Optimal. Leaf size=215 \[ \frac{4 \cot ^{11}(c+d x)}{11 a^3 d}-\frac{\cot ^9(c+d x)}{9 a^3 d}+\frac{\cot ^7(c+d x)}{7 a^3 d}-\frac{\cot ^5(c+d x)}{5 a^3 d}+\frac{\cot ^3(c+d x)}{3 a^3 d}-\frac{\cot (c+d x)}{a^3 d}-\frac{4 \csc ^{11}(c+d x)}{11 a^3 d}+\frac{19 \csc ^9(c+d x)}{9 a^3 d}-\frac{36 \csc ^7(c+d x)}{7 a^3 d}+\frac{34 \csc ^5(c+d x)}{5 a^3 d}-\frac{16 \csc ^3(c+d x)}{3 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}-\frac{x}{a^3} \]
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Rubi [A] time = 0.278093, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {3888, 3886, 3473, 8, 2606, 194, 2607, 30, 270} \[ \frac{4 \cot ^{11}(c+d x)}{11 a^3 d}-\frac{\cot ^9(c+d x)}{9 a^3 d}+\frac{\cot ^7(c+d x)}{7 a^3 d}-\frac{\cot ^5(c+d x)}{5 a^3 d}+\frac{\cot ^3(c+d x)}{3 a^3 d}-\frac{\cot (c+d x)}{a^3 d}-\frac{4 \csc ^{11}(c+d x)}{11 a^3 d}+\frac{19 \csc ^9(c+d x)}{9 a^3 d}-\frac{36 \csc ^7(c+d x)}{7 a^3 d}+\frac{34 \csc ^5(c+d x)}{5 a^3 d}-\frac{16 \csc ^3(c+d x)}{3 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}-\frac{x}{a^3} \]
Antiderivative was successfully verified.
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Rule 3888
Rule 3886
Rule 3473
Rule 8
Rule 2606
Rule 194
Rule 2607
Rule 30
Rule 270
Rubi steps
\begin{align*} \int \frac{\cot ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=\frac{\int \cot ^{12}(c+d x) (-a+a \sec (c+d x))^3 \, dx}{a^6}\\ &=\frac{\int \left (-a^3 \cot ^{12}(c+d x)+3 a^3 \cot ^{11}(c+d x) \csc (c+d x)-3 a^3 \cot ^{10}(c+d x) \csc ^2(c+d x)+a^3 \cot ^9(c+d x) \csc ^3(c+d x)\right ) \, dx}{a^6}\\ &=-\frac{\int \cot ^{12}(c+d x) \, dx}{a^3}+\frac{\int \cot ^9(c+d x) \csc ^3(c+d x) \, dx}{a^3}+\frac{3 \int \cot ^{11}(c+d x) \csc (c+d x) \, dx}{a^3}-\frac{3 \int \cot ^{10}(c+d x) \csc ^2(c+d x) \, dx}{a^3}\\ &=\frac{\cot ^{11}(c+d x)}{11 a^3 d}+\frac{\int \cot ^{10}(c+d x) \, dx}{a^3}-\frac{\operatorname{Subst}\left (\int x^2 \left (-1+x^2\right )^4 \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac{3 \operatorname{Subst}\left (\int x^{10} \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac{3 \operatorname{Subst}\left (\int \left (-1+x^2\right )^5 \, dx,x,\csc (c+d x)\right )}{a^3 d}\\ &=-\frac{\cot ^9(c+d x)}{9 a^3 d}+\frac{4 \cot ^{11}(c+d x)}{11 a^3 d}-\frac{\int \cot ^8(c+d x) \, dx}{a^3}-\frac{\operatorname{Subst}\left (\int \left (x^2-4 x^4+6 x^6-4 x^8+x^{10}\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac{3 \operatorname{Subst}\left (\int \left (-1+5 x^2-10 x^4+10 x^6-5 x^8+x^{10}\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}\\ &=\frac{\cot ^7(c+d x)}{7 a^3 d}-\frac{\cot ^9(c+d x)}{9 a^3 d}+\frac{4 \cot ^{11}(c+d x)}{11 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}-\frac{16 \csc ^3(c+d x)}{3 a^3 d}+\frac{34 \csc ^5(c+d x)}{5 a^3 d}-\frac{36 \csc ^7(c+d x)}{7 a^3 d}+\frac{19 \csc ^9(c+d x)}{9 a^3 d}-\frac{4 \csc ^{11}(c+d x)}{11 a^3 d}+\frac{\int \cot ^6(c+d x) \, dx}{a^3}\\ &=-\frac{\cot ^5(c+d x)}{5 a^3 d}+\frac{\cot ^7(c+d x)}{7 a^3 d}-\frac{\cot ^9(c+d x)}{9 a^3 d}+\frac{4 \cot ^{11}(c+d x)}{11 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}-\frac{16 \csc ^3(c+d x)}{3 a^3 d}+\frac{34 \csc ^5(c+d x)}{5 a^3 d}-\frac{36 \csc ^7(c+d x)}{7 a^3 d}+\frac{19 \csc ^9(c+d x)}{9 a^3 d}-\frac{4 \csc ^{11}(c+d x)}{11 a^3 d}-\frac{\int \cot ^4(c+d x) \, dx}{a^3}\\ &=\frac{\cot ^3(c+d x)}{3 a^3 d}-\frac{\cot ^5(c+d x)}{5 a^3 d}+\frac{\cot ^7(c+d x)}{7 a^3 d}-\frac{\cot ^9(c+d x)}{9 a^3 d}+\frac{4 \cot ^{11}(c+d x)}{11 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}-\frac{16 \csc ^3(c+d x)}{3 a^3 d}+\frac{34 \csc ^5(c+d x)}{5 a^3 d}-\frac{36 \csc ^7(c+d x)}{7 a^3 d}+\frac{19 \csc ^9(c+d x)}{9 a^3 d}-\frac{4 \csc ^{11}(c+d x)}{11 a^3 d}+\frac{\int \cot ^2(c+d x) \, dx}{a^3}\\ &=-\frac{\cot (c+d x)}{a^3 d}+\frac{\cot ^3(c+d x)}{3 a^3 d}-\frac{\cot ^5(c+d x)}{5 a^3 d}+\frac{\cot ^7(c+d x)}{7 a^3 d}-\frac{\cot ^9(c+d x)}{9 a^3 d}+\frac{4 \cot ^{11}(c+d x)}{11 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}-\frac{16 \csc ^3(c+d x)}{3 a^3 d}+\frac{34 \csc ^5(c+d x)}{5 a^3 d}-\frac{36 \csc ^7(c+d x)}{7 a^3 d}+\frac{19 \csc ^9(c+d x)}{9 a^3 d}-\frac{4 \csc ^{11}(c+d x)}{11 a^3 d}-\frac{\int 1 \, dx}{a^3}\\ &=-\frac{x}{a^3}-\frac{\cot (c+d x)}{a^3 d}+\frac{\cot ^3(c+d x)}{3 a^3 d}-\frac{\cot ^5(c+d x)}{5 a^3 d}+\frac{\cot ^7(c+d x)}{7 a^3 d}-\frac{\cot ^9(c+d x)}{9 a^3 d}+\frac{4 \cot ^{11}(c+d x)}{11 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}-\frac{16 \csc ^3(c+d x)}{3 a^3 d}+\frac{34 \csc ^5(c+d x)}{5 a^3 d}-\frac{36 \csc ^7(c+d x)}{7 a^3 d}+\frac{19 \csc ^9(c+d x)}{9 a^3 d}-\frac{4 \csc ^{11}(c+d x)}{11 a^3 d}\\ \end{align*}
Mathematica [A] time = 3.59001, size = 394, normalized size = 1.83 \[ -\frac{\tan \left (\frac{c}{2}\right ) \cos ^6\left (\frac{1}{2} (c+d x)\right ) \sec ^3(c+d x) \left (315 \sec ^{10}\left (\frac{1}{2} (c+d x)\right )-5425 \sec ^8\left (\frac{1}{2} (c+d x)\right )+41320 \sec ^6\left (\frac{1}{2} (c+d x)\right )-184650 \sec ^4\left (\frac{1}{2} (c+d x)\right )+561145 \sec ^2\left (\frac{1}{2} (c+d x)\right )+6468 \sin (c) \csc ^3\left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \csc ^3\left (\frac{1}{2} (c+d x)\right )+231 \cot ^2\left (\frac{c}{2}\right ) (28 \cos (c+d x)-25) \csc ^4\left (\frac{1}{2} (c+d x)\right )+231 \cot \left (\frac{c}{2}\right ) \left (3840 d x-\csc \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \csc \left (\frac{1}{2} (c+d x)\right ) \left (3 \csc ^4\left (\frac{1}{2} (c+d x)\right )+743\right )\right )+315 \csc \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \sec ^{11}\left (\frac{1}{2} (c+d x)\right )-5425 \csc \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \sec ^9\left (\frac{1}{2} (c+d x)\right )+41320 \csc \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \sec ^7\left (\frac{1}{2} (c+d x)\right )-184650 \csc \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \sec ^5\left (\frac{1}{2} (c+d x)\right )+561145 \csc \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \sec ^3\left (\frac{1}{2} (c+d x)\right )-1736335 \csc \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \sec \left (\frac{1}{2} (c+d x)\right )\right )}{110880 a^3 d (\sec (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.082, size = 189, normalized size = 0.9 \begin{align*} -{\frac{1}{2816\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{11}}+{\frac{5}{1152\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9}}-{\frac{23}{896\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{13}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{1}{3\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{191}{128\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{3}}}-{\frac{1}{1280\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}}+{\frac{5}{384\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}-{\frac{23}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.55563, size = 294, normalized size = 1.37 \begin{align*} \frac{\frac{5 \,{\left (\frac{264726 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{59136 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{18018 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{4554 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{770 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac{63 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}\right )}}{a^{3}} - \frac{1774080 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} + \frac{231 \,{\left (\frac{50 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{690 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 3\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}{a^{3} \sin \left (d x + c\right )^{5}}}{887040 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59559, size = 757, normalized size = 3.52 \begin{align*} -\frac{7453 \, \cos \left (d x + c\right )^{8} + 11964 \, \cos \left (d x + c\right )^{7} - 11866 \, \cos \left (d x + c\right )^{6} - 30542 \, \cos \left (d x + c\right )^{5} + 90 \, \cos \left (d x + c\right )^{4} + 26438 \, \cos \left (d x + c\right )^{3} + 8539 \, \cos \left (d x + c\right )^{2} + 3465 \,{\left (d x \cos \left (d x + c\right )^{7} + 3 \, d x \cos \left (d x + c\right )^{6} + d x \cos \left (d x + c\right )^{5} - 5 \, d x \cos \left (d x + c\right )^{4} - 5 \, d x \cos \left (d x + c\right )^{3} + d x \cos \left (d x + c\right )^{2} + 3 \, d x \cos \left (d x + c\right ) + d x\right )} \sin \left (d x + c\right ) - 7671 \, \cos \left (d x + c\right ) - 3712}{3465 \,{\left (a^{3} d \cos \left (d x + c\right )^{7} + 3 \, a^{3} d \cos \left (d x + c\right )^{6} + a^{3} d \cos \left (d x + c\right )^{5} - 5 \, a^{3} d \cos \left (d x + c\right )^{4} - 5 \, a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} 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 = 1.56771, size = 216, normalized size = 1. \begin{align*} -\frac{\frac{887040 \,{\left (d x + c\right )}}{a^{3}} + \frac{231 \,{\left (690 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 50 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 3\right )}}{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}} + \frac{5 \,{\left (63 \, a^{30} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 770 \, a^{30} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 4554 \, a^{30} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 18018 \, a^{30} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 59136 \, a^{30} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 264726 \, a^{30} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{a^{33}}}{887040 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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