Optimal. Leaf size=141 \[ -\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^7(c+d x)}{7 d}+\frac{11 a^3 \csc ^5(c+d x)}{5 d}-\frac{10 a^3 \csc ^3(c+d x)}{3 d}+\frac{3 a^3 \csc (c+d x)}{d}+a^3 x \]
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Rubi [A] time = 0.178499, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 15, 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, 270} \[ -\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^7(c+d x)}{7 d}+\frac{11 a^3 \csc ^5(c+d x)}{5 d}-\frac{10 a^3 \csc ^3(c+d x)}{3 d}+\frac{3 a^3 \csc (c+d x)}{d}+a^3 x \]
Antiderivative was successfully verified.
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Rule 3886
Rule 3473
Rule 8
Rule 2606
Rule 194
Rule 2607
Rule 30
Rule 270
Rubi steps
\begin{align*} \int \cot ^8(c+d x) (a+a \sec (c+d x))^3 \, dx &=\int \left (a^3 \cot ^8(c+d x)+3 a^3 \cot ^7(c+d x) \csc (c+d x)+3 a^3 \cot ^6(c+d x) \csc ^2(c+d x)+a^3 \cot ^5(c+d x) \csc ^3(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^8(c+d x) \, dx+a^3 \int \cot ^5(c+d x) \csc ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^7(c+d x) \csc (c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx\\ &=-\frac{a^3 \cot ^7(c+d x)}{7 d}-a^3 \int \cot ^6(c+d x) \, dx-\frac{a^3 \operatorname{Subst}\left (\int x^2 \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{d}+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (-1+x^2\right )^3 \, dx,x,\csc (c+d x)\right )}{d}\\ &=\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+a^3 \int \cot ^4(c+d x) \, dx-\frac{a^3 \operatorname{Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\csc (c+d x)\right )}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (-1+3 x^2-3 x^4+x^6\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{3 a^3 \csc (c+d x)}{d}-\frac{10 a^3 \csc ^3(c+d x)}{3 d}+\frac{11 a^3 \csc ^5(c+d x)}{5 d}-\frac{4 a^3 \csc ^7(c+d x)}{7 d}-a^3 \int \cot ^2(c+d x) \, dx\\ &=\frac{a^3 \cot (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{3 a^3 \csc (c+d x)}{d}-\frac{10 a^3 \csc ^3(c+d x)}{3 d}+\frac{11 a^3 \csc ^5(c+d x)}{5 d}-\frac{4 a^3 \csc ^7(c+d x)}{7 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{a^3 \cot ^5(c+d x)}{5 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{3 a^3 \csc (c+d x)}{d}-\frac{10 a^3 \csc ^3(c+d x)}{3 d}+\frac{11 a^3 \csc ^5(c+d x)}{5 d}-\frac{4 a^3 \csc ^7(c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.986898, size = 252, normalized size = 1.79 \[ \frac{a^3 \csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) \csc ^7\left (\frac{1}{2} (c+d x)\right ) \sec \left (\frac{1}{2} (c+d x)\right ) (-23282 \sin (c+d x)+23282 \sin (2 (c+d x))-9978 \sin (3 (c+d x))+1663 \sin (4 (c+d x))-13720 \sin (2 c+d x)+15512 \sin (c+2 d x)+9240 \sin (3 c+2 d x)-8088 \sin (2 c+3 d x)-2520 \sin (4 c+3 d x)+1768 \sin (3 c+4 d x)-5880 d x \cos (2 c+d x)-5880 d x \cos (c+2 d x)+5880 d x \cos (3 c+2 d x)+2520 d x \cos (2 c+3 d x)-2520 d x \cos (4 c+3 d x)-420 d x \cos (3 c+4 d x)+420 d x \cos (5 c+4 d x)+4200 \sin (c)-11032 \sin (d x)+5880 d x \cos (d x))}{215040 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.08, size = 293, normalized size = 2.1 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( -{\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 ) +3\,{a}^{3} \left ( -1/7\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{ \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}+1/35\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-1/35\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+1/7\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{\sin \left ( dx+c \right ) }}+1/7\, \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+6/5\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}+8/5\, \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 ) ^{7}}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}+{a}^{3} \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{35\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{105\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{35\,\sin \left ( dx+c \right ) }}-{\frac{\sin \left ( dx+c \right ) }{35} \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 = 1.67757, size = 205, normalized size = 1.45 \begin{align*} \frac{{\left (105 \, d x + 105 \, c + \frac{105 \, \tan \left (d x + c\right )^{6} - 35 \, \tan \left (d x + c\right )^{4} + 21 \, \tan \left (d x + c\right )^{2} - 15}{\tan \left (d x + c\right )^{7}}\right )} a^{3} + \frac{9 \,{\left (35 \, \sin \left (d x + c\right )^{6} - 35 \, \sin \left (d x + c\right )^{4} + 21 \, \sin \left (d x + c\right )^{2} - 5\right )} a^{3}}{\sin \left (d x + c\right )^{7}} - \frac{{\left (35 \, \sin \left (d x + c\right )^{4} - 42 \, \sin \left (d x + c\right )^{2} + 15\right )} a^{3}}{\sin \left (d x + c\right )^{7}} - \frac{45 \, a^{3}}{\tan \left (d x + c\right )^{7}}}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.13793, size = 402, normalized size = 2.85 \begin{align*} \frac{221 \, a^{3} \cos \left (d x + c\right )^{4} - 348 \, a^{3} \cos \left (d x + c\right )^{3} - 25 \, a^{3} \cos \left (d x + c\right )^{2} + 303 \, a^{3} \cos \left (d x + c\right ) - 136 \, a^{3} + 105 \,{\left (a^{3} d x \cos \left (d x + c\right )^{3} - 3 \, a^{3} d x \cos \left (d x + c\right )^{2} + 3 \, a^{3} d x \cos \left (d x + c\right ) - a^{3} d x\right )} \sin \left (d x + c\right )}{105 \,{\left (d \cos \left (d x + c\right )^{3} - 3 \, d \cos \left (d x + c\right )^{2} + 3 \, 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 = 1.58607, size = 130, normalized size = 0.92 \begin{align*} \frac{1680 \,{\left (d x + c\right )} a^{3} - 105 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{2730 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 560 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 126 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7}}}{1680 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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