Optimal. Leaf size=153 \[ -\frac{a^2 \cot ^7(c+d x)}{7 d}+\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}+a^2 x-\frac{2 a b \csc ^7(c+d x)}{7 d}+\frac{6 a b \csc ^5(c+d x)}{5 d}-\frac{2 a b \csc ^3(c+d x)}{d}+\frac{2 a b \csc (c+d x)}{d}-\frac{b^2 \cot ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.147713, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3886, 3473, 8, 2606, 194, 2607, 30} \[ -\frac{a^2 \cot ^7(c+d x)}{7 d}+\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}+a^2 x-\frac{2 a b \csc ^7(c+d x)}{7 d}+\frac{6 a b \csc ^5(c+d x)}{5 d}-\frac{2 a b \csc ^3(c+d x)}{d}+\frac{2 a b \csc (c+d x)}{d}-\frac{b^2 \cot ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 3886
Rule 3473
Rule 8
Rule 2606
Rule 194
Rule 2607
Rule 30
Rubi steps
\begin{align*} \int \cot ^8(c+d x) (a+b \sec (c+d x))^2 \, dx &=\int \left (a^2 \cot ^8(c+d x)+2 a b \cot ^7(c+d x) \csc (c+d x)+b^2 \cot ^6(c+d x) \csc ^2(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^8(c+d x) \, dx+(2 a b) \int \cot ^7(c+d x) \csc (c+d x) \, dx+b^2 \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx\\ &=-\frac{a^2 \cot ^7(c+d x)}{7 d}-a^2 \int \cot ^6(c+d x) \, dx-\frac{(2 a b) \operatorname{Subst}\left (\int \left (-1+x^2\right )^3 \, dx,x,\csc (c+d x)\right )}{d}+\frac{b^2 \operatorname{Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{a^2 \cot ^7(c+d x)}{7 d}-\frac{b^2 \cot ^7(c+d x)}{7 d}+a^2 \int \cot ^4(c+d x) \, dx-\frac{(2 a b) \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^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{a^2 \cot ^7(c+d x)}{7 d}-\frac{b^2 \cot ^7(c+d x)}{7 d}+\frac{2 a b \csc (c+d x)}{d}-\frac{2 a b \csc ^3(c+d x)}{d}+\frac{6 a b \csc ^5(c+d x)}{5 d}-\frac{2 a b \csc ^7(c+d x)}{7 d}-a^2 \int \cot ^2(c+d x) \, dx\\ &=\frac{a^2 \cot (c+d x)}{d}-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{a^2 \cot ^7(c+d x)}{7 d}-\frac{b^2 \cot ^7(c+d x)}{7 d}+\frac{2 a b \csc (c+d x)}{d}-\frac{2 a b \csc ^3(c+d x)}{d}+\frac{6 a b \csc ^5(c+d x)}{5 d}-\frac{2 a b \csc ^7(c+d x)}{7 d}+a^2 \int 1 \, dx\\ &=a^2 x+\frac{a^2 \cot (c+d x)}{d}-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{a^2 \cot ^7(c+d x)}{7 d}-\frac{b^2 \cot ^7(c+d x)}{7 d}+\frac{2 a b \csc (c+d x)}{d}-\frac{2 a b \csc ^3(c+d x)}{d}+\frac{6 a b \csc ^5(c+d x)}{5 d}-\frac{2 a b \csc ^7(c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.814682, size = 257, normalized size = 1.68 \[ -\frac{\csc ^7(c+d x) \left (-3675 a^2 c \sin (c+d x)-3675 a^2 d x \sin (c+d x)+2205 a^2 c \sin (3 (c+d x))+2205 a^2 d x \sin (3 (c+d x))-735 a^2 c \sin (5 (c+d x))-735 a^2 d x \sin (5 (c+d x))+105 a^2 c \sin (7 (c+d x))+105 a^2 d x \sin (7 (c+d x))+1176 a^2 \cos (3 (c+d x))-392 a^2 \cos (5 (c+d x))+176 a^2 \cos (7 (c+d x))+3612 a b \cos (2 (c+d x))-840 a b \cos (4 (c+d x))+420 a b \cos (6 (c+d x))-1272 a b+525 b^2 \cos (c+d x)+315 b^2 \cos (3 (c+d x))+105 b^2 \cos (5 (c+d x))+15 b^2 \cos (7 (c+d x))\right )}{6720 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 187, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({a}^{2} \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 ) +2\,ab \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{{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.53239, size = 157, normalized size = 1.03 \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^{2} + \frac{6 \,{\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 b}{\sin \left (d x + c\right )^{7}} - \frac{15 \, b^{2}}{\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 = 0.95232, size = 524, normalized size = 3.42 \begin{align*} \frac{210 \, a b \cos \left (d x + c\right )^{6} +{\left (176 \, a^{2} + 15 \, b^{2}\right )} \cos \left (d x + c\right )^{7} - 406 \, a^{2} \cos \left (d x + c\right )^{5} - 420 \, a b \cos \left (d x + c\right )^{4} + 350 \, a^{2} \cos \left (d x + c\right )^{3} + 336 \, a b \cos \left (d x + c\right )^{2} - 105 \, a^{2} \cos \left (d x + c\right ) - 96 \, a b + 105 \,{\left (a^{2} d x \cos \left (d x + c\right )^{6} - 3 \, a^{2} d x \cos \left (d x + c\right )^{4} + 3 \, a^{2} d x \cos \left (d x + c\right )^{2} - a^{2} d x\right )} \sin \left (d x + c\right )}{105 \,{\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - 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 [B] time = 1.29837, size = 494, normalized size = 3.23 \begin{align*} \frac{15 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 30 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 15 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 189 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 294 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 105 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1295 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 1470 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 315 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 13440 \,{\left (d x + c\right )} a^{2} - 9765 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 7350 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 525 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{9765 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 7350 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 525 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 1295 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 1470 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 315 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 189 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 294 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 105 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, a^{2} - 30 \, a b - 15 \, b^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7}}}{13440 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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