Optimal. Leaf size=121 \[ \frac{a \cos ^5(c+d x)}{5 d}+\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \cos (c+d x)}{d}-\frac{15 a \cot (c+d x)}{8 d}+\frac{a \cos ^4(c+d x) \cot (c+d x)}{4 d}+\frac{5 a \cos ^2(c+d x) \cot (c+d x)}{8 d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}-\frac{15 a x}{8} \]
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Rubi [A] time = 0.133799, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {2838, 2591, 288, 321, 203, 2592, 302, 206} \[ \frac{a \cos ^5(c+d x)}{5 d}+\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \cos (c+d x)}{d}-\frac{15 a \cot (c+d x)}{8 d}+\frac{a \cos ^4(c+d x) \cot (c+d x)}{4 d}+\frac{5 a \cos ^2(c+d x) \cot (c+d x)}{8 d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}-\frac{15 a x}{8} \]
Antiderivative was successfully verified.
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Rule 2838
Rule 2591
Rule 288
Rule 321
Rule 203
Rule 2592
Rule 302
Rule 206
Rubi steps
\begin{align*} \int \cos ^4(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cos ^5(c+d x) \cot (c+d x) \, dx+a \int \cos ^4(c+d x) \cot ^2(c+d x) \, dx\\ &=-\frac{a \operatorname{Subst}\left (\int \frac{x^6}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}-\frac{a \operatorname{Subst}\left (\int \frac{x^6}{\left (1+x^2\right )^3} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac{a \cos ^4(c+d x) \cot (c+d x)}{4 d}-\frac{a \operatorname{Subst}\left (\int \left (-1-x^2-x^4+\frac{1}{1-x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac{(5 a) \operatorname{Subst}\left (\int \frac{x^4}{\left (1+x^2\right )^2} \, dx,x,\cot (c+d x)\right )}{4 d}\\ &=\frac{a \cos (c+d x)}{d}+\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \cos ^5(c+d x)}{5 d}+\frac{5 a \cos ^2(c+d x) \cot (c+d x)}{8 d}+\frac{a \cos ^4(c+d x) \cot (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{(15 a) \operatorname{Subst}\left (\int \frac{x^2}{1+x^2} \, dx,x,\cot (c+d x)\right )}{8 d}\\ &=-\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{a \cos ^5(c+d x)}{5 d}-\frac{15 a \cot (c+d x)}{8 d}+\frac{5 a \cos ^2(c+d x) \cot (c+d x)}{8 d}+\frac{a \cos ^4(c+d x) \cot (c+d x)}{4 d}+\frac{(15 a) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{8 d}\\ &=-\frac{15 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{a \cos ^5(c+d x)}{5 d}-\frac{15 a \cot (c+d x)}{8 d}+\frac{5 a \cos ^2(c+d x) \cot (c+d x)}{8 d}+\frac{a \cos ^4(c+d x) \cot (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.259947, size = 98, normalized size = 0.81 \[ -\frac{a \left (240 \sin (2 (c+d x))+15 \sin (4 (c+d x))-660 \cos (c+d x)-70 \cos (3 (c+d x))-6 \cos (5 (c+d x))+480 \cot (c+d x)-480 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+480 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+900 c+900 d x\right )}{480 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 153, normalized size = 1.3 \begin{align*}{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{5\,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}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{d\sin \left ( dx+c \right ) }}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) }{d}}-{\frac{5\,a \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }{4\,d}}-{\frac{15\,\cos \left ( dx+c \right ) a\sin \left ( dx+c \right ) }{8\,d}}-{\frac{15\,ax}{8}}-{\frac{15\,ca}{8\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50333, size = 163, normalized size = 1.35 \begin{align*} \frac{4 \,{\left (6 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 30 \, \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 - 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}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.18991, size = 375, normalized size = 3.1 \begin{align*} \frac{30 \, a \cos \left (d x + c\right )^{5} + 75 \, a \cos \left (d x + c\right )^{3} - 60 \, a \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) + 60 \, a \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 225 \, a \cos \left (d x + c\right ) +{\left (24 \, a \cos \left (d x + c\right )^{5} + 40 \, a \cos \left (d x + c\right )^{3} - 225 \, a d x + 120 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, d \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 = 1.18444, size = 267, normalized size = 2.21 \begin{align*} -\frac{225 \,{\left (d x + c\right )} a - 120 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 60 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{60 \,{\left (2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a\right )}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )} - \frac{2 \,{\left (135 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 360 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 150 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 720 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 1120 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 150 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 560 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 135 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 184 \, a\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{5}}}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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