Integrand size = 8, antiderivative size = 27 \[ \int \cot ^4(a+b x) \, dx=x+\frac {\cot (a+b x)}{b}-\frac {\cot ^3(a+b x)}{3 b} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3554, 8} \[ \int \cot ^4(a+b x) \, dx=-\frac {\cot ^3(a+b x)}{3 b}+\frac {\cot (a+b x)}{b}+x \]
[In]
[Out]
Rule 8
Rule 3554
Rubi steps \begin{align*} \text {integral}& = -\frac {\cot ^3(a+b x)}{3 b}-\int \cot ^2(a+b x) \, dx \\ & = \frac {\cot (a+b x)}{b}-\frac {\cot ^3(a+b x)}{3 b}+\int 1 \, dx \\ & = x+\frac {\cot (a+b x)}{b}-\frac {\cot ^3(a+b x)}{3 b} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \cot ^4(a+b x) \, dx=-\frac {\cot ^3(a+b x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\tan ^2(a+b x)\right )}{3 b} \]
[In]
[Out]
Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(\frac {-\frac {\left (\cot ^{3}\left (b x +a \right )\right )}{3}+\cot \left (b x +a \right )+b x +a}{b}\) | \(26\) |
default | \(\frac {-\frac {\left (\cot ^{3}\left (b x +a \right )\right )}{3}+\cot \left (b x +a \right )+b x +a}{b}\) | \(26\) |
risch | \(x +\frac {4 i \left (3 \,{\mathrm e}^{4 i \left (b x +a \right )}-3 \,{\mathrm e}^{2 i \left (b x +a \right )}+2\right )}{3 b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3}}\) | \(46\) |
parallelrisch | \(\frac {-\left (\cot ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )+24 b x +15 \cot \left (\frac {b x}{2}+\frac {a}{2}\right )-15 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{24 b}\) | \(57\) |
norman | \(\frac {x \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\frac {1}{24 b}+\frac {5 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8 b}-\frac {5 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8 b}+\frac {\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )}{24 b}}{\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}\) | \(80\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (25) = 50\).
Time = 0.31 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.56 \[ \int \cot ^4(a+b x) \, dx=\frac {4 \, \cos \left (b x + a\right )^{3} + 3 \, {\left (b x \cos \left (b x + a\right )^{2} - b x\right )} \sin \left (b x + a\right ) - 3 \, \cos \left (b x + a\right )}{3 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).
Time = 0.51 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \cot ^4(a+b x) \, dx=\begin {cases} x + \frac {\cos {\left (a + b x \right )}}{b \sin {\left (a + b x \right )}} - \frac {\cos ^{3}{\left (a + b x \right )}}{3 b \sin ^{3}{\left (a + b x \right )}} & \text {for}\: b \neq 0 \\\frac {x \cos ^{4}{\left (a \right )}}{\sin ^{4}{\left (a \right )}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \cot ^4(a+b x) \, dx=\frac {3 \, b x + 3 \, a + \frac {3 \, \tan \left (b x + a\right )^{2} - 1}{\tan \left (b x + a\right )^{3}}}{3 \, b} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (25) = 50\).
Time = 0.39 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.30 \[ \int \cot ^4(a+b x) \, dx=\frac {\tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{3} + 24 \, b x + 24 \, a + \frac {15 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - 1}{\tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{3}} - 15 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}{24 \, b} \]
[In]
[Out]
Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \cot ^4(a+b x) \, dx=x+\frac {{\mathrm {tan}\left (a+b\,x\right )}^2-\frac {1}{3}}{b\,{\mathrm {tan}\left (a+b\,x\right )}^3} \]
[In]
[Out]