Optimal. Leaf size=231 \[ -\frac{a \left (a^2-17 b^2\right ) \cos (c+d x)}{2 d}-\frac{\left (a^2-4 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 a^2 d}-\frac{\left (a^2-6 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{4 a d}-\frac{b \left (2 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a \left (a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3}{8} b x \left (12 a^2-b^2\right )-\frac{b \cot (c+d x) (a+b \sin (c+d x))^4}{a^2 d}-\frac{\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^4}{2 a d} \]
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Rubi [A] time = 0.691722, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2893, 3049, 3033, 3023, 2735, 3770} \[ -\frac{a \left (a^2-17 b^2\right ) \cos (c+d x)}{2 d}-\frac{\left (a^2-4 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 a^2 d}-\frac{\left (a^2-6 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{4 a d}-\frac{b \left (2 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a \left (a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3}{8} b x \left (12 a^2-b^2\right )-\frac{b \cot (c+d x) (a+b \sin (c+d x))^4}{a^2 d}-\frac{\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^4}{2 a d} \]
Antiderivative was successfully verified.
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Rule 2893
Rule 3049
Rule 3033
Rule 3023
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^3 \, dx &=-\frac{b \cot (c+d x) (a+b \sin (c+d x))^4}{a^2 d}-\frac{\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^4}{2 a d}-\frac{\int \csc (c+d x) (a+b \sin (c+d x))^3 \left (3 \left (a^2-2 b^2\right )+3 a b \sin (c+d x)-2 \left (a^2-4 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{2 a^2}\\ &=-\frac{\left (a^2-4 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 a^2 d}-\frac{b \cot (c+d x) (a+b \sin (c+d x))^4}{a^2 d}-\frac{\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^4}{2 a d}-\frac{\int \csc (c+d x) (a+b \sin (c+d x))^2 \left (12 a \left (a^2-2 b^2\right )+18 a^2 b \sin (c+d x)-6 a \left (a^2-6 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{8 a^2}\\ &=-\frac{\left (a^2-6 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{4 a d}-\frac{\left (a^2-4 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 a^2 d}-\frac{b \cot (c+d x) (a+b \sin (c+d x))^4}{a^2 d}-\frac{\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^4}{2 a d}-\frac{\int \csc (c+d x) (a+b \sin (c+d x)) \left (36 a^2 \left (a^2-2 b^2\right )+78 a^3 b \sin (c+d x)-6 a^2 \left (2 a^2-21 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{24 a^2}\\ &=-\frac{b \left (2 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}-\frac{\left (a^2-6 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{4 a d}-\frac{\left (a^2-4 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 a^2 d}-\frac{b \cot (c+d x) (a+b \sin (c+d x))^4}{a^2 d}-\frac{\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^4}{2 a d}-\frac{\int \csc (c+d x) \left (72 a^3 \left (a^2-2 b^2\right )+18 a^2 b \left (12 a^2-b^2\right ) \sin (c+d x)-24 a^3 \left (a^2-17 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{48 a^2}\\ &=-\frac{a \left (a^2-17 b^2\right ) \cos (c+d x)}{2 d}-\frac{b \left (2 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}-\frac{\left (a^2-6 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{4 a d}-\frac{\left (a^2-4 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 a^2 d}-\frac{b \cot (c+d x) (a+b \sin (c+d x))^4}{a^2 d}-\frac{\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^4}{2 a d}-\frac{\int \csc (c+d x) \left (72 a^3 \left (a^2-2 b^2\right )+18 a^2 b \left (12 a^2-b^2\right ) \sin (c+d x)\right ) \, dx}{48 a^2}\\ &=-\frac{3}{8} b \left (12 a^2-b^2\right ) x-\frac{a \left (a^2-17 b^2\right ) \cos (c+d x)}{2 d}-\frac{b \left (2 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}-\frac{\left (a^2-6 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{4 a d}-\frac{\left (a^2-4 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 a^2 d}-\frac{b \cot (c+d x) (a+b \sin (c+d x))^4}{a^2 d}-\frac{\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^4}{2 a d}-\frac{1}{2} \left (3 a \left (a^2-2 b^2\right )\right ) \int \csc (c+d x) \, dx\\ &=-\frac{3}{8} b \left (12 a^2-b^2\right ) x+\frac{3 a \left (a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \left (a^2-17 b^2\right ) \cos (c+d x)}{2 d}-\frac{b \left (2 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}-\frac{\left (a^2-6 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{4 a d}-\frac{\left (a^2-4 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 a^2 d}-\frac{b \cot (c+d x) (a+b \sin (c+d x))^4}{a^2 d}-\frac{\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^4}{2 a d}\\ \end{align*}
Mathematica [A] time = 6.16045, size = 252, normalized size = 1.09 \[ \frac{3 b \left (b^2-12 a^2\right ) (c+d x)}{8 d}+\frac{b \left (b^2-3 a^2\right ) \sin (2 (c+d x))}{4 d}-\frac{a \left (4 a^2-15 b^2\right ) \cos (c+d x)}{4 d}-\frac{3 \left (a^3-2 a b^2\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}+\frac{3 \left (a^3-2 a b^2\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}+\frac{3 a^2 b \tan \left (\frac{1}{2} (c+d x)\right )}{2 d}-\frac{3 a^2 b \cot \left (\frac{1}{2} (c+d x)\right )}{2 d}-\frac{a^3 \csc ^2\left (\frac{1}{2} (c+d x)\right )}{8 d}+\frac{a^3 \sec ^2\left (\frac{1}{2} (c+d x)\right )}{8 d}+\frac{a b^2 \cos (3 (c+d x))}{4 d}+\frac{b^3 \sin (4 (c+d x))}{32 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.103, size = 279, normalized size = 1.2 \begin{align*} -{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{2\,d}}-{\frac{3\,{a}^{3}\cos \left ( dx+c \right ) }{2\,d}}-{\frac{3\,{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{2\,d}}-3\,{\frac{{a}^{2}b \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{d\sin \left ( dx+c \right ) }}-3\,{\frac{{a}^{2}b\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{d}}-{\frac{9\,{a}^{2}b\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}-{\frac{9\,{a}^{2}bx}{2}}-{\frac{9\,{a}^{2}bc}{2\,d}}+{\frac{a{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{d}}+3\,{\frac{a{b}^{2}\cos \left ( dx+c \right ) }{d}}+3\,{\frac{a{b}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}}+{\frac{{b}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }{4\,d}}+{\frac{3\,{b}^{3}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{8\,d}}+{\frac{3\,{b}^{3}x}{8}}+{\frac{3\,{b}^{3}c}{8\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.54, size = 251, normalized size = 1.09 \begin{align*} -\frac{48 \,{\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} b - 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 b^{2} -{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} b^{3} - 8 \, a^{3}{\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 )}}{32 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99461, size = 616, normalized size = 2.67 \begin{align*} \frac{8 \, a b^{2} \cos \left (d x + c\right )^{5} - 3 \,{\left (12 \, a^{2} b - b^{3}\right )} d x \cos \left (d x + c\right )^{2} - 8 \,{\left (a^{3} - 2 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (12 \, a^{2} b - b^{3}\right )} d x + 12 \,{\left (a^{3} - 2 \, a b^{2}\right )} \cos \left (d x + c\right ) - 6 \,{\left (a^{3} - 2 \, a b^{2} -{\left (a^{3} - 2 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 6 \,{\left (a^{3} - 2 \, a b^{2} -{\left (a^{3} - 2 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) +{\left (2 \, b^{3} \cos \left (d x + c\right )^{5} -{\left (12 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (12 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \,{\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 = 1.32357, size = 540, normalized size = 2.34 \begin{align*} \frac{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 12 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 3 \,{\left (12 \, a^{2} b - b^{3}\right )}{\left (d x + c\right )} - 12 \,{\left (a^{3} - 2 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + \frac{18 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 36 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 12 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}} + \frac{2 \,{\left (12 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 5 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 8 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 48 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 12 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 24 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 96 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 12 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 3 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 24 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 80 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 12 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 5 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 8 \, a^{3} + 32 \, a b^{2}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{4}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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