Optimal. Leaf size=140 \[ \frac{4 a^4 \cos ^3(c+d x)}{3 d}-\frac{a^4 \cot ^3(c+d x)}{3 d}-\frac{5 a^4 \cot (c+d x)}{d}-\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac{19 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{2 a^4 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac{61 a^4 x}{8} \]
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Rubi [A] time = 0.228222, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {2709, 3770, 3767, 8, 3768, 2638, 2635, 2633} \[ \frac{4 a^4 \cos ^3(c+d x)}{3 d}-\frac{a^4 \cot ^3(c+d x)}{3 d}-\frac{5 a^4 \cot (c+d x)}{d}-\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac{19 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{2 a^4 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac{61 a^4 x}{8} \]
Antiderivative was successfully verified.
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Rule 2709
Rule 3770
Rule 3767
Rule 8
Rule 3768
Rule 2638
Rule 2635
Rule 2633
Rubi steps
\begin{align*} \int \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx &=\frac{\int \left (-10 a^8-4 a^8 \csc (c+d x)+4 a^8 \csc ^2(c+d x)+4 a^8 \csc ^3(c+d x)+a^8 \csc ^4(c+d x)-4 a^8 \sin (c+d x)+4 a^8 \sin ^2(c+d x)+4 a^8 \sin ^3(c+d x)+a^8 \sin ^4(c+d x)\right ) \, dx}{a^4}\\ &=-10 a^4 x+a^4 \int \csc ^4(c+d x) \, dx+a^4 \int \sin ^4(c+d x) \, dx-\left (4 a^4\right ) \int \csc (c+d x) \, dx+\left (4 a^4\right ) \int \csc ^2(c+d x) \, dx+\left (4 a^4\right ) \int \csc ^3(c+d x) \, dx-\left (4 a^4\right ) \int \sin (c+d x) \, dx+\left (4 a^4\right ) \int \sin ^2(c+d x) \, dx+\left (4 a^4\right ) \int \sin ^3(c+d x) \, dx\\ &=-10 a^4 x+\frac{4 a^4 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{4 a^4 \cos (c+d x)}{d}-\frac{2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac{2 a^4 \cos (c+d x) \sin (c+d x)}{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+\left (2 a^4\right ) \int 1 \, dx+\left (2 a^4\right ) \int \csc (c+d x) \, dx-\frac{a^4 \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}(\int 1 \, dx,x,\cot (c+d x))}{d}-\frac{\left (4 a^4\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-8 a^4 x+\frac{2 a^4 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{4 a^4 \cos ^3(c+d x)}{3 d}-\frac{5 a^4 \cot (c+d x)}{d}-\frac{a^4 \cot ^3(c+d x)}{3 d}-\frac{2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac{19 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{61 a^4 x}{8}+\frac{2 a^4 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{4 a^4 \cos ^3(c+d x)}{3 d}-\frac{5 a^4 \cot (c+d x)}{d}-\frac{a^4 \cot ^3(c+d x)}{3 d}-\frac{2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac{19 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 = 5.29066, size = 209, normalized size = 1.49 \[ \frac{a^4 (\sin (c+d x)+1)^4 \left (-732 (c+d x)-120 \sin (2 (c+d x))+3 \sin (4 (c+d x))+96 \cos (c+d x)+32 \cos (3 (c+d x))+224 \tan \left (\frac{1}{2} (c+d x)\right )-224 \cot \left (\frac{1}{2} (c+d x)\right )-48 \csc ^2\left (\frac{1}{2} (c+d x)\right )+48 \sec ^2\left (\frac{1}{2} (c+d x)\right )-192 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+192 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+32 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-2 \sin (c+d x) \csc ^4\left (\frac{1}{2} (c+d x)\right )\right )}{96 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.091, size = 190, normalized size = 1.4 \begin{align*} -{\frac{23\,{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }{4\,d}}-{\frac{69\,{a}^{4}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{8\,d}}-{\frac{61\,{a}^{4}x}{8}}-{\frac{61\,{a}^{4}c}{8\,d}}-{\frac{2\,{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-2\,{\frac{{a}^{4}\cos \left ( dx+c \right ) }{d}}-2\,{\frac{{a}^{4}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}}-6\,{\frac{{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{d\sin \left ( dx+c \right ) }}-2\,{\frac{{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{{a}^{4}\cot \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.64189, size = 294, normalized size = 2.1 \begin{align*} \frac{64 \,{\left (2 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right ) - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{4} + 3 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} - 288 \,{\left (3 \, d x + 3 \, c + \frac{3 \, \tan \left (d x + c\right )^{2} + 2}{\tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{4} + 32 \,{\left (3 \, d x + 3 \, c + \frac{3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a^{4} + 96 \, a^{4}{\left (\frac{2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - 4 \, \cos \left (d x + c\right ) + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.26403, size = 560, normalized size = 4. \begin{align*} -\frac{6 \, a^{4} \cos \left (d x + c\right )^{7} - 75 \, a^{4} \cos \left (d x + c\right )^{5} + 244 \, a^{4} \cos \left (d x + c\right )^{3} - 183 \, a^{4} \cos \left (d x + c\right ) - 24 \,{\left (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 ) + 24 \,{\left (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 ) -{\left (32 \, a^{4} \cos \left (d x + c\right )^{5} - 183 \, a^{4} d x \cos \left (d x + c\right )^{2} - 32 \, a^{4} \cos \left (d x + c\right )^{3} + 183 \, a^{4} d x + 48 \, a^{4} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \,{\left (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 [B] time = 1.50484, size = 370, normalized size = 2.64 \begin{align*} \frac{a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 12 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 183 \,{\left (d x + c\right )} a^{4} - 48 \, a^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 57 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{88 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 57 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 12 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - a^{4}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3}} + \frac{2 \,{\left (57 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 96 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 81 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 96 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 81 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 32 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 57 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 32 \, a^{4}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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