Optimal. Leaf size=94 \[ -\frac{3 a \cos (c+d x)}{2 d}-\frac{3 a \cot (c+d x)}{2 d}+\frac{a \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac{a \cos (c+d x) \cot ^2(c+d x)}{2 d}+\frac{3 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3 a x}{2} \]
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Rubi [A] time = 0.109099, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {2838, 2592, 288, 321, 206, 2591, 203} \[ -\frac{3 a \cos (c+d x)}{2 d}-\frac{3 a \cot (c+d x)}{2 d}+\frac{a \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac{a \cos (c+d x) \cot ^2(c+d x)}{2 d}+\frac{3 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3 a x}{2} \]
Antiderivative was successfully verified.
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Rule 2838
Rule 2592
Rule 288
Rule 321
Rule 206
Rule 2591
Rule 203
Rubi steps
\begin{align*} \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cos ^2(c+d x) \cot ^2(c+d x) \, dx+a \int \cos (c+d x) \cot ^3(c+d x) \, dx\\ &=-\frac{a \operatorname{Subst}\left (\int \frac{x^4}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{d}-\frac{a \operatorname{Subst}\left (\int \frac{x^4}{\left (1+x^2\right )^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac{a \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac{a \cos (c+d x) \cot ^2(c+d x)}{2 d}+\frac{(3 a) \operatorname{Subst}\left (\int \frac{x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{2 d}-\frac{(3 a) \operatorname{Subst}\left (\int \frac{x^2}{1+x^2} \, dx,x,\cot (c+d x)\right )}{2 d}\\ &=-\frac{3 a \cos (c+d x)}{2 d}-\frac{3 a \cot (c+d x)}{2 d}+\frac{a \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac{a \cos (c+d x) \cot ^2(c+d x)}{2 d}+\frac{(3 a) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{2 d}+\frac{(3 a) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{2 d}\\ &=-\frac{3 a x}{2}+\frac{3 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3 a \cos (c+d x)}{2 d}-\frac{3 a \cot (c+d x)}{2 d}+\frac{a \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac{a \cos (c+d x) \cot ^2(c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.795045, size = 94, normalized size = 1. \[ -\frac{a \left (2 \sin (2 (c+d x))+8 \cos (c+d x)+8 \cot (c+d x)+\csc ^2\left (\frac{1}{2} (c+d x)\right )-\sec ^2\left (\frac{1}{2} (c+d x)\right )+12 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-12 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+12 c+12 d x\right )}{8 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 143, normalized size = 1.5 \begin{align*} -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{d\sin \left ( dx+c \right ) }}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }{d}}-{\frac{3\,\cos \left ( dx+c \right ) a\sin \left ( dx+c \right ) }{2\,d}}-{\frac{3\,ax}{2}}-{\frac{3\,ca}{2\,d}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{2\,d}}-{\frac{3\,\cos \left ( dx+c \right ) a}{2\,d}}-{\frac{3\,a\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 = 1.55659, size = 136, normalized size = 1.45 \begin{align*} -\frac{2 \,{\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 - a{\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 )}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57165, size = 365, normalized size = 3.88 \begin{align*} -\frac{6 \, a d x \cos \left (d x + c\right )^{2} + 4 \, a \cos \left (d x + c\right )^{3} - 6 \, a d x - 6 \, a \cos \left (d x + c\right ) - 3 \,{\left (a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 3 \,{\left (a \cos \left (d x + c\right )^{2} - a\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 2 \,{\left (a \cos \left (d x + c\right )^{3} - 3 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \,{\left (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 = 1.41398, size = 220, normalized size = 2.34 \begin{align*} \frac{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 12 \,{\left (d x + c\right )} a - 12 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 4 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{6 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 4 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 5 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 16 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 12 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 4 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - a}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}^{2}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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