Optimal. Leaf size=96 \[ \frac {\left (a^2-2 b^2\right ) \cot (c+d x)}{3 d}+\frac {a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac {a b \cot (c+d x) \csc (c+d x)}{3 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}+b^2 (-x) \]
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Rubi [A] time = 0.39, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2889, 3048, 3031, 3021, 2735, 3770} \[ \frac {\left (a^2-2 b^2\right ) \cot (c+d x)}{3 d}+\frac {a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac {a b \cot (c+d x) \csc (c+d x)}{3 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}+b^2 (-x) \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2889
Rule 3021
Rule 3031
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))^2 \, dx &=\int \csc ^4(c+d x) (a+b \sin (c+d x))^2 \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))^2}{3 d}+\frac {1}{3} \int \csc ^3(c+d x) (a+b \sin (c+d x)) \left (2 b-a \sin (c+d x)-3 b \sin ^2(c+d x)\right ) \, dx\\ &=-\frac {a b \cot (c+d x) \csc (c+d x)}{3 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}-\frac {1}{6} \int \csc ^2(c+d x) \left (2 \left (a^2-2 b^2\right )+6 a b \sin (c+d x)+6 b^2 \sin ^2(c+d x)\right ) \, dx\\ &=\frac {\left (a^2-2 b^2\right ) \cot (c+d x)}{3 d}-\frac {a b \cot (c+d x) \csc (c+d x)}{3 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}-\frac {1}{6} \int \csc (c+d x) \left (6 a b+6 b^2 \sin (c+d x)\right ) \, dx\\ &=-b^2 x+\frac {\left (a^2-2 b^2\right ) \cot (c+d x)}{3 d}-\frac {a b \cot (c+d x) \csc (c+d x)}{3 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}-(a b) \int \csc (c+d x) \, dx\\ &=-b^2 x+\frac {a b \tanh ^{-1}(\cos (c+d x))}{d}+\frac {\left (a^2-2 b^2\right ) \cot (c+d x)}{3 d}-\frac {a b \cot (c+d x) \csc (c+d x)}{3 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}\\ \end {align*}
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Mathematica [B] time = 6.19, size = 538, normalized size = 5.60 \[ \frac {\sin ^2(c+d x) \csc \left (\frac {1}{2} (c+d x)\right ) \left (a^2 \cos \left (\frac {1}{2} (c+d x)\right )-3 b^2 \cos \left (\frac {1}{2} (c+d x)\right )\right ) (a \csc (c+d x)+b)^2}{6 d (a+b \sin (c+d x))^2}+\frac {\sin ^2(c+d x) \sec \left (\frac {1}{2} (c+d x)\right ) \left (3 b^2 \sin \left (\frac {1}{2} (c+d x)\right )-a^2 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \csc (c+d x)+b)^2}{6 d (a+b \sin (c+d x))^2}-\frac {a^2 \sin ^2(c+d x) \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^2}{24 d (a+b \sin (c+d x))^2}+\frac {a^2 \sin ^2(c+d x) \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^2}{24 d (a+b \sin (c+d x))^2}-\frac {b^2 (c+d x) \sin ^2(c+d x) (a \csc (c+d x)+b)^2}{d (a+b \sin (c+d x))^2}-\frac {a b \sin ^2(c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^2}{4 d (a+b \sin (c+d x))^2}-\frac {a b \sin ^2(c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \csc (c+d x)+b)^2}{d (a+b \sin (c+d x))^2}+\frac {a b \sin ^2(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^2}{4 d (a+b \sin (c+d x))^2}+\frac {a b \sin ^2(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) (a \csc (c+d x)+b)^2}{d (a+b \sin (c+d x))^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 167, normalized size = 1.74 \[ \frac {2 \, {\left (a^{2} - 3 \, b^{2}\right )} \cos \left (d x + c\right )^{3} + 6 \, b^{2} \cos \left (d x + c\right ) + 3 \, {\left (a b \cos \left (d x + c\right )^{2} - a b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 3 \, {\left (a b \cos \left (d x + c\right )^{2} - a b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 6 \, {\left (b^{2} d x \cos \left (d x + c\right )^{2} - b^{2} d x - a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 167, normalized size = 1.74 \[ \frac {a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, {\left (d x + c\right )} b^{2} - 24 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {44 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 114, normalized size = 1.19 \[ -\frac {a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{3 d \sin \left (d x +c \right )^{3}}-\frac {a b \left (\cos ^{3}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )^{2}}-\frac {a b \cos \left (d x +c \right )}{d}-\frac {a b \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d}-b^{2} x -\frac {b^{2} \cot \left (d x +c \right )}{d}-\frac {b^{2} c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 82, normalized size = 0.85 \[ -\frac {6 \, {\left (d x + c + \frac {1}{\tan \left (d x + c\right )}\right )} b^{2} - 3 \, a b {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + \log \left (\cos \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {2 \, a^{2}}{\tan \left (d x + c\right )^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.48, size = 231, normalized size = 2.41 \[ \frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}+\frac {a^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}-\frac {b^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}-\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}+\frac {b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}-\frac {2\,b^2\,\mathrm {atan}\left (\frac {b\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-b\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {a\,b\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,d}+\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,d}-\frac {a\,b\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
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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