a^3 \cot ^9(c+d x)}{9 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{41 a^3 \tanh ^{-1}(\cos (c+d x))}{1024 d}-\frac{a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}+\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}-\frac{35 a^3 \cot (c+d x) \csc ^5(c+d x)}{384 d}+\frac{41 a^3 \cot (c+d x) \csc ^3(c+d x)}{1536 d}+\frac{41 a^3 \cot (c+d x) \csc (c+d x)}{1024 d} \]
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Rubi [A] time = 0.464142, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2873, 2607, 14, 2611, 3768, 3770, 270} \[ -\frac{3 a^3 \cot ^{11}(c+d x)}{11 d}-\frac{7 a^3 \cot ^9(c+d x)}{9 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{41 a^3 \tanh ^{-1}(\cos (c+d x))}{1024 d}-\frac{a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}+\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}-\frac{35 a^3 \cot (c+d x) \csc ^5(c+d x)}{384 d}+\frac{41 a^3 \cot (c+d x) \csc ^3(c+d x)}{1536 d}+\frac{41 a^3 \cot (c+d x) \csc (c+d x)}{1024 d} \]
Antiderivative was successfully verified.
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Rule 2873
Rule 2607
Rule 14
Rule 2611
Rule 3768
Rule 3770
Rule 270
Rubi steps
\begin{align*} \int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (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)+3 a^3 \cot ^6(c+d x) \csc ^6(c+d x)+a^3 \cot ^6(c+d x) \csc ^7(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^6(c+d x) \csc ^4(c+d x) \, dx+a^3 \int \cot ^6(c+d x) \csc ^7(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^5(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^6(c+d x) \, dx\\ &=-\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 ^7(c+d x)}{12 d}-\frac{1}{12} \left (5 a^3\right ) \int \cot ^4(c+d x) \csc ^7(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 ) \, 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 )^2 \, dx,x,-\cot (c+d x)\right )}{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{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac{a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}+\frac{1}{8} a^3 \int \cot ^2(c+d x) \csc ^7(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+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (x^6+2 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{4 a^3 \cot ^7(c+d x)}{7 d}-\frac{7 a^3 \cot ^9(c+d x)}{9 d}-\frac{3 a^3 \cot ^{11}(c+d x)}{11 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{a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac{a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac{1}{64} a^3 \int \csc ^7(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{7 a^3 \cot ^9(c+d x)}{9 d}-\frac{3 a^3 \cot ^{11}(c+d x)}{11 d}+\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac{35 a^3 \cot (c+d x) \csc ^5(c+d x)}{384 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{a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac{a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac{1}{384} \left (5 a^3\right ) \int \csc ^5(c+d x) \, dx-\frac{1}{128} \left (9 a^3\right ) \int \csc ^3(c+d x) \, dx\\ &=-\frac{4 a^3 \cot ^7(c+d x)}{7 d}-\frac{7 a^3 \cot ^9(c+d x)}{9 d}-\frac{3 a^3 \cot ^{11}(c+d x)}{11 d}+\frac{9 a^3 \cot (c+d x) \csc (c+d x)}{256 d}+\frac{41 a^3 \cot (c+d x) \csc ^3(c+d x)}{1536 d}-\frac{35 a^3 \cot (c+d x) \csc ^5(c+d x)}{384 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{a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac{a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac{1}{512} \left (5 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac{1}{256} \left (9 a^3\right ) \int \csc (c+d x) \, dx\\ &=\frac{9 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}-\frac{7 a^3 \cot ^9(c+d x)}{9 d}-\frac{3 a^3 \cot ^{11}(c+d x)}{11 d}+\frac{41 a^3 \cot (c+d x) \csc (c+d x)}{1024 d}+\frac{41 a^3 \cot (c+d x) \csc ^3(c+d x)}{1536 d}-\frac{35 a^3 \cot (c+d x) \csc ^5(c+d x)}{384 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{a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac{a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac{\left (5 a^3\right ) \int \csc (c+d x) \, dx}{1024}\\ &=\frac{41 a^3 \tanh ^{-1}(\cos (c+d x))}{1024 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}-\frac{7 a^3 \cot ^9(c+d x)}{9 d}-\frac{3 a^3 \cot ^{11}(c+d x)}{11 d}+\frac{41 a^3 \cot (c+d x) \csc (c+d x)}{1024 d}+\frac{41 a^3 \cot (c+d x) \csc ^3(c+d x)}{1536 d}-\frac{35 a^3 \cot (c+d x) \csc ^5(c+d x)}{384 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{a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}-\frac{a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}\\ \end{align*}
Mathematica [A] time = 4.70877, size = 197, normalized size = 0.73 \[ \frac{a^3 (\sin (c+d x)+1)^3 \left (72737280 \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )-\cot (c+d x) \csc ^{11}(c+d x) (49776640 \sin (c+d x)+84039680 \sin (3 (c+d x))+38118400 \sin (5 (c+d x))+2206720 \sin (7 (c+d x))-1530880 \sin (9 (c+d x))+117760 \sin (11 (c+d x))+62609778 \cos (2 (c+d x))+22551144 \cos (4 (c+d x))-23426403 \cos (6 (c+d x))-1799490 \cos (8 (c+d x))+142065 \cos (10 (c+d x))+91311066)\right )}{1816657920 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.1, size = 288, normalized size = 1.1 \begin{align*} -{\frac{23\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{99\,d \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{46\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{693\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{41\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{120\,d \left ( \sin \left ( dx+c \right ) \right ) ^{10}}}-{\frac{41\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{320\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{41\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{1920\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{41\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{7680\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{41\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{5120\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{41\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{5120\,d}}-{\frac{41\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3072\,d}}-{\frac{41\,{a}^{3}\cos \left ( dx+c \right ) }{1024\,d}}-{\frac{41\,{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{1024\,d}}-{\frac{3\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{11\,d \left ( \sin \left ( dx+c \right ) \right ) ^{11}}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{12\,d \left ( \sin \left ( dx+c \right ) \right ) ^{12}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03957, size = 470, normalized size = 1.74 \begin{align*} -\frac{1155 \, a^{3}{\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 )} + 8316 \, 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 )} + \frac{112640 \,{\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a^{3}}{\tan \left (d x + c\right )^{9}} + \frac{30720 \,{\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}}}{7096320 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33993, size = 1053, normalized size = 3.9 \begin{align*} -\frac{284130 \, a^{3} \cos \left (d x + c\right )^{11} - 1610070 \, a^{3} \cos \left (d x + c\right )^{9} - 507276 \, a^{3} \cos \left (d x + c\right )^{7} + 3750516 \, a^{3} \cos \left (d x + c\right )^{5} - 1610070 \, a^{3} \cos \left (d x + c\right )^{3} + 284130 \, a^{3} \cos \left (d x + c\right ) - 142065 \,{\left (a^{3} \cos \left (d x + c\right )^{12} - 6 \, a^{3} \cos \left (d x + c\right )^{10} + 15 \, a^{3} \cos \left (d x + c\right )^{8} - 20 \, a^{3} \cos \left (d x + c\right )^{6} + 15 \, a^{3} \cos \left (d x + c\right )^{4} - 6 \, 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 ) + 142065 \,{\left (a^{3} \cos \left (d x + c\right )^{12} - 6 \, a^{3} \cos \left (d x + c\right )^{10} + 15 \, a^{3} \cos \left (d x + c\right )^{8} - 20 \, a^{3} \cos \left (d x + c\right )^{6} + 15 \, a^{3} \cos \left (d x + c\right )^{4} - 6 \, 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 ) + 10240 \,{\left (46 \, a^{3} \cos \left (d x + c\right )^{11} - 253 \, a^{3} \cos \left (d x + c\right )^{9} + 396 \, a^{3} \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 )}} \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 = 1.56032, size = 567, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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