Optimal. Leaf size=157 \[ \frac{a^2 \cos (c+d x)}{d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}+\frac{2 a^2 \cot ^3(c+d x)}{3 d}-\frac{2 a^2 \cot (c+d x)}{d}-\frac{25 a^2 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac{7 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac{7 a^2 \cot (c+d x) \csc (c+d x)}{16 d}-2 a^2 x \]
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Rubi [A] time = 0.234526, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2872, 3770, 3767, 8, 3768, 2638} \[ \frac{a^2 \cos (c+d x)}{d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}+\frac{2 a^2 \cot ^3(c+d x)}{3 d}-\frac{2 a^2 \cot (c+d x)}{d}-\frac{25 a^2 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac{7 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac{7 a^2 \cot (c+d x) \csc (c+d x)}{16 d}-2 a^2 x \]
Antiderivative was successfully verified.
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Rule 2872
Rule 3770
Rule 3767
Rule 8
Rule 3768
Rule 2638
Rubi steps
\begin{align*} \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac{\int \left (-2 a^8+2 a^8 \csc (c+d x)+6 a^8 \csc ^2(c+d x)-6 a^8 \csc ^4(c+d x)-2 a^8 \csc ^5(c+d x)+2 a^8 \csc ^6(c+d x)+a^8 \csc ^7(c+d x)-a^8 \sin (c+d x)\right ) \, dx}{a^6}\\ &=-2 a^2 x+a^2 \int \csc ^7(c+d x) \, dx-a^2 \int \sin (c+d x) \, dx+\left (2 a^2\right ) \int \csc (c+d x) \, dx-\left (2 a^2\right ) \int \csc ^5(c+d x) \, dx+\left (2 a^2\right ) \int \csc ^6(c+d x) \, dx+\left (6 a^2\right ) \int \csc ^2(c+d x) \, dx-\left (6 a^2\right ) \int \csc ^4(c+d x) \, dx\\ &=-2 a^2 x-\frac{2 a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a^2 \cos (c+d x)}{d}+\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{2 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac{1}{6} \left (5 a^2\right ) \int \csc ^5(c+d x) \, dx-\frac{1}{2} \left (3 a^2\right ) \int \csc ^3(c+d x) \, dx-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac{\left (6 a^2\right ) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}+\frac{\left (6 a^2\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=-2 a^2 x-\frac{2 a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a^2 \cos (c+d x)}{d}-\frac{2 a^2 \cot (c+d x)}{d}+\frac{2 a^2 \cot ^3(c+d x)}{3 d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}+\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{4 d}+\frac{7 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac{1}{8} \left (5 a^2\right ) \int \csc ^3(c+d x) \, dx-\frac{1}{4} \left (3 a^2\right ) \int \csc (c+d x) \, dx\\ &=-2 a^2 x-\frac{5 a^2 \tanh ^{-1}(\cos (c+d x))}{4 d}+\frac{a^2 \cos (c+d x)}{d}-\frac{2 a^2 \cot (c+d x)}{d}+\frac{2 a^2 \cot ^3(c+d x)}{3 d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}+\frac{7 a^2 \cot (c+d x) \csc (c+d x)}{16 d}+\frac{7 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac{1}{16} \left (5 a^2\right ) \int \csc (c+d x) \, dx\\ &=-2 a^2 x-\frac{25 a^2 \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac{a^2 \cos (c+d x)}{d}-\frac{2 a^2 \cot (c+d x)}{d}+\frac{2 a^2 \cot ^3(c+d x)}{3 d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}+\frac{7 a^2 \cot (c+d x) \csc (c+d x)}{16 d}+\frac{7 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 1.49425, size = 270, normalized size = 1.72 \[ -\frac{a^2 \sin (c+d x) (\sin (c+d x)+1)^2 \left (-1920 \cot (c+d x)+\csc ^2\left (\frac{1}{2} (c+d x)\right ) (1472-210 \csc (c+d x))+\csc ^6\left (\frac{1}{2} (c+d x)\right ) (5 \csc (c+d x)+12)-2 \csc ^4\left (\frac{1}{2} (c+d x)\right ) (15 \csc (c+d x)+82)-2 (327 \cos (c+d x)+92 \cos (2 (c+d x))+241) \sec ^6\left (\frac{1}{2} (c+d x)\right )-320 \sin ^6\left (\frac{1}{2} (c+d x)\right ) \csc ^7(c+d x)+480 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^5(c+d x)+840 \sin ^2\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)+120 \csc (c+d x) \left (32 (c+d x)-25 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+25 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{1920 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.083, size = 205, normalized size = 1.3 \begin{align*} -{\frac{5\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{24\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{5\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{16\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{5\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{16\,d}}+{\frac{25\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{48\,d}}+{\frac{25\,{a}^{2}\cos \left ( dx+c \right ) }{16\,d}}+{\frac{25\,{a}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{16\,d}}-{\frac{2\,{a}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{2\,{a}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-2\,{\frac{{a}^{2}\cot \left ( dx+c \right ) }{d}}-2\,{a}^{2}x-2\,{\frac{c{a}^{2}}{d}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{6\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.64454, size = 297, normalized size = 1.89 \begin{align*} -\frac{64 \,{\left (15 \, d x + 15 \, c + \frac{15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{2} - 5 \, a^{2}{\left (\frac{2 \,{\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 30 \, a^{2}{\left (\frac{2 \,{\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.25213, size = 782, normalized size = 4.98 \begin{align*} -\frac{960 \, a^{2} d x \cos \left (d x + c\right )^{6} - 480 \, a^{2} \cos \left (d x + c\right )^{7} - 2880 \, a^{2} d x \cos \left (d x + c\right )^{4} + 1650 \, a^{2} \cos \left (d x + c\right )^{5} + 2880 \, a^{2} d x \cos \left (d x + c\right )^{2} - 2000 \, a^{2} \cos \left (d x + c\right )^{3} - 960 \, a^{2} d x + 750 \, a^{2} \cos \left (d x + c\right ) + 375 \,{\left (a^{2} \cos \left (d x + c\right )^{6} - 3 \, a^{2} \cos \left (d x + c\right )^{4} + 3 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 375 \,{\left (a^{2} \cos \left (d x + c\right )^{6} - 3 \, a^{2} \cos \left (d x + c\right )^{4} + 3 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 64 \,{\left (23 \, a^{2} \cos \left (d x + c\right )^{5} - 35 \, a^{2} \cos \left (d x + c\right )^{3} + 15 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \,{\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 )}} \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 = 1.35652, size = 350, normalized size = 2.23 \begin{align*} \frac{5 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 24 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 15 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 280 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 255 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3840 \,{\left (d x + c\right )} a^{2} + 3000 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 2640 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{3840 \, a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1} - \frac{7350 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 2640 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 255 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 280 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 15 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 24 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 5 \, a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6}}}{1920 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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