Integrand size = 17, antiderivative size = 57 \[ \int \cos ^2(a+b x) \cot ^4(a+b x) \, dx=\frac {5 x}{2}+\frac {5 \cot (a+b x)}{2 b}-\frac {5 \cot ^3(a+b x)}{6 b}+\frac {\cos ^2(a+b x) \cot ^3(a+b x)}{2 b} \]
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Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2671, 294, 308, 209} \[ \int \cos ^2(a+b x) \cot ^4(a+b x) \, dx=-\frac {5 \cot ^3(a+b x)}{6 b}+\frac {5 \cot (a+b x)}{2 b}+\frac {\cos ^2(a+b x) \cot ^3(a+b x)}{2 b}+\frac {5 x}{2} \]
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Rule 209
Rule 294
Rule 308
Rule 2671
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^2} \, dx,x,\cot (a+b x)\right )}{b} \\ & = \frac {\cos ^2(a+b x) \cot ^3(a+b x)}{2 b}-\frac {5 \text {Subst}\left (\int \frac {x^4}{1+x^2} \, dx,x,\cot (a+b x)\right )}{2 b} \\ & = \frac {\cos ^2(a+b x) \cot ^3(a+b x)}{2 b}-\frac {5 \text {Subst}\left (\int \left (-1+x^2+\frac {1}{1+x^2}\right ) \, dx,x,\cot (a+b x)\right )}{2 b} \\ & = \frac {5 \cot (a+b x)}{2 b}-\frac {5 \cot ^3(a+b x)}{6 b}+\frac {\cos ^2(a+b x) \cot ^3(a+b x)}{2 b}-\frac {5 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\cot (a+b x)\right )}{2 b} \\ & = \frac {5 x}{2}+\frac {5 \cot (a+b x)}{2 b}-\frac {5 \cot ^3(a+b x)}{6 b}+\frac {\cos ^2(a+b x) \cot ^3(a+b x)}{2 b} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.75 \[ \int \cos ^2(a+b x) \cot ^4(a+b x) \, dx=\frac {30 (a+b x)-4 \cot (a+b x) \left (-7+\csc ^2(a+b x)\right )+3 \sin (2 (a+b x))}{12 b} \]
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Result contains complex when optimal does not.
Time = 0.15 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.37
method | result | size |
risch | \(\frac {5 x}{2}-\frac {i {\mathrm e}^{2 i \left (b x +a \right )}}{8 b}+\frac {i {\mathrm e}^{-2 i \left (b x +a \right )}}{8 b}+\frac {2 i \left (9 \,{\mathrm e}^{4 i \left (b x +a \right )}-12 \,{\mathrm e}^{2 i \left (b x +a \right )}+7\right )}{3 b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3}}\) | \(78\) |
parallelrisch | \(\frac {\left (180 b x \sin \left (b x +a \right )-60 b x \sin \left (3 b x +3 a \right )+30 \cos \left (b x +a \right )-65 \cos \left (3 b x +3 a \right )+3 \cos \left (5 b x +5 a \right )\right ) \left (\sec ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \left (\csc ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{768 b}\) | \(82\) |
derivativedivides | \(\frac {-\frac {\cos ^{7}\left (b x +a \right )}{3 \sin \left (b x +a \right )^{3}}+\frac {4 \left (\cos ^{7}\left (b x +a \right )\right )}{3 \sin \left (b x +a \right )}+\frac {4 \left (\cos ^{5}\left (b x +a \right )+\frac {5 \left (\cos ^{3}\left (b x +a \right )\right )}{4}+\frac {15 \cos \left (b x +a \right )}{8}\right ) \sin \left (b x +a \right )}{3}+\frac {5 b x}{2}+\frac {5 a}{2}}{b}\) | \(84\) |
default | \(\frac {-\frac {\cos ^{7}\left (b x +a \right )}{3 \sin \left (b x +a \right )^{3}}+\frac {4 \left (\cos ^{7}\left (b x +a \right )\right )}{3 \sin \left (b x +a \right )}+\frac {4 \left (\cos ^{5}\left (b x +a \right )+\frac {5 \left (\cos ^{3}\left (b x +a \right )\right )}{4}+\frac {15 \cos \left (b x +a \right )}{8}\right ) \sin \left (b x +a \right )}{3}+\frac {5 b x}{2}+\frac {5 a}{2}}{b}\) | \(84\) |
norman | \(\frac {-\frac {1}{24 b}+\frac {25 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{24 b}+\frac {25 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{12 b}-\frac {25 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{12 b}-\frac {25 \left (\tan ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{24 b}+\frac {\tan ^{10}\left (\frac {b x}{2}+\frac {a}{2}\right )}{24 b}+\frac {5 x \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2}+5 x \left (\tan ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\frac {5 x \left (\tan ^{7}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}\) | \(156\) |
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Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.39 \[ \int \cos ^2(a+b x) \cot ^4(a+b x) \, dx=-\frac {3 \, \cos \left (b x + a\right )^{5} - 20 \, \cos \left (b x + a\right )^{3} - 15 \, {\left (b x \cos \left (b x + a\right )^{2} - b x\right )} \sin \left (b x + a\right ) + 15 \, \cos \left (b x + a\right )}{6 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \]
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Time = 0.73 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.70 \[ \int \cos ^2(a+b x) \cot ^4(a+b x) \, dx=\begin {cases} \frac {5 x \sin ^{2}{\left (a + b x \right )}}{2} + \frac {5 x \cos ^{2}{\left (a + b x \right )}}{2} + \frac {5 \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{2 b} + \frac {5 \cos ^{3}{\left (a + b x \right )}}{3 b \sin {\left (a + b x \right )}} - \frac {\cos ^{5}{\left (a + b x \right )}}{3 b \sin ^{3}{\left (a + b x \right )}} & \text {for}\: b \neq 0 \\\frac {x \cos ^{6}{\left (a \right )}}{\sin ^{4}{\left (a \right )}} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.96 \[ \int \cos ^2(a+b x) \cot ^4(a+b x) \, dx=\frac {15 \, b x + 15 \, a + \frac {15 \, \tan \left (b x + a\right )^{4} + 10 \, \tan \left (b x + a\right )^{2} - 2}{\tan \left (b x + a\right )^{5} + \tan \left (b x + a\right )^{3}}}{6 \, b} \]
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Time = 0.32 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.96 \[ \int \cos ^2(a+b x) \cot ^4(a+b x) \, dx=\frac {15 \, b x + 15 \, a + \frac {3 \, \tan \left (b x + a\right )}{\tan \left (b x + a\right )^{2} + 1} + \frac {2 \, {\left (6 \, \tan \left (b x + a\right )^{2} - 1\right )}}{\tan \left (b x + a\right )^{3}}}{6 \, b} \]
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Time = 0.51 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.81 \[ \int \cos ^2(a+b x) \cot ^4(a+b x) \, dx=\frac {5\,x}{2}+\frac {{\cos \left (a+b\,x\right )}^2\,\left (\frac {5\,{\mathrm {tan}\left (a+b\,x\right )}^4}{2}+\frac {5\,{\mathrm {tan}\left (a+b\,x\right )}^2}{3}-\frac {1}{3}\right )}{b\,{\mathrm {tan}\left (a+b\,x\right )}^3} \]
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