Optimal. Leaf size=89 \[ \frac{a \cos ^3(c+d x)}{3 d}+\frac{a \cos (c+d x)}{d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}+\frac{3 a x}{8} \]
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Rubi [A] time = 0.086916, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {2838, 2592, 302, 206, 2635, 8} \[ \frac{a \cos ^3(c+d x)}{3 d}+\frac{a \cos (c+d x)}{d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}+\frac{3 a x}{8} \]
Antiderivative was successfully verified.
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Rule 2838
Rule 2592
Rule 302
Rule 206
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cos ^4(c+d x) \, dx+a \int \cos ^3(c+d x) \cot (c+d x) \, dx\\ &=\frac{a \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac{1}{4} (3 a) \int \cos ^2(c+d x) \, dx-\frac{a \operatorname{Subst}\left (\int \frac{x^4}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{3 a \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac{1}{8} (3 a) \int 1 \, dx-\frac{a \operatorname{Subst}\left (\int \left (-1-x^2+\frac{1}{1-x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{3 a x}{8}+\frac{a \cos (c+d x)}{d}+\frac{a \cos ^3(c+d x)}{3 d}+\frac{3 a \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac{a \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{3 a x}{8}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a \cos (c+d x)}{d}+\frac{a \cos ^3(c+d x)}{3 d}+\frac{3 a \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a \cos ^3(c+d x) \sin (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.146551, size = 81, normalized size = 0.91 \[ \frac{a \left (120 \cos (c+d x)+8 \cos (3 (c+d x))+3 \left (8 \sin (2 (c+d x))+\sin (4 (c+d x))+32 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-32 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+12 c+12 d x\right )\right )}{96 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 97, normalized size = 1.1 \begin{align*}{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }{4\,d}}+{\frac{3\,\cos \left ( dx+c \right ) a\sin \left ( dx+c \right ) }{8\,d}}+{\frac{3\,ax}{8}}+{\frac{3\,ca}{8\,d}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{\cos \left ( dx+c \right ) a}{d}}+{\frac{a\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14849, size = 109, normalized size = 1.22 \begin{align*} \frac{16 \,{\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 + 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}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54472, size = 252, normalized size = 2.83 \begin{align*} \frac{8 \, a \cos \left (d x + c\right )^{3} + 9 \, a d x + 24 \, a \cos \left (d x + c\right ) - 12 \, a \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 12 \, a \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 3 \,{\left (2 \, a \cos \left (d x + c\right )^{3} + 3 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \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 = 1.46771, size = 196, normalized size = 2.2 \begin{align*} \frac{9 \,{\left (d x + c\right )} a + 24 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - \frac{2 \,{\left (15 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 48 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 9 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 96 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 9 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 80 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 32 \, a\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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