Optimal. Leaf size=98 \[ -\frac{2 a^2 \cos (c+d x)}{d}-\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{d}-\frac{a^2 x}{2} \]
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Rubi [A] time = 0.155112, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2709, 3770, 3767, 8, 3768, 2638, 2635} \[ -\frac{2 a^2 \cos (c+d x)}{d}-\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{d}-\frac{a^2 x}{2} \]
Antiderivative was successfully verified.
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Rule 2709
Rule 3770
Rule 3767
Rule 8
Rule 3768
Rule 2638
Rule 2635
Rubi steps
\begin{align*} \int \cot ^4(c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac{\int \left (-a^6-4 a^6 \csc (c+d x)-a^6 \csc ^2(c+d x)+2 a^6 \csc ^3(c+d x)+a^6 \csc ^4(c+d x)+2 a^6 \sin (c+d x)+a^6 \sin ^2(c+d x)\right ) \, dx}{a^4}\\ &=-a^2 x-a^2 \int \csc ^2(c+d x) \, dx+a^2 \int \csc ^4(c+d x) \, dx+a^2 \int \sin ^2(c+d x) \, dx+\left (2 a^2\right ) \int \csc ^3(c+d x) \, dx+\left (2 a^2\right ) \int \sin (c+d x) \, dx-\left (4 a^2\right ) \int \csc (c+d x) \, dx\\ &=-a^2 x+\frac{4 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{2 a^2 \cos (c+d x)}{d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{d}-\frac{a^2 \cos (c+d x) \sin (c+d x)}{2 d}+\frac{1}{2} a^2 \int 1 \, dx+a^2 \int \csc (c+d x) \, dx+\frac{a^2 \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}-\frac{a^2 \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac{a^2 x}{2}+\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{2 a^2 \cos (c+d x)}{d}-\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{d}-\frac{a^2 \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 5.3888, size = 191, normalized size = 1.95 \[ \frac{a^2 (\sin (c+d x)+1)^2 \left (-12 (c+d x)-6 \sin (2 (c+d x))-48 \cos (c+d x)-4 \tan \left (\frac{1}{2} (c+d x)\right )+4 \cot \left (\frac{1}{2} (c+d x)\right )-6 \csc ^2\left (\frac{1}{2} (c+d x)\right )+6 \sec ^2\left (\frac{1}{2} (c+d x)\right )-72 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+72 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+8 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-\frac{1}{2} \sin (c+d x) \csc ^4\left (\frac{1}{2} (c+d x)\right )\right )}{24 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.078, size = 190, normalized size = 1.9 \begin{align*} -{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{d\sin \left ( dx+c \right ) }}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }{d}}-{\frac{3\,{a}^{2}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}-{\frac{{a}^{2}x}{2}}-{\frac{c{a}^{2}}{2\,d}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{d}}-3\,{\frac{{a}^{2}\cos \left ( dx+c \right ) }{d}}-3\,{\frac{{a}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{{a}^{2}\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 = 1.74867, size = 188, normalized size = 1.92 \begin{align*} -\frac{3 \,{\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^{2} - 2 \,{\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^{2} - 3 \, a^{2}{\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 )}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.43615, size = 474, normalized size = 4.84 \begin{align*} \frac{3 \, a^{2} \cos \left (d x + c\right )^{5} - 4 \, a^{2} \cos \left (d x + c\right )^{3} + 3 \, a^{2} \cos \left (d x + c\right ) + 9 \,{\left (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 ) \sin \left (d x + c\right ) - 9 \,{\left (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 ) \sin \left (d x + c\right ) - 3 \,{\left (a^{2} d x \cos \left (d x + c\right )^{2} + 4 \, a^{2} \cos \left (d x + c\right )^{3} - a^{2} d x - 6 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \,{\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 = 1.31259, size = 282, normalized size = 2.88 \begin{align*} \frac{a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 12 \,{\left (d x + c\right )} a^{2} - 72 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{24 \,{\left (a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 4 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 4 \, a^{2}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{2}} + \frac{132 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 6 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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