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 = 0.02 (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 = 0.02 (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 = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07
method | result | size |
parallelrisch | \(\frac {-\cot \left (b x +a \right )^{3}+3 b x +3 \cot \left (b x +a \right )}{3 b}\) | \(29\) |
derivativedivides | \(\frac {-\frac {\cot \left (b x +a \right )^{3}}{3}+\cot \left (b x +a \right )-\frac {\pi }{2}+\operatorname {arccot}\left (\cot \left (b x +a \right )\right )}{b}\) | \(32\) |
default | \(\frac {-\frac {\cot \left (b x +a \right )^{3}}{3}+\cot \left (b x +a \right )-\frac {\pi }{2}+\operatorname {arccot}\left (\cot \left (b x +a \right )\right )}{b}\) | \(32\) |
norman | \(\frac {\frac {\tan \left (b x +a \right )^{2}}{b}+x \tan \left (b x +a \right )^{3}-\frac {1}{3 b}}{\tan \left (b x +a \right )^{3}}\) | \(38\) |
risch | \(x +\frac {4 i \left (3 \,{\mathrm e}^{4 i \left (b x +a \right )}-3 \,{\mathrm e}^{2 i \left (b x +a \right )}+2\right )}{3 b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3}}\) | \(46\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.11 \[ \int \cot ^4(a+b x) \, dx=\frac {4 \, \cos \left (2 \, b x + 2 \, a\right )^{2} + 3 \, {\left (b x \cos \left (2 \, b x + 2 \, a\right ) - b x\right )} \sin \left (2 \, b x + 2 \, a\right ) + 2 \, \cos \left (2 \, b x + 2 \, a\right ) - 2}{3 \, {\left (b \cos \left (2 \, b x + 2 \, a\right ) - b\right )} \sin \left (2 \, b x + 2 \, a\right )} \]
[In]
[Out]
Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \cot ^4(a+b x) \, dx=\begin {cases} x - \frac {\cot ^{3}{\left (a + b x \right )}}{3 b} + \frac {\cot {\left (a + b x \right )}}{b} & \text {for}\: b \neq 0 \\x \cot ^{4}{\left (a \right )} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.33 (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 = 0.27 (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 = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \cot ^4(a+b x) \, dx=x+\frac {\mathrm {cot}\left (a+b\,x\right )-\frac {{\mathrm {cot}\left (a+b\,x\right )}^3}{3}}{b} \]
[In]
[Out]