Optimal. Leaf size=178 \[ -\frac{b^2 \left (39 a^2+2 b^2\right ) \cos (c+d x)}{24 a^2 d}-\frac{3 \left (a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{12 a^2 d}+\frac{17 a b \cot (c+d x)}{12 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 a d}+\frac{5 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{8 d}+2 a b x \]
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Rubi [A] time = 0.462778, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2893, 3047, 3031, 3023, 2735, 3770} \[ -\frac{b^2 \left (39 a^2+2 b^2\right ) \cos (c+d x)}{24 a^2 d}-\frac{3 \left (a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{12 a^2 d}+\frac{17 a b \cot (c+d x)}{12 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 a d}+\frac{5 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{8 d}+2 a b x \]
Antiderivative was successfully verified.
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Rule 2893
Rule 3047
Rule 3031
Rule 3023
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx &=\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{12 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 a d}-\frac{\int \csc ^3(c+d x) (a+b \sin (c+d x))^2 \left (15 a^2+2 a b \sin (c+d x)-\left (12 a^2+b^2\right ) \sin ^2(c+d x)\right ) \, dx}{12 a^2}\\ &=\frac{5 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{8 d}+\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{12 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 a d}-\frac{\int \csc ^2(c+d x) (a+b \sin (c+d x)) \left (34 a^2 b-a \left (9 a^2-2 b^2\right ) \sin (c+d x)-b \left (39 a^2+2 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{24 a^2}\\ &=\frac{17 a b \cot (c+d x)}{12 d}+\frac{5 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{8 d}+\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{12 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 a d}+\frac{\int \csc (c+d x) \left (9 a^2 \left (a^2-4 b^2\right )+48 a^3 b \sin (c+d x)+b^2 \left (39 a^2+2 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{24 a^2}\\ &=-\frac{b^2 \left (39 a^2+2 b^2\right ) \cos (c+d x)}{24 a^2 d}+\frac{17 a b \cot (c+d x)}{12 d}+\frac{5 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{8 d}+\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{12 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 a d}+\frac{\int \csc (c+d x) \left (9 a^2 \left (a^2-4 b^2\right )+48 a^3 b \sin (c+d x)\right ) \, dx}{24 a^2}\\ &=2 a b x-\frac{b^2 \left (39 a^2+2 b^2\right ) \cos (c+d x)}{24 a^2 d}+\frac{17 a b \cot (c+d x)}{12 d}+\frac{5 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{8 d}+\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{12 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 a d}+\frac{1}{8} \left (3 \left (a^2-4 b^2\right )\right ) \int \csc (c+d x) \, dx\\ &=2 a b x-\frac{3 \left (a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{b^2 \left (39 a^2+2 b^2\right ) \cos (c+d x)}{24 a^2 d}+\frac{17 a b \cot (c+d x)}{12 d}+\frac{5 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{8 d}+\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{12 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 a d}\\ \end{align*}
Mathematica [A] time = 2.71773, size = 270, normalized size = 1.52 \[ \frac{-3 a^2 \csc ^4\left (\frac{1}{2} (c+d x)\right )+30 a^2 \csc ^2\left (\frac{1}{2} (c+d x)\right )+3 a^2 \sec ^4\left (\frac{1}{2} (c+d x)\right )-30 a^2 \sec ^2\left (\frac{1}{2} (c+d x)\right )+72 a^2 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-72 a^2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-256 a b \tan \left (\frac{1}{2} (c+d x)\right )+256 a b \cot \left (\frac{1}{2} (c+d x)\right )+128 a b \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-8 a b \sin (c+d x) \csc ^4\left (\frac{1}{2} (c+d x)\right )+384 a b c+384 a b d x-192 b^2 \cos (c+d x)-24 b^2 \csc ^2\left (\frac{1}{2} (c+d x)\right )+24 b^2 \sec ^2\left (\frac{1}{2} (c+d x)\right )-288 b^2 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+288 b^2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{192 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.088, size = 223, normalized size = 1.3 \begin{align*} -{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{8\,d}}+{\frac{3\,{a}^{2}\cos \left ( dx+c \right ) }{8\,d}}+{\frac{3\,{a}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{8\,d}}-{\frac{2\,ab \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+2\,{\frac{ab\cot \left ( dx+c \right ) }{d}}+2\,abx+2\,{\frac{abc}{d}}-{\frac{{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{2\,d}}-{\frac{3\,{b}^{2}\cos \left ( dx+c \right ) }{2\,d}}-{\frac{3\,{b}^{2}\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 = 1.61168, size = 224, normalized size = 1.26 \begin{align*} \frac{32 \,{\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 b - 3 \, a^{2}{\left (\frac{2 \,{\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 12 \, b^{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 )}}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83193, size = 666, normalized size = 3.74 \begin{align*} \frac{96 \, a b d x \cos \left (d x + c\right )^{4} - 48 \, b^{2} \cos \left (d x + c\right )^{5} - 192 \, a b d x \cos \left (d x + c\right )^{2} + 96 \, a b d x - 30 \,{\left (a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{3} + 18 \,{\left (a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right ) - 9 \,{\left ({\left (a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \,{\left (a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} - 4 \, b^{2}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 9 \,{\left ({\left (a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \,{\left (a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} - 4 \, b^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 32 \,{\left (4 \, a b \cos \left (d x + c\right )^{3} - 3 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, 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 = 1.37084, size = 329, normalized size = 1.85 \begin{align*} \frac{3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 16 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 24 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 24 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 384 \,{\left (d x + c\right )} a b - 240 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 72 \,{\left (a^{2} - 4 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - \frac{384 \, b^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1} - \frac{150 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 600 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 240 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 24 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 24 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 16 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4}}}{192 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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