Optimal. Leaf size=84 \[ -\frac{\cot ^5(c+d x) (a \sec (c+d x)+a)}{5 d}+\frac{\cot ^3(c+d x) (4 a \sec (c+d x)+5 a)}{15 d}-\frac{\cot (c+d x) (8 a \sec (c+d x)+15 a)}{15 d}-a x \]
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Rubi [A] time = 0.0809355, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3882, 8} \[ -\frac{\cot ^5(c+d x) (a \sec (c+d x)+a)}{5 d}+\frac{\cot ^3(c+d x) (4 a \sec (c+d x)+5 a)}{15 d}-\frac{\cot (c+d x) (8 a \sec (c+d x)+15 a)}{15 d}-a x \]
Antiderivative was successfully verified.
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Rule 3882
Rule 8
Rubi steps
\begin{align*} \int \cot ^6(c+d x) (a+a \sec (c+d x)) \, dx &=-\frac{\cot ^5(c+d x) (a+a \sec (c+d x))}{5 d}+\frac{1}{5} \int \cot ^4(c+d x) (-5 a-4 a \sec (c+d x)) \, dx\\ &=-\frac{\cot ^5(c+d x) (a+a \sec (c+d x))}{5 d}+\frac{\cot ^3(c+d x) (5 a+4 a \sec (c+d x))}{15 d}+\frac{1}{15} \int \cot ^2(c+d x) (15 a+8 a \sec (c+d x)) \, dx\\ &=-\frac{\cot ^5(c+d x) (a+a \sec (c+d x))}{5 d}+\frac{\cot ^3(c+d x) (5 a+4 a \sec (c+d x))}{15 d}-\frac{\cot (c+d x) (15 a+8 a \sec (c+d x))}{15 d}+\frac{1}{15} \int -15 a \, dx\\ &=-a x-\frac{\cot ^5(c+d x) (a+a \sec (c+d x))}{5 d}+\frac{\cot ^3(c+d x) (5 a+4 a \sec (c+d x))}{15 d}-\frac{\cot (c+d x) (15 a+8 a \sec (c+d x))}{15 d}\\ \end{align*}
Mathematica [C] time = 0.0495894, size = 79, normalized size = 0.94 \[ -\frac{a \cot ^5(c+d x) \text{Hypergeometric2F1}\left (-\frac{5}{2},1,-\frac{3}{2},-\tan ^2(c+d x)\right )}{5 d}-\frac{a \csc ^5(c+d x)}{5 d}+\frac{2 a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.067, size = 129, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ( a \left ( -{\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 ) ^{6}}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{15\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{5\,\sin \left ( dx+c \right ) }}-{\frac{\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) } \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.65846, size = 107, normalized size = 1.27 \begin{align*} -\frac{{\left (15 \, d x + 15 \, c + \frac{15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a + \frac{{\left (15 \, \sin \left (d x + c\right )^{4} - 10 \, \sin \left (d x + c\right )^{2} + 3\right )} a}{\sin \left (d x + c\right )^{5}}}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.986682, size = 356, normalized size = 4.24 \begin{align*} -\frac{23 \, a \cos \left (d x + c\right )^{4} - 8 \, a \cos \left (d x + c\right )^{3} - 27 \, a \cos \left (d x + c\right )^{2} + 7 \, a \cos \left (d x + c\right ) + 15 \,{\left (a d x \cos \left (d x + c\right )^{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 ) + 8 \, a}{15 \,{\left (d \cos \left (d x + c\right )^{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 = 1.37042, size = 112, normalized size = 1.33 \begin{align*} -\frac{5 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 240 \,{\left (d x + c\right )} a - 90 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{3 \,{\left (80 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 10 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a\right )}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}}}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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