Optimal. Leaf size=152 \[ \frac{b \left (2 a^2-b^2\right ) \cot (c+d x)}{2 d}+\frac{a \left (a^2+12 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{a \left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}+b^3 (-x) \]
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Rubi [A] time = 0.510627, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2889, 3048, 3047, 3031, 3021, 2735, 3770} \[ \frac{b \left (2 a^2-b^2\right ) \cot (c+d x)}{2 d}+\frac{a \left (a^2+12 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{a \left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}+b^3 (-x) \]
Antiderivative was successfully verified.
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Rule 2889
Rule 3048
Rule 3047
Rule 3031
Rule 3021
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int \cot ^2(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3 \, dx &=\int \csc ^5(c+d x) (a+b \sin (c+d x))^3 \left (1-\sin ^2(c+d x)\right ) \, dx\\ &=-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}+\frac{1}{4} \int \csc ^4(c+d x) (a+b \sin (c+d x))^2 \left (3 b-a \sin (c+d x)-4 b \sin ^2(c+d x)\right ) \, dx\\ &=-\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}+\frac{1}{12} \int \csc ^3(c+d x) (a+b \sin (c+d x)) \left (-3 \left (a^2-2 b^2\right )-9 a b \sin (c+d x)-12 b^2 \sin ^2(c+d x)\right ) \, dx\\ &=\frac{a \left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}-\frac{1}{24} \int \csc ^2(c+d x) \left (12 b \left (2 a^2-b^2\right )+3 a \left (a^2+12 b^2\right ) \sin (c+d x)+24 b^3 \sin ^2(c+d x)\right ) \, dx\\ &=\frac{b \left (2 a^2-b^2\right ) \cot (c+d x)}{2 d}+\frac{a \left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}-\frac{1}{24} \int \csc (c+d x) \left (3 a \left (a^2+12 b^2\right )+24 b^3 \sin (c+d x)\right ) \, dx\\ &=-b^3 x+\frac{b \left (2 a^2-b^2\right ) \cot (c+d x)}{2 d}+\frac{a \left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}-\frac{1}{8} \left (a \left (a^2+12 b^2\right )\right ) \int \csc (c+d x) \, dx\\ &=-b^3 x+\frac{a \left (a^2+12 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{b \left (2 a^2-b^2\right ) \cot (c+d x)}{2 d}+\frac{a \left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}\\ \end{align*}
Mathematica [B] time = 6.21487, size = 690, normalized size = 4.54 \[ \frac{\left (a^3-12 a b^2\right ) \sin ^3(c+d x) \csc ^2\left (\frac{1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^3}{32 d (a+b \sin (c+d x))^3}+\frac{\sin ^3(c+d x) \csc \left (\frac{1}{2} (c+d x)\right ) \left (a^2 b \cos \left (\frac{1}{2} (c+d x)\right )-b^3 \cos \left (\frac{1}{2} (c+d x)\right )\right ) (a \csc (c+d x)+b)^3}{2 d (a+b \sin (c+d x))^3}+\frac{\left (-a^3-12 a b^2\right ) \sin ^3(c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right ) (a \csc (c+d x)+b)^3}{8 d (a+b \sin (c+d x))^3}+\frac{\left (12 a b^2-a^3\right ) \sin ^3(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^3}{32 d (a+b \sin (c+d x))^3}+\frac{\sin ^3(c+d x) \sec \left (\frac{1}{2} (c+d x)\right ) \left (b^3 \sin \left (\frac{1}{2} (c+d x)\right )-a^2 b \sin \left (\frac{1}{2} (c+d x)\right )\right ) (a \csc (c+d x)+b)^3}{2 d (a+b \sin (c+d x))^3}+\frac{\left (a^3+12 a b^2\right ) \sin ^3(c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right ) (a \csc (c+d x)+b)^3}{8 d (a+b \sin (c+d x))^3}-\frac{a^3 \sin ^3(c+d x) \csc ^4\left (\frac{1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^3}{64 d (a+b \sin (c+d x))^3}-\frac{a^2 b \sin ^3(c+d x) \cot \left (\frac{1}{2} (c+d x)\right ) \csc ^2\left (\frac{1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^3}{8 d (a+b \sin (c+d x))^3}+\frac{a^3 \sin ^3(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^3}{64 d (a+b \sin (c+d x))^3}+\frac{a^2 b \sin ^3(c+d x) \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^3}{8 d (a+b \sin (c+d x))^3}-\frac{b^3 (c+d x) \sin ^3(c+d x) (a \csc (c+d x)+b)^3}{d (a+b \sin (c+d x))^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.098, size = 207, normalized size = 1.4 \begin{align*} -{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{a}^{3}\cos \left ( dx+c \right ) }{8\,d}}-{\frac{{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{8\,d}}-{\frac{{a}^{2}b \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{3\,a{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3\,a{b}^{2}\cos \left ( dx+c \right ) }{2\,d}}-{\frac{3\,a{b}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{2\,d}}-{b}^{3}x-{\frac{\cot \left ( dx+c \right ){b}^{3}}{d}}-{\frac{{b}^{3}c}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.63156, size = 201, normalized size = 1.32 \begin{align*} -\frac{16 \,{\left (d x + c + \frac{1}{\tan \left (d x + c\right )}\right )} b^{3} + a^{3}{\left (\frac{2 \,{\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 12 \, a b^{2}{\left (\frac{2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + \log \left (\cos \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac{16 \, a^{2} b}{\tan \left (d x + c\right )^{3}}}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63152, size = 666, normalized size = 4.38 \begin{align*} -\frac{16 \, b^{3} d x \cos \left (d x + c\right )^{4} - 32 \, b^{3} d x \cos \left (d x + c\right )^{2} + 16 \, b^{3} d x + 2 \,{\left (a^{3} - 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 2 \,{\left (a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right ) -{\left ({\left (a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + a^{3} + 12 \, a b^{2} - 2 \,{\left (a^{3} + 12 \, 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 ({\left (a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + a^{3} + 12 \, a b^{2} - 2 \,{\left (a^{3} + 12 \, 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 ) + 16 \,{\left (b^{3} \cos \left (d x + c\right ) +{\left (a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{16 \,{\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 = 1.35072, size = 316, normalized size = 2.08 \begin{align*} \frac{3 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 24 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 72 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 192 \,{\left (d x + c\right )} b^{3} - 72 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 96 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 24 \,{\left (a^{3} + 12 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + \frac{50 \, 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} + 72 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 96 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 72 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 24 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 3 \, a^{3}}{\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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