Optimal. Leaf size=108 \[ -\frac {\cot (c+d x)}{a^4 d}+\frac {4 \tanh ^{-1}(\cos (c+d x))}{a^4 d}-\frac {104 \cot (c+d x)}{15 a^4 d (\csc (c+d x)+1)}+\frac {31 \cot (c+d x)}{15 a^4 d (\csc (c+d x)+1)^2}-\frac {2 \cot (c+d x)}{5 a^4 d (\csc (c+d x)+1)^3} \]
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Rubi [A] time = 0.32, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {2709, 3770, 3767, 8, 3777, 3922, 3919, 3794} \[ -\frac {\cot (c+d x)}{a^4 d}+\frac {4 \tanh ^{-1}(\cos (c+d x))}{a^4 d}-\frac {104 \cot (c+d x)}{15 a^4 d (\csc (c+d x)+1)}+\frac {31 \cot (c+d x)}{15 a^4 d (\csc (c+d x)+1)^2}-\frac {2 \cot (c+d x)}{5 a^4 d (\csc (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2709
Rule 3767
Rule 3770
Rule 3777
Rule 3794
Rule 3919
Rule 3922
Rubi steps
\begin {align*} \int \frac {\cot ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac {\int \left (\frac {9}{a^2}-\frac {4 \csc (c+d x)}{a^2}+\frac {\csc ^2(c+d x)}{a^2}-\frac {2}{a^2 (1+\csc (c+d x))^3}+\frac {9}{a^2 (1+\csc (c+d x))^2}-\frac {16}{a^2 (1+\csc (c+d x))}\right ) \, dx}{a^2}\\ &=\frac {9 x}{a^4}+\frac {\int \csc ^2(c+d x) \, dx}{a^4}-\frac {2 \int \frac {1}{(1+\csc (c+d x))^3} \, dx}{a^4}-\frac {4 \int \csc (c+d x) \, dx}{a^4}+\frac {9 \int \frac {1}{(1+\csc (c+d x))^2} \, dx}{a^4}-\frac {16 \int \frac {1}{1+\csc (c+d x)} \, dx}{a^4}\\ &=\frac {9 x}{a^4}+\frac {4 \tanh ^{-1}(\cos (c+d x))}{a^4 d}-\frac {2 \cot (c+d x)}{5 a^4 d (1+\csc (c+d x))^3}+\frac {3 \cot (c+d x)}{a^4 d (1+\csc (c+d x))^2}-\frac {16 \cot (c+d x)}{a^4 d (1+\csc (c+d x))}+\frac {2 \int \frac {-5+2 \csc (c+d x)}{(1+\csc (c+d x))^2} \, dx}{5 a^4}-\frac {3 \int \frac {-3+\csc (c+d x)}{1+\csc (c+d x)} \, dx}{a^4}+\frac {16 \int -1 \, dx}{a^4}-\frac {\operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^4 d}\\ &=\frac {2 x}{a^4}+\frac {4 \tanh ^{-1}(\cos (c+d x))}{a^4 d}-\frac {\cot (c+d x)}{a^4 d}-\frac {2 \cot (c+d x)}{5 a^4 d (1+\csc (c+d x))^3}+\frac {31 \cot (c+d x)}{15 a^4 d (1+\csc (c+d x))^2}-\frac {16 \cot (c+d x)}{a^4 d (1+\csc (c+d x))}-\frac {2 \int \frac {15-7 \csc (c+d x)}{1+\csc (c+d x)} \, dx}{15 a^4}-\frac {12 \int \frac {\csc (c+d x)}{1+\csc (c+d x)} \, dx}{a^4}\\ &=\frac {4 \tanh ^{-1}(\cos (c+d x))}{a^4 d}-\frac {\cot (c+d x)}{a^4 d}-\frac {2 \cot (c+d x)}{5 a^4 d (1+\csc (c+d x))^3}+\frac {31 \cot (c+d x)}{15 a^4 d (1+\csc (c+d x))^2}-\frac {4 \cot (c+d x)}{a^4 d (1+\csc (c+d x))}+\frac {44 \int \frac {\csc (c+d x)}{1+\csc (c+d x)} \, dx}{15 a^4}\\ &=\frac {4 \tanh ^{-1}(\cos (c+d x))}{a^4 d}-\frac {\cot (c+d x)}{a^4 d}-\frac {2 \cot (c+d x)}{5 a^4 d (1+\csc (c+d x))^3}+\frac {31 \cot (c+d x)}{15 a^4 d (1+\csc (c+d x))^2}-\frac {104 \cot (c+d x)}{15 a^4 d (1+\csc (c+d x))}\\ \end {align*}
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Mathematica [B] time = 0.42, size = 315, normalized size = 2.92 \[ \frac {\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (24 \sin \left (\frac {1}{2} (c+d x)\right )+316 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^4-38 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3+76 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2-12 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+120 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^5-120 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^5+15 \tan \left (\frac {1}{2} (c+d x)\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^5-15 \cot \left (\frac {1}{2} (c+d x)\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^5\right )}{30 d (a \sin (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 369, normalized size = 3.42 \[ \frac {94 \, \cos \left (d x + c\right )^{4} + 222 \, \cos \left (d x + c\right )^{3} - 115 \, \cos \left (d x + c\right )^{2} + 30 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) - 4\right )} \sin \left (d x + c\right ) + 2 \, \cos \left (d x + c\right ) + 4\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 30 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) - 4\right )} \sin \left (d x + c\right ) + 2 \, \cos \left (d x + c\right ) + 4\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (94 \, \cos \left (d x + c\right )^{3} - 128 \, \cos \left (d x + c\right )^{2} - 243 \, \cos \left (d x + c\right ) - 6\right )} \sin \left (d x + c\right ) - 237 \, \cos \left (d x + c\right ) + 6}{15 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} - 2 \, a^{4} d \cos \left (d x + c\right )^{3} - 5 \, a^{4} d \cos \left (d x + c\right )^{2} + 2 \, a^{4} d \cos \left (d x + c\right ) + 4 \, a^{4} d - {\left (a^{4} d \cos \left (d x + c\right )^{3} + 3 \, a^{4} d \cos \left (d x + c\right )^{2} - 2 \, a^{4} d \cos \left (d x + c\right ) - 4 \, a^{4} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.70, size = 135, normalized size = 1.25 \[ -\frac {\frac {120 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{4}} - \frac {15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{4}} - \frac {15 \, {\left (8 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}}{a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} + \frac {4 \, {\left (135 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 435 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 605 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 385 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 104\right )}}{a^{4} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}}}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 161, normalized size = 1.49 \[ \frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{4} d}-\frac {1}{2 a^{4} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4} d}-\frac {16}{5 a^{4} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {8}{a^{4} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {44}{3 a^{4} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {14}{a^{4} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {18}{a^{4} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 288, normalized size = 2.67 \[ -\frac {\frac {\frac {491 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {1690 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {2570 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1815 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {555 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + 15}{\frac {a^{4} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {5 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, a^{4} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {10 \, a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {5 \, a^{4} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {a^{4} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {120 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}} - \frac {15 \, \sin \left (d x + c\right )}{a^{4} {\left (\cos \left (d x + c\right ) + 1\right )}}}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.05, size = 203, normalized size = 1.88 \[ \frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^4\,d}-\frac {37\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+121\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {514\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {338\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+\frac {491\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{15}+1}{d\,\left (2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+20\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+20\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+10\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}-\frac {4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cot ^{2}{\left (c + d x \right )}}{\sin ^{4}{\left (c + d x \right )} + 4 \sin ^{3}{\left (c + d x \right )} + 6 \sin ^{2}{\left (c + d x \right )} + 4 \sin {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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