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 = 0.48, antiderivative size = 138, normalized size of antiderivative = 1.00, 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 2735
Rule 2889
Rule 3023
Rule 3031
Rule 3047
Rule 3048
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*}
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Mathematica [B] time = 6.19, size = 615, normalized size = 4.46 \[ \frac {\sin ^3(c+d x) \csc \left (\frac {1}{2} (c+d x)\right ) \left (a^3 \cos \left (\frac {1}{2} (c+d x)\right )-9 a b^2 \cos \left (\frac {1}{2} (c+d x)\right )\right ) (a \csc (c+d x)+b)^3}{6 d (a+b \sin (c+d x))^3}+\frac {\sin ^3(c+d x) \sec \left (\frac {1}{2} (c+d x)\right ) \left (9 a b^2 \sin \left (\frac {1}{2} (c+d x)\right )-a^3 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \csc (c+d x)+b)^3}{6 d (a+b \sin (c+d x))^3}-\frac {a^3 \sin ^3(c+d x) \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^3}{24 d (a+b \sin (c+d x))^3}+\frac {a^3 \sin ^3(c+d x) \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^3}{24 d (a+b \sin (c+d x))^3}+\frac {\left (2 b^3-3 a^2 b\right ) \sin ^3(c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \csc (c+d x)+b)^3}{2 d (a+b \sin (c+d x))^3}+\frac {\left (3 a^2 b-2 b^3\right ) \sin ^3(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) (a \csc (c+d x)+b)^3}{2 d (a+b \sin (c+d x))^3}-\frac {3 a^2 b \sin ^3(c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^3}{8 d (a+b \sin (c+d x))^3}+\frac {3 a^2 b \sin ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^3}{8 d (a+b \sin (c+d x))^3}+\frac {b^3 \sin ^3(c+d x) \cos (c+d x) (a \csc (c+d x)+b)^3}{d (a+b \sin (c+d x))^3}-\frac {3 a b^2 (c+d x) \sin ^3(c+d x) (a \csc (c+d x)+b)^3}{d (a+b \sin (c+d x))^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.74, size = 231, normalized size = 1.67 \[ \frac {36 \, a b^{2} \cos \left (d x + c\right ) + 4 \, {\left (a^{3} - 9 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (3 \, a^{2} b - 2 \, b^{3} - {\left (3 \, a^{2} b - 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 3 \, {\left (3 \, a^{2} b - 2 \, b^{3} - {\left (3 \, a^{2} b - 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 6 \, {\left (6 \, a b^{2} d x \cos \left (d x + c\right )^{2} - 2 \, b^{3} \cos \left (d x + c\right )^{3} - 6 \, a b^{2} d x - {\left (3 \, a^{2} b - 2 \, b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 222, normalized size = 1.61 \[ \frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 72 \, {\left (d x + c\right )} a b^{2} - 3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {48 \, b^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} - 12 \, {\left (3 \, a^{2} b - 2 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {66 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 44 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 36 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 159, normalized size = 1.15 \[ -\frac {a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{3 d \sin \left (d x +c \right )^{3}}-\frac {3 a^{2} b \left (\cos ^{3}\left (d x +c \right )\right )}{2 d \sin \left (d x +c \right )^{2}}-\frac {3 a^{2} b \cos \left (d x +c \right )}{2 d}-\frac {3 a^{2} b \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2 d}-3 a \,b^{2} x -\frac {3 a \,b^{2} \cot \left (d x +c \right )}{d}-\frac {3 a \,b^{2} c}{d}+\frac {b^{3} \cos \left (d x +c \right )}{d}+\frac {b^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 119, normalized size = 0.86 \[ -\frac {36 \, {\left (d x + c + \frac {1}{\tan \left (d x + c\right )}\right )} a b^{2} - 9 \, a^{2} b {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + \log \left (\cos \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 6 \, b^{3} {\left (2 \, \cos \left (d x + c\right ) - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {4 \, a^{3}}{\tan \left (d x + c\right )^{3}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.71, size = 477, normalized size = 3.46 \[ -\frac {\frac {a^3\,\cos \left (c+d\,x\right )}{4}-\frac {3\,b^3\,\sin \left (c+d\,x\right )}{4}+\frac {a^3\,\cos \left (3\,c+3\,d\,x\right )}{12}-\frac {b^3\,\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {b^3\,\sin \left (3\,c+3\,d\,x\right )}{4}+\frac {b^3\,\sin \left (4\,c+4\,d\,x\right )}{8}-\frac {3\,a\,b^2\,\cos \left (3\,c+3\,d\,x\right )}{4}-\frac {3\,b^3\,\sin \left (c+d\,x\right )\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4}+\frac {3\,a^2\,b\,\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {b^3\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\sin \left (3\,c+3\,d\,x\right )}{4}+\frac {3\,a\,b^2\,\cos \left (c+d\,x\right )}{4}-\frac {3\,a^2\,b\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\sin \left (3\,c+3\,d\,x\right )}{8}-\frac {9\,a\,b^2\,\mathrm {atan}\left (\frac {3\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+6\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b-2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}{-3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b+2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}\right )\,\sin \left (c+d\,x\right )}{2}+\frac {3\,a\,b^2\,\mathrm {atan}\left (\frac {3\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+6\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b-2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}{-3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b+2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}\right )\,\sin \left (3\,c+3\,d\,x\right )}{2}+\frac {9\,a^2\,b\,\sin \left (c+d\,x\right )\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{8}}{d\,{\sin \left (c+d\,x\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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