Optimal. Leaf size=138 \[ \frac{a \left (a^2-3 b^2\right ) \cot (c+d x)}{3 d}+\frac{b \left (3 a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 d}-3 a b^2 x-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{3 d}-\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{2 d}+\frac{11 b^3 \cos (c+d x)}{6 d} \]
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Rubi [A] time = 0.482681, antiderivative size = 138, 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, 3023, 2735, 3770} \[ \frac{a \left (a^2-3 b^2\right ) \cot (c+d x)}{3 d}+\frac{b \left (3 a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 d}-3 a b^2 x-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{3 d}-\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{2 d}+\frac{11 b^3 \cos (c+d x)}{6 d} \]
Antiderivative was successfully verified.
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Rule 2889
Rule 3048
Rule 3047
Rule 3031
Rule 3023
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int \cot ^2(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3 \, dx &=\int \csc ^4(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 ^2(c+d x) (a+b \sin (c+d x))^3}{3 d}+\frac{1}{3} \int \csc ^3(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 (c+d x) (a+b \sin (c+d x))^2}{2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{3 d}+\frac{1}{6} \int \csc ^2(c+d x) (a+b \sin (c+d x)) \left (-2 \left (a^2-3 b^2\right )-7 a b \sin (c+d x)-11 b^2 \sin ^2(c+d x)\right ) \, dx\\ &=\frac{a \left (a^2-3 b^2\right ) \cot (c+d x)}{3 d}-\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{3 d}-\frac{1}{6} \int \csc (c+d x) \left (3 b \left (3 a^2-2 b^2\right )+18 a b^2 \sin (c+d x)+11 b^3 \sin ^2(c+d x)\right ) \, dx\\ &=\frac{11 b^3 \cos (c+d x)}{6 d}+\frac{a \left (a^2-3 b^2\right ) \cot (c+d x)}{3 d}-\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{3 d}-\frac{1}{6} \int \csc (c+d x) \left (3 b \left (3 a^2-2 b^2\right )+18 a b^2 \sin (c+d x)\right ) \, dx\\ &=-3 a b^2 x+\frac{11 b^3 \cos (c+d x)}{6 d}+\frac{a \left (a^2-3 b^2\right ) \cot (c+d x)}{3 d}-\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{3 d}-\frac{1}{2} \left (b \left (3 a^2-2 b^2\right )\right ) \int \csc (c+d x) \, dx\\ &=-3 a b^2 x+\frac{b \left (3 a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 d}+\frac{11 b^3 \cos (c+d x)}{6 d}+\frac{a \left (a^2-3 b^2\right ) \cot (c+d x)}{3 d}-\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{3 d}\\ \end{align*}
Mathematica [B] time = 6.19941, size = 615, normalized size = 4.46 \[ \frac{\sin ^3(c+d x) \csc \left (\frac{1}{2} (c+d x)\right ) \left (a^3 \cos \left (\frac{1}{2} (c+d x)\right )-9 a b^2 \cos \left (\frac{1}{2} (c+d x)\right )\right ) (a \csc (c+d x)+b)^3}{6 d (a+b \sin (c+d x))^3}+\frac{\left (2 b^3-3 a^2 b\right ) \sin ^3(c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right ) (a \csc (c+d x)+b)^3}{2 d (a+b \sin (c+d x))^3}+\frac{\sin ^3(c+d x) \sec \left (\frac{1}{2} (c+d x)\right ) \left (9 a b^2 \sin \left (\frac{1}{2} (c+d x)\right )-a^3 \sin \left (\frac{1}{2} (c+d x)\right )\right ) (a \csc (c+d x)+b)^3}{6 d (a+b \sin (c+d x))^3}+\frac{\left (3 a^2 b-2 b^3\right ) \sin ^3(c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right ) (a \csc (c+d x)+b)^3}{2 d (a+b \sin (c+d x))^3}-\frac{3 a^2 b \sin ^3(c+d x) \csc ^2\left (\frac{1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^3}{8 d (a+b \sin (c+d x))^3}-\frac{a^3 \sin ^3(c+d x) \cot \left (\frac{1}{2} (c+d x)\right ) \csc ^2\left (\frac{1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^3}{24 d (a+b \sin (c+d x))^3}+\frac{3 a^2 b \sin ^3(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^3}{8 d (a+b \sin (c+d x))^3}+\frac{a^3 \sin ^3(c+d x) \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^3}{24 d (a+b \sin (c+d x))^3}-\frac{3 a b^2 (c+d x) \sin ^3(c+d x) (a \csc (c+d x)+b)^3}{d (a+b \sin (c+d x))^3}+\frac{b^3 \sin ^3(c+d x) \cos (c+d x) (a \csc (c+d x)+b)^3}{d (a+b \sin (c+d x))^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.101, size = 159, normalized size = 1.2 \begin{align*} -{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{3\,{a}^{2}b \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3\,{a}^{2}b\cos \left ( dx+c \right ) }{2\,d}}-{\frac{3\,{a}^{2}b\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{2\,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}\cos \left ( dx+c \right ) }{d}}+{\frac{{b}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.64517, size = 161, normalized size = 1.17 \begin{align*} -\frac{36 \,{\left (d x + c + \frac{1}{\tan \left (d x + c\right )}\right )} a b^{2} - 9 \, a^{2} b{\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 )} - 6 \, b^{3}{\left (2 \, \cos \left (d x + c\right ) - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac{4 \, a^{3}}{\tan \left (d x + c\right )^{3}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60852, size = 562, normalized size = 4.07 \begin{align*} \frac{36 \, a b^{2} \cos \left (d x + c\right ) + 4 \,{\left (a^{3} - 9 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} - 3 \,{\left (3 \, a^{2} b - 2 \, b^{3} -{\left (3 \, a^{2} b - 2 \, 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 ) + 3 \,{\left (3 \, a^{2} b - 2 \, b^{3} -{\left (3 \, a^{2} b - 2 \, 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 ) - 6 \,{\left (6 \, a b^{2} d x \cos \left (d x + c\right )^{2} - 2 \, b^{3} \cos \left (d x + c\right )^{3} - 6 \, a b^{2} d x -{\left (3 \, a^{2} b - 2 \, b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \,{\left (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 = 1.30907, size = 300, normalized size = 2.17 \begin{align*} \frac{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 9 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 72 \,{\left (d x + c\right )} a b^{2} - 3 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 36 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{48 \, b^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1} - 12 \,{\left (3 \, a^{2} b - 2 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + \frac{66 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 44 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 3 \, 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} - 9 \, 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 )^{3}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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