Optimal. Leaf size=227 \[ -\frac{b^3 \left (83 a^2+2 b^2\right ) \cos (c+d x)}{40 a^2 d}-\frac{a \left (4 a^2-29 b^2\right ) \cot (c+d x)}{20 d}-\frac{3 b \left (3 a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}+3 a b^2 x-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}+\frac{2 \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac{27 b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{40 d} \]
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Rubi [A] time = 0.711053, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2893, 3047, 3031, 3023, 2735, 3770} \[ -\frac{b^3 \left (83 a^2+2 b^2\right ) \cos (c+d x)}{40 a^2 d}-\frac{a \left (4 a^2-29 b^2\right ) \cot (c+d x)}{20 d}-\frac{3 b \left (3 a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}+3 a b^2 x-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}+\frac{2 \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac{27 b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{40 d} \]
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 ^2(c+d x) (a+b \sin (c+d x))^3 \, dx &=\frac{b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}-\frac{\int \csc ^4(c+d x) (a+b \sin (c+d x))^3 \left (24 a^2+3 a b \sin (c+d x)-\left (20 a^2+b^2\right ) \sin ^2(c+d x)\right ) \, dx}{20 a^2}\\ &=\frac{2 \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac{b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}-\frac{\int \csc ^3(c+d x) (a+b \sin (c+d x))^2 \left (81 a^2 b-6 a \left (2 a^2-b^2\right ) \sin (c+d x)-3 b \left (28 a^2+b^2\right ) \sin ^2(c+d x)\right ) \, dx}{60 a^2}\\ &=\frac{27 b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac{2 \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac{b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}-\frac{\int \csc ^2(c+d x) (a+b \sin (c+d x)) \left (-6 a^2 \left (4 a^2-29 b^2\right )-3 a b \left (37 a^2-2 b^2\right ) \sin (c+d x)-3 b^2 \left (83 a^2+2 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{120 a^2}\\ &=-\frac{a \left (4 a^2-29 b^2\right ) \cot (c+d x)}{20 d}+\frac{27 b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac{2 \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac{b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}+\frac{\int \csc (c+d x) \left (45 a^2 b \left (3 a^2-4 b^2\right )+360 a^3 b^2 \sin (c+d x)+3 b^3 \left (83 a^2+2 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{120 a^2}\\ &=-\frac{b^3 \left (83 a^2+2 b^2\right ) \cos (c+d x)}{40 a^2 d}-\frac{a \left (4 a^2-29 b^2\right ) \cot (c+d x)}{20 d}+\frac{27 b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac{2 \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac{b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}+\frac{\int \csc (c+d x) \left (45 a^2 b \left (3 a^2-4 b^2\right )+360 a^3 b^2 \sin (c+d x)\right ) \, dx}{120 a^2}\\ &=3 a b^2 x-\frac{b^3 \left (83 a^2+2 b^2\right ) \cos (c+d x)}{40 a^2 d}-\frac{a \left (4 a^2-29 b^2\right ) \cot (c+d x)}{20 d}+\frac{27 b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac{2 \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac{b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}+\frac{1}{8} \left (3 b \left (3 a^2-4 b^2\right )\right ) \int \csc (c+d x) \, dx\\ &=3 a b^2 x-\frac{3 b \left (3 a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{b^3 \left (83 a^2+2 b^2\right ) \cos (c+d x)}{40 a^2 d}-\frac{a \left (4 a^2-29 b^2\right ) \cot (c+d x)}{20 d}+\frac{27 b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac{2 \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac{b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}\\ \end{align*}
Mathematica [A] time = 1.24748, size = 405, normalized size = 1.78 \[ \frac{-32 \left (a^3-20 a b^2\right ) \cot \left (\frac{1}{2} (c+d x)\right )-15 a^2 b \csc ^4\left (\frac{1}{2} (c+d x)\right )+150 a^2 b \csc ^2\left (\frac{1}{2} (c+d x)\right )+15 a^2 b \sec ^4\left (\frac{1}{2} (c+d x)\right )-150 a^2 b \sec ^2\left (\frac{1}{2} (c+d x)\right )+360 a^2 b \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-360 a^2 b \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+32 a^3 \tan \left (\frac{1}{2} (c+d x)\right )+64 a^3 \sin ^6\left (\frac{1}{2} (c+d x)\right ) \csc ^5(c+d x)-112 a^3 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-a^3 \sin (c+d x) \csc ^6\left (\frac{1}{2} (c+d x)\right )+7 a^3 \sin (c+d x) \csc ^4\left (\frac{1}{2} (c+d x)\right )-640 a b^2 \tan \left (\frac{1}{2} (c+d x)\right )+320 a b^2 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-20 a b^2 \sin (c+d x) \csc ^4\left (\frac{1}{2} (c+d x)\right )+960 a b^2 c+960 a b^2 d x-320 b^3 \cos (c+d x)-40 b^3 \csc ^2\left (\frac{1}{2} (c+d x)\right )+40 b^3 \sec ^2\left (\frac{1}{2} (c+d x)\right )-480 b^3 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+480 b^3 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{320 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.109, size = 260, normalized size = 1.2 \begin{align*} -{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{3\,{a}^{2}b \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{3\,{a}^{2}b \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,{a}^{2}b \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{8\,d}}+{\frac{9\,{a}^{2}b\cos \left ( dx+c \right ) }{8\,d}}+{\frac{9\,{a}^{2}b\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{8\,d}}-{\frac{a{b}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{d}}+3\,a{b}^{2}x+3\,{\frac{a{b}^{2}\cot \left ( dx+c \right ) }{d}}+3\,{\frac{a{b}^{2}c}{d}}-{\frac{{b}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{b}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{2\,d}}-{\frac{3\,{b}^{3}\cos \left ( dx+c \right ) }{2\,d}}-{\frac{3\,{b}^{3}\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 = 1.68249, size = 246, normalized size = 1.08 \begin{align*} \frac{80 \,{\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^{2} - 15 \, a^{2} b{\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 )} + 20 \, b^{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 )} - \frac{16 \, a^{3}}{\tan \left (d x + c\right )^{5}}}{80 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92508, size = 832, normalized size = 3.67 \begin{align*} -\frac{560 \, a b^{2} \cos \left (d x + c\right )^{3} + 16 \,{\left (a^{3} - 20 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} - 240 \, a b^{2} \cos \left (d x + c\right ) + 15 \,{\left ({\left (3 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{4} + 3 \, a^{2} b - 4 \, b^{3} - 2 \,{\left (3 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 15 \,{\left ({\left (3 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{4} + 3 \, a^{2} b - 4 \, b^{3} - 2 \,{\left (3 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 10 \,{\left (24 \, a b^{2} d x \cos \left (d x + c\right )^{4} - 8 \, b^{3} \cos \left (d x + c\right )^{5} - 48 \, a b^{2} d x \cos \left (d x + c\right )^{2} + 24 \, a b^{2} d x - 5 \,{\left (3 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (3 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \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 = 1.43498, size = 481, normalized size = 2.12 \begin{align*} \frac{2 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 15 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 10 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 40 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 120 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 40 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 960 \,{\left (d x + c\right )} a b^{2} + 20 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 600 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{640 \, b^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1} + 120 \,{\left (3 \, a^{2} b - 4 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - \frac{822 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 1096 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 20 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 600 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 120 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 40 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 10 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 40 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 15 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \, a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}}}{320 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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