Optimal. Leaf size=179 \[ -\frac{4 a^3 \cot ^9(c+d x)}{9 d}+\frac{a^3 \cot ^7(c+d x)}{7 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^9(c+d x)}{9 d}+\frac{15 a^3 \csc ^7(c+d x)}{7 d}-\frac{21 a^3 \csc ^5(c+d x)}{5 d}+\frac{13 a^3 \csc ^3(c+d x)}{3 d}-\frac{3 a^3 \csc (c+d x)}{d}-a^3 x \]
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Rubi [A] time = 0.199362, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 16, 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} \[ -\frac{4 a^3 \cot ^9(c+d x)}{9 d}+\frac{a^3 \cot ^7(c+d x)}{7 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^9(c+d x)}{9 d}+\frac{15 a^3 \csc ^7(c+d x)}{7 d}-\frac{21 a^3 \csc ^5(c+d x)}{5 d}+\frac{13 a^3 \csc ^3(c+d x)}{3 d}-\frac{3 a^3 \csc (c+d x)}{d}-a^3 x \]
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 ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx &=\int \left (a^3 \cot ^{10}(c+d x)+3 a^3 \cot ^9(c+d x) \csc (c+d x)+3 a^3 \cot ^8(c+d x) \csc ^2(c+d x)+a^3 \cot ^7(c+d x) \csc ^3(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^{10}(c+d x) \, dx+a^3 \int \cot ^7(c+d x) \csc ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^9(c+d x) \csc (c+d x) \, dx+\left (3 a^3\right ) \int \cot ^8(c+d x) \csc ^2(c+d x) \, dx\\ &=-\frac{a^3 \cot ^9(c+d x)}{9 d}-a^3 \int \cot ^8(c+d x) \, dx-\frac{a^3 \operatorname{Subst}\left (\int x^2 \left (-1+x^2\right )^3 \, dx,x,\csc (c+d x)\right )}{d}+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int x^8 \, dx,x,-\cot (c+d x)\right )}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (-1+x^2\right )^4 \, dx,x,\csc (c+d x)\right )}{d}\\ &=\frac{a^3 \cot ^7(c+d x)}{7 d}-\frac{4 a^3 \cot ^9(c+d x)}{9 d}+a^3 \int \cot ^6(c+d x) \, dx-\frac{a^3 \operatorname{Subst}\left (\int \left (-x^2+3 x^4-3 x^6+x^8\right ) \, dx,x,\csc (c+d x)\right )}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (1-4 x^2+6 x^4-4 x^6+x^8\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^7(c+d x)}{7 d}-\frac{4 a^3 \cot ^9(c+d x)}{9 d}-\frac{3 a^3 \csc (c+d x)}{d}+\frac{13 a^3 \csc ^3(c+d x)}{3 d}-\frac{21 a^3 \csc ^5(c+d x)}{5 d}+\frac{15 a^3 \csc ^7(c+d x)}{7 d}-\frac{4 a^3 \csc ^9(c+d x)}{9 d}-a^3 \int \cot ^4(c+d x) \, dx\\ &=\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^7(c+d x)}{7 d}-\frac{4 a^3 \cot ^9(c+d x)}{9 d}-\frac{3 a^3 \csc (c+d x)}{d}+\frac{13 a^3 \csc ^3(c+d x)}{3 d}-\frac{21 a^3 \csc ^5(c+d x)}{5 d}+\frac{15 a^3 \csc ^7(c+d x)}{7 d}-\frac{4 a^3 \csc ^9(c+d x)}{9 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{a^3 \cot ^7(c+d x)}{7 d}-\frac{4 a^3 \cot ^9(c+d x)}{9 d}-\frac{3 a^3 \csc (c+d x)}{d}+\frac{13 a^3 \csc ^3(c+d x)}{3 d}-\frac{21 a^3 \csc ^5(c+d x)}{5 d}+\frac{15 a^3 \csc ^7(c+d x)}{7 d}-\frac{4 a^3 \csc ^9(c+d x)}{9 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{a^3 \cot ^7(c+d x)}{7 d}-\frac{4 a^3 \cot ^9(c+d x)}{9 d}-\frac{3 a^3 \csc (c+d x)}{d}+\frac{13 a^3 \csc ^3(c+d x)}{3 d}-\frac{21 a^3 \csc ^5(c+d x)}{5 d}+\frac{15 a^3 \csc ^7(c+d x)}{7 d}-\frac{4 a^3 \csc ^9(c+d x)}{9 d}\\ \end{align*}
Mathematica [B] time = 1.36962, size = 370, normalized size = 2.07 \[ \frac{a^3 \csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) \csc ^9\left (\frac{1}{2} (c+d x)\right ) \sec ^3\left (\frac{1}{2} (c+d x)\right ) (675036 \sin (c+d x)-506277 \sin (2 (c+d x))-37502 \sin (3 (c+d x))+225012 \sin (4 (c+d x))-112506 \sin (5 (c+d x))+18751 \sin (6 (c+d x))+431424 \sin (2 c+d x)-375552 \sin (c+2 d x)-201600 \sin (3 c+2 d x)+41248 \sin (2 c+3 d x)-84000 \sin (4 c+3 d x)+155712 \sin (3 c+4 d x)+100800 \sin (5 c+4 d x)-98016 \sin (4 c+5 d x)-30240 \sin (6 c+5 d x)+21376 \sin (5 c+6 d x)+181440 d x \cos (2 c+d x)+136080 d x \cos (c+2 d x)-136080 d x \cos (3 c+2 d x)+10080 d x \cos (2 c+3 d x)-10080 d x \cos (4 c+3 d x)-60480 d x \cos (3 c+4 d x)+60480 d x \cos (5 c+4 d x)+30240 d x \cos (4 c+5 d x)-30240 d x \cos (6 c+5 d x)-5040 d x \cos (5 c+6 d x)+5040 d x \cos (7 c+6 d x)-169344 \sin (c)+338112 \sin (d x)-181440 d x \cos (d x))}{41287680 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.086, size = 364, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.69417, size = 246, normalized size = 1.37 \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^{3} + \frac{3 \,{\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^{3}}{\sin \left (d x + c\right )^{9}} - \frac{{\left (105 \, \sin \left (d x + c\right )^{6} - 189 \, \sin \left (d x + c\right )^{4} + 135 \, \sin \left (d x + c\right )^{2} - 35\right )} a^{3}}{\sin \left (d x + c\right )^{9}} + \frac{105 \, a^{3}}{\tan \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 [A] time = 1.17243, size = 605, normalized size = 3.38 \begin{align*} -\frac{668 \, a^{3} \cos \left (d x + c\right )^{6} - 1059 \, a^{3} \cos \left (d x + c\right )^{5} - 573 \, a^{3} \cos \left (d x + c\right )^{4} + 1813 \, a^{3} \cos \left (d x + c\right )^{3} - 393 \, a^{3} \cos \left (d x + c\right )^{2} - 789 \, a^{3} \cos \left (d x + c\right ) + 368 \, a^{3} + 315 \,{\left (a^{3} d x \cos \left (d x + c\right )^{5} - 3 \, a^{3} d x \cos \left (d x + c\right )^{4} + 2 \, a^{3} d x \cos \left (d x + c\right )^{3} + 2 \, 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 )}{315 \,{\left (d \cos \left (d x + c\right )^{5} - 3 \, d \cos \left (d x + c\right )^{4} + 2 \, d \cos \left (d x + c\right )^{3} + 2 \, 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 = 1.55588, size = 173, normalized size = 0.97 \begin{align*} -\frac{105 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 20160 \,{\left (d x + c\right )} a^{3} - 2520 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{31185 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 6720 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 1827 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 360 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 35 \, a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9}}}{20160 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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