Optimal. Leaf size=140 \[ -\frac{\cot ^9(c+d x) (a \sec (c+d x)+a)}{9 d}+\frac{\cot ^7(c+d x) (8 a \sec (c+d x)+9 a)}{63 d}-\frac{\cot ^5(c+d x) (16 a \sec (c+d x)+21 a)}{105 d}+\frac{\cot ^3(c+d x) (64 a \sec (c+d x)+105 a)}{315 d}-\frac{\cot (c+d x) (128 a \sec (c+d x)+315 a)}{315 d}-a x \]
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Rubi [A] time = 0.144726, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3882, 8} \[ -\frac{\cot ^9(c+d x) (a \sec (c+d x)+a)}{9 d}+\frac{\cot ^7(c+d x) (8 a \sec (c+d x)+9 a)}{63 d}-\frac{\cot ^5(c+d x) (16 a \sec (c+d x)+21 a)}{105 d}+\frac{\cot ^3(c+d x) (64 a \sec (c+d x)+105 a)}{315 d}-\frac{\cot (c+d x) (128 a \sec (c+d x)+315 a)}{315 d}-a x \]
Antiderivative was successfully verified.
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Rule 3882
Rule 8
Rubi steps
\begin{align*} \int \cot ^{10}(c+d x) (a+a \sec (c+d x)) \, dx &=-\frac{\cot ^9(c+d x) (a+a \sec (c+d x))}{9 d}+\frac{1}{9} \int \cot ^8(c+d x) (-9 a-8 a \sec (c+d x)) \, dx\\ &=-\frac{\cot ^9(c+d x) (a+a \sec (c+d x))}{9 d}+\frac{\cot ^7(c+d x) (9 a+8 a \sec (c+d x))}{63 d}+\frac{1}{63} \int \cot ^6(c+d x) (63 a+48 a \sec (c+d x)) \, dx\\ &=-\frac{\cot ^9(c+d x) (a+a \sec (c+d x))}{9 d}+\frac{\cot ^7(c+d x) (9 a+8 a \sec (c+d x))}{63 d}-\frac{\cot ^5(c+d x) (21 a+16 a \sec (c+d x))}{105 d}+\frac{1}{315} \int \cot ^4(c+d x) (-315 a-192 a \sec (c+d x)) \, dx\\ &=-\frac{\cot ^9(c+d x) (a+a \sec (c+d x))}{9 d}+\frac{\cot ^7(c+d x) (9 a+8 a \sec (c+d x))}{63 d}-\frac{\cot ^5(c+d x) (21 a+16 a \sec (c+d x))}{105 d}+\frac{\cot ^3(c+d x) (105 a+64 a \sec (c+d x))}{315 d}+\frac{1}{945} \int \cot ^2(c+d x) (945 a+384 a \sec (c+d x)) \, dx\\ &=-\frac{\cot ^9(c+d x) (a+a \sec (c+d x))}{9 d}+\frac{\cot ^7(c+d x) (9 a+8 a \sec (c+d x))}{63 d}-\frac{\cot ^5(c+d x) (21 a+16 a \sec (c+d x))}{105 d}+\frac{\cot ^3(c+d x) (105 a+64 a \sec (c+d x))}{315 d}-\frac{\cot (c+d x) (315 a+128 a \sec (c+d x))}{315 d}+\frac{1}{945} \int -945 a \, dx\\ &=-a x-\frac{\cot ^9(c+d x) (a+a \sec (c+d x))}{9 d}+\frac{\cot ^7(c+d x) (9 a+8 a \sec (c+d x))}{63 d}-\frac{\cot ^5(c+d x) (21 a+16 a \sec (c+d x))}{105 d}+\frac{\cot ^3(c+d x) (105 a+64 a \sec (c+d x))}{315 d}-\frac{\cot (c+d x) (315 a+128 a \sec (c+d x))}{315 d}\\ \end{align*}
Mathematica [C] time = 0.0636214, size = 111, normalized size = 0.79 \[ -\frac{a \cot ^9(c+d x) \text{Hypergeometric2F1}\left (-\frac{9}{2},1,-\frac{7}{2},-\tan ^2(c+d x)\right )}{9 d}-\frac{a \csc ^9(c+d x)}{9 d}+\frac{4 a \csc ^7(c+d x)}{7 d}-\frac{6 a \csc ^5(c+d x)}{5 d}+\frac{4 a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.136, size = 205, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{9}}{9}}+{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{7}}{7}}-{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5}}+{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3}}-\cot \left ( dx+c \right ) -dx-c \right ) +a \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{10}}{9\, \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{10}}{63\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{10}}{105\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{10}}{63\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{10}}{9\,\sin \left ( dx+c \right ) }}-{\frac{\sin \left ( dx+c \right ) }{9} \left ({\frac{128}{35}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{8}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{7}}+{\frac{48\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{35}}+{\frac{64\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{35}} \right ) } \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.69648, size = 161, normalized size = 1.15 \begin{align*} -\frac{{\left (315 \, d x + 315 \, c + \frac{315 \, \tan \left (d x + c\right )^{8} - 105 \, \tan \left (d x + c\right )^{6} + 63 \, \tan \left (d x + c\right )^{4} - 45 \, \tan \left (d x + c\right )^{2} + 35}{\tan \left (d x + c\right )^{9}}\right )} a + \frac{{\left (315 \, \sin \left (d x + c\right )^{8} - 420 \, \sin \left (d x + c\right )^{6} + 378 \, \sin \left (d x + c\right )^{4} - 180 \, \sin \left (d x + c\right )^{2} + 35\right )} a}{\sin \left (d x + c\right )^{9}}}{315 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.907772, size = 745, normalized size = 5.32 \begin{align*} -\frac{563 \, a \cos \left (d x + c\right )^{8} - 248 \, a \cos \left (d x + c\right )^{7} - 1498 \, a \cos \left (d x + c\right )^{6} + 658 \, a \cos \left (d x + c\right )^{5} + 1610 \, a \cos \left (d x + c\right )^{4} - 602 \, a \cos \left (d x + c\right )^{3} - 763 \, a \cos \left (d x + c\right )^{2} + 187 \, a \cos \left (d x + c\right ) + 315 \,{\left (a d x \cos \left (d x + c\right )^{7} - a d x \cos \left (d x + c\right )^{6} - 3 \, a d x \cos \left (d x + c\right )^{5} + 3 \, a d x \cos \left (d x + c\right )^{4} + 3 \, a d x \cos \left (d x + c\right )^{3} - 3 \, a d x \cos \left (d x + c\right )^{2} - a d x \cos \left (d x + c\right ) + a d x\right )} \sin \left (d x + c\right ) + 128 \, a}{315 \,{\left (d \cos \left (d x + c\right )^{7} - d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{5} + 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{3} - 3 \, d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + 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 = 1.51179, size = 189, normalized size = 1.35 \begin{align*} -\frac{45 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 630 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 4830 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 80640 \,{\left (d x + c\right )} a - 40950 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{80640 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 13650 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 2898 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 450 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 35 \, a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9}}}{80640 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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