Optimal. Leaf size=73 \[ \frac{\cot ^7(c+d x)}{7 a d}+\frac{2 \cot ^5(c+d x)}{5 a d}+\frac{\cot ^3(c+d x)}{3 a d}-\frac{\csc ^7(c+d x)}{7 a d} \]
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Rubi [A] time = 0.147004, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3872, 2839, 2606, 30, 2607, 270} \[ \frac{\cot ^7(c+d x)}{7 a d}+\frac{2 \cot ^5(c+d x)}{5 a d}+\frac{\cot ^3(c+d x)}{3 a d}-\frac{\csc ^7(c+d x)}{7 a d} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2839
Rule 2606
Rule 30
Rule 2607
Rule 270
Rubi steps
\begin{align*} \int \frac{\csc ^6(c+d x)}{a+a \sec (c+d x)} \, dx &=-\int \frac{\cot (c+d x) \csc ^5(c+d x)}{-a-a \cos (c+d x)} \, dx\\ &=-\frac{\int \cot ^2(c+d x) \csc ^6(c+d x) \, dx}{a}+\frac{\int \cot (c+d x) \csc ^7(c+d x) \, dx}{a}\\ &=-\frac{\operatorname{Subst}\left (\int x^6 \, dx,x,\csc (c+d x)\right )}{a d}-\frac{\operatorname{Subst}\left (\int x^2 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=-\frac{\csc ^7(c+d x)}{7 a d}-\frac{\operatorname{Subst}\left (\int \left (x^2+2 x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=\frac{\cot ^3(c+d x)}{3 a d}+\frac{2 \cot ^5(c+d x)}{5 a d}+\frac{\cot ^7(c+d x)}{7 a d}-\frac{\csc ^7(c+d x)}{7 a d}\\ \end{align*}
Mathematica [B] time = 0.618358, size = 158, normalized size = 2.16 \[ \frac{\csc (c) (1500 \sin (c+d x)+375 \sin (2 (c+d x))-750 \sin (3 (c+d x))-300 \sin (4 (c+d x))+150 \sin (5 (c+d x))+75 \sin (6 (c+d x))+640 \sin (c+2 d x)-1280 \sin (2 c+3 d x)-512 \sin (3 c+4 d x)+256 \sin (4 c+5 d x)+128 \sin (5 c+6 d x)-8960 \sin (c)+2560 \sin (d x)) \csc ^5(c+d x) \sec (c+d x)}{53760 a d (\sec (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 88, normalized size = 1.2 \begin{align*}{\frac{1}{64\,da} \left ( -{\frac{1}{7} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{4}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{5}{3} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{4}{3} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}-5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{-1}-{\frac{1}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.997052, size = 184, normalized size = 2.52 \begin{align*} -\frac{\frac{\frac{175 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{84 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a} + \frac{7 \,{\left (\frac{20 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{75 \, \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 \sin \left (d x + c\right )^{5}}}{6720 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.67308, size = 347, normalized size = 4.75 \begin{align*} -\frac{8 \, \cos \left (d x + c\right )^{6} + 8 \, \cos \left (d x + c\right )^{5} - 20 \, \cos \left (d x + c\right )^{4} - 20 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )^{2} + 15 \, \cos \left (d x + c\right ) + 15}{105 \,{\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{3} - 2 \, a d \cos \left (d x + c\right )^{2} + a d \cos \left (d x + c\right ) + a 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 = 1.3124, size = 139, normalized size = 1.9 \begin{align*} -\frac{\frac{7 \,{\left (75 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 20 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 3\right )}}{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}} + \frac{15 \, a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 84 \, a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 175 \, a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3}}{a^{7}}}{6720 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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