Optimal. Leaf size=107 \[ -\frac{4 a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^5(c+d x)}{5 d}+\frac{7 a^3 \csc ^3(c+d x)}{3 d}-\frac{3 a^3 \csc (c+d x)}{d}-a^3 x \]
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Rubi [A] time = 0.164053, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {3886, 3473, 8, 2606, 194, 2607, 30, 14} \[ -\frac{4 a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^5(c+d x)}{5 d}+\frac{7 a^3 \csc ^3(c+d x)}{3 d}-\frac{3 a^3 \csc (c+d x)}{d}-a^3 x \]
Antiderivative was successfully verified.
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Rule 3886
Rule 3473
Rule 8
Rule 2606
Rule 194
Rule 2607
Rule 30
Rule 14
Rubi steps
\begin{align*} \int \cot ^6(c+d x) (a+a \sec (c+d x))^3 \, dx &=\int \left (a^3 \cot ^6(c+d x)+3 a^3 \cot ^5(c+d x) \csc (c+d x)+3 a^3 \cot ^4(c+d x) \csc ^2(c+d x)+a^3 \cot ^3(c+d x) \csc ^3(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^6(c+d x) \, dx+a^3 \int \cot ^3(c+d x) \csc ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^5(c+d x) \csc (c+d x) \, dx+\left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^2(c+d x) \, dx\\ &=-\frac{a^3 \cot ^5(c+d x)}{5 d}-a^3 \int \cot ^4(c+d x) \, dx-\frac{a^3 \operatorname{Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\csc (c+d x)\right )}{d}+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int x^4 \, dx,x,-\cot (c+d x)\right )}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{d}\\ &=\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}+a^3 \int \cot ^2(c+d x) \, dx-\frac{a^3 \operatorname{Subst}\left (\int \left (-x^2+x^4\right ) \, dx,x,\csc (c+d x)\right )}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac{a^3 \cot (c+d x)}{d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{3 a^3 \csc (c+d x)}{d}+\frac{7 a^3 \csc ^3(c+d x)}{3 d}-\frac{4 a^3 \csc ^5(c+d x)}{5 d}-a^3 \int 1 \, dx\\ &=-a^3 x-\frac{a^3 \cot (c+d x)}{d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{3 a^3 \csc (c+d x)}{d}+\frac{7 a^3 \csc ^3(c+d x)}{3 d}-\frac{4 a^3 \csc ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.662737, size = 112, normalized size = 1.05 \[ -\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left (\frac{1}{2} (c+d x)\right ) \left (\cot \left (\frac{c}{2}\right ) (13 \cos (c+d x)-10) \csc ^4\left (\frac{1}{2} (c+d x)\right )+\csc \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) (51 \cos (c+d x)-16 \cos (2 (c+d x))-38) \csc ^5\left (\frac{1}{2} (c+d x)\right )+60 d x\right )}{480 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.075, size = 232, normalized size = 2.2 \begin{align*}{\frac{1}{d} \left ({a}^{3} \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 ) +3\,{a}^{3} \left ( -1/5\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+1/15\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-1/5\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{\sin \left ( dx+c \right ) }}-1/5\, \left ( 8/3+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+4/3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) \right ) -{\frac{3\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{a}^{3} \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{15\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{15\,\sin \left ( dx+c \right ) }}+{\frac{ \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{15}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.76012, size = 165, normalized size = 1.54 \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^{3} + \frac{3 \,{\left (15 \, \sin \left (d x + c\right )^{4} - 10 \, \sin \left (d x + c\right )^{2} + 3\right )} a^{3}}{\sin \left (d x + c\right )^{5}} - \frac{{\left (5 \, \sin \left (d x + c\right )^{2} - 3\right )} a^{3}}{\sin \left (d x + c\right )^{5}} + \frac{9 \, a^{3}}{\tan \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 = 1.1057, size = 298, normalized size = 2.79 \begin{align*} -\frac{32 \, a^{3} \cos \left (d x + c\right )^{3} - 19 \, a^{3} \cos \left (d x + c\right )^{2} - 29 \, a^{3} \cos \left (d x + c\right ) + 22 \, a^{3} + 15 \,{\left (a^{3} d x \cos \left (d x + c\right )^{2} - 2 \, a^{3} d x \cos \left (d x + c\right ) + a^{3} d x\right )} \sin \left (d x + c\right )}{15 \,{\left (d \cos \left (d x + c\right )^{2} - 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 = 1.43085, size = 89, normalized size = 0.83 \begin{align*} -\frac{60 \,{\left (d x + c\right )} a^{3} + \frac{105 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 20 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 3 \, a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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