Optimal. Leaf size=198 \[ -\frac{4 a^4 \cos ^3(c+d x)}{3 d}-\frac{4 a^4 \cos (c+d x)}{d}-\frac{a^4 \cot ^5(c+d x)}{5 d}-\frac{5 a^4 \cot ^3(c+d x)}{3 d}+\frac{10 a^4 \cot (c+d x)}{d}+\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac{15 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5 a^4 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^4 \cot (c+d x) \csc ^3(c+d x)}{d}+\frac{5 a^4 \cot (c+d x) \csc (c+d x)}{2 d}+\frac{97 a^4 x}{8} \]
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Rubi [A] time = 0.427749, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {2709, 3767, 8, 3768, 3770, 2638, 2635, 2633} \[ -\frac{4 a^4 \cos ^3(c+d x)}{3 d}-\frac{4 a^4 \cos (c+d x)}{d}-\frac{a^4 \cot ^5(c+d x)}{5 d}-\frac{5 a^4 \cot ^3(c+d x)}{3 d}+\frac{10 a^4 \cot (c+d x)}{d}+\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac{15 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5 a^4 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^4 \cot (c+d x) \csc ^3(c+d x)}{d}+\frac{5 a^4 \cot (c+d x) \csc (c+d x)}{2 d}+\frac{97 a^4 x}{8} \]
Antiderivative was successfully verified.
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Rule 2709
Rule 3767
Rule 8
Rule 3768
Rule 3770
Rule 2638
Rule 2635
Rule 2633
Rubi steps
\begin{align*} \int \cot ^6(c+d x) (a+a \sin (c+d x))^4 \, dx &=\frac{\int \left (14 a^{10}-14 a^{10} \csc ^2(c+d x)-8 a^{10} \csc ^3(c+d x)+3 a^{10} \csc ^4(c+d x)+4 a^{10} \csc ^5(c+d x)+a^{10} \csc ^6(c+d x)+8 a^{10} \sin (c+d x)-3 a^{10} \sin ^2(c+d x)-4 a^{10} \sin ^3(c+d x)-a^{10} \sin ^4(c+d x)\right ) \, dx}{a^6}\\ &=14 a^4 x+a^4 \int \csc ^6(c+d x) \, dx-a^4 \int \sin ^4(c+d x) \, dx+\left (3 a^4\right ) \int \csc ^4(c+d x) \, dx-\left (3 a^4\right ) \int \sin ^2(c+d x) \, dx+\left (4 a^4\right ) \int \csc ^5(c+d x) \, dx-\left (4 a^4\right ) \int \sin ^3(c+d x) \, dx-\left (8 a^4\right ) \int \csc ^3(c+d x) \, dx+\left (8 a^4\right ) \int \sin (c+d x) \, dx-\left (14 a^4\right ) \int \csc ^2(c+d x) \, dx\\ &=14 a^4 x-\frac{8 a^4 \cos (c+d x)}{d}+\frac{4 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac{a^4 \cot (c+d x) \csc ^3(c+d x)}{d}+\frac{3 a^4 \cos (c+d x) \sin (c+d x)}{2 d}+\frac{a^4 \cos (c+d x) \sin ^3(c+d x)}{4 d}-\frac{1}{4} \left (3 a^4\right ) \int \sin ^2(c+d x) \, dx-\frac{1}{2} \left (3 a^4\right ) \int 1 \, dx+\left (3 a^4\right ) \int \csc ^3(c+d x) \, dx-\left (4 a^4\right ) \int \csc (c+d x) \, dx-\frac{a^4 \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac{\left (3 a^4\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}+\frac{\left (4 a^4\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac{\left (14 a^4\right ) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}\\ &=\frac{25 a^4 x}{2}+\frac{4 a^4 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{4 a^4 \cos (c+d x)}{d}-\frac{4 a^4 \cos ^3(c+d x)}{3 d}+\frac{10 a^4 \cot (c+d x)}{d}-\frac{5 a^4 \cot ^3(c+d x)}{3 d}-\frac{a^4 \cot ^5(c+d x)}{5 d}+\frac{5 a^4 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{a^4 \cot (c+d x) \csc ^3(c+d x)}{d}+\frac{15 a^4 \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a^4 \cos (c+d x) \sin ^3(c+d x)}{4 d}-\frac{1}{8} \left (3 a^4\right ) \int 1 \, dx+\frac{1}{2} \left (3 a^4\right ) \int \csc (c+d x) \, dx\\ &=\frac{97 a^4 x}{8}+\frac{5 a^4 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{4 a^4 \cos (c+d x)}{d}-\frac{4 a^4 \cos ^3(c+d x)}{3 d}+\frac{10 a^4 \cot (c+d x)}{d}-\frac{5 a^4 \cot ^3(c+d x)}{3 d}-\frac{a^4 \cot ^5(c+d x)}{5 d}+\frac{5 a^4 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{a^4 \cot (c+d x) \csc ^3(c+d x)}{d}+\frac{15 a^4 \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a^4 \cos (c+d x) \sin ^3(c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 1.5202, size = 283, normalized size = 1.43 \[ \frac{a^4 (\sin (c+d x)+1)^4 \left (5820 (c+d x)+480 \sin (2 (c+d x))-15 \sin (4 (c+d x))-2400 \cos (c+d x)-160 \cos (3 (c+d x))-2752 \tan \left (\frac{1}{2} (c+d x)\right )+2752 \cot \left (\frac{1}{2} (c+d x)\right )-30 \csc ^4\left (\frac{1}{2} (c+d x)\right )+300 \csc ^2\left (\frac{1}{2} (c+d x)\right )+30 \sec ^4\left (\frac{1}{2} (c+d x)\right )-300 \sec ^2\left (\frac{1}{2} (c+d x)\right )-1200 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+1200 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+96 \sin ^6\left (\frac{1}{2} (c+d x)\right ) \csc ^5(c+d x)+632 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-\frac{3}{2} \sin (c+d x) \csc ^6\left (\frac{1}{2} (c+d x)\right )-\frac{79}{2} \sin (c+d x) \csc ^4\left (\frac{1}{2} (c+d x)\right )\right )}{480 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 293, normalized size = 1.5 \begin{align*} -{\frac{5\,{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{6\,d}}+7\,{\frac{{a}^{4}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{d}}-{\frac{5\,{a}^{4}\cos \left ( dx+c \right ) }{2\,d}}-{\frac{{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{2\,d}}-{\frac{{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-2\,{\frac{{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{97\,{a}^{4}x}{8}}+7\,{\frac{{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{d\sin \left ( dx+c \right ) }}+{\frac{35\,{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }{4\,d}}+{\frac{105\,{a}^{4}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{8\,d}}-{\frac{{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{{a}^{4}\cot \left ( dx+c \right ) }{d}}+{\frac{97\,{a}^{4}c}{8\,d}}-{\frac{5\,{a}^{4}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.69153, size = 423, normalized size = 2.14 \begin{align*} -\frac{40 \,{\left (4 \, \cos \left (d x + c\right )^{3} - \frac{6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \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 )} a^{4} + 15 \,{\left (15 \, d x + 15 \, c + \frac{15 \, \tan \left (d x + c\right )^{4} + 25 \, \tan \left (d x + c\right )^{2} + 8}{\tan \left (d x + c\right )^{5} + 2 \, \tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{4} - 120 \,{\left (15 \, d x + 15 \, c + \frac{15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} a^{4} + 8 \,{\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^{4} + 30 \, a^{4}{\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 )}}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08524, size = 770, normalized size = 3.89 \begin{align*} \frac{30 \, a^{4} \cos \left (d x + c\right )^{9} - 345 \, a^{4} \cos \left (d x + c\right )^{7} + 2231 \, a^{4} \cos \left (d x + c\right )^{5} - 3395 \, a^{4} \cos \left (d x + c\right )^{3} + 1455 \, a^{4} \cos \left (d x + c\right ) + 150 \,{\left (a^{4} \cos \left (d x + c\right )^{4} - 2 \, a^{4} \cos \left (d x + c\right )^{2} + a^{4}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 150 \,{\left (a^{4} \cos \left (d x + c\right )^{4} - 2 \, a^{4} \cos \left (d x + c\right )^{2} + a^{4}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 5 \,{\left (32 \, a^{4} \cos \left (d x + c\right )^{7} - 291 \, a^{4} d x \cos \left (d x + c\right )^{4} + 32 \, a^{4} \cos \left (d x + c\right )^{5} + 582 \, a^{4} d x \cos \left (d x + c\right )^{2} - 100 \, a^{4} \cos \left (d x + c\right )^{3} - 291 \, a^{4} d x + 60 \, a^{4} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, 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 [A] time = 1.70284, size = 458, normalized size = 2.31 \begin{align*} \frac{3 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 30 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 85 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 240 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 5820 \,{\left (d x + c\right )} a^{4} - 1200 \, a^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 2670 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{40 \,{\left (45 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 192 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 69 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 384 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 69 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 320 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 45 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 128 \, a^{4}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{4}} + \frac{2740 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 2670 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 240 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 85 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 30 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 3 \, a^{4}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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