Integrand size = 17, antiderivative size = 80 \[ \int \cos ^4(a+b x) \cot ^4(a+b x) \, dx=\frac {35 x}{8}+\frac {35 \cot (a+b x)}{8 b}-\frac {35 \cot ^3(a+b x)}{24 b}+\frac {7 \cos ^2(a+b x) \cot ^3(a+b x)}{8 b}+\frac {\cos ^4(a+b x) \cot ^3(a+b x)}{4 b} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2671, 294, 308, 209} \[ \int \cos ^4(a+b x) \cot ^4(a+b x) \, dx=-\frac {35 \cot ^3(a+b x)}{24 b}+\frac {35 \cot (a+b x)}{8 b}+\frac {\cos ^4(a+b x) \cot ^3(a+b x)}{4 b}+\frac {7 \cos ^2(a+b x) \cot ^3(a+b x)}{8 b}+\frac {35 x}{8} \]
[In]
[Out]
Rule 209
Rule 294
Rule 308
Rule 2671
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {x^8}{\left (1+x^2\right )^3} \, dx,x,\cot (a+b x)\right )}{b} \\ & = \frac {\cos ^4(a+b x) \cot ^3(a+b x)}{4 b}-\frac {7 \text {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^2} \, dx,x,\cot (a+b x)\right )}{4 b} \\ & = \frac {7 \cos ^2(a+b x) \cot ^3(a+b x)}{8 b}+\frac {\cos ^4(a+b x) \cot ^3(a+b x)}{4 b}-\frac {35 \text {Subst}\left (\int \frac {x^4}{1+x^2} \, dx,x,\cot (a+b x)\right )}{8 b} \\ & = \frac {7 \cos ^2(a+b x) \cot ^3(a+b x)}{8 b}+\frac {\cos ^4(a+b x) \cot ^3(a+b x)}{4 b}-\frac {35 \text {Subst}\left (\int \left (-1+x^2+\frac {1}{1+x^2}\right ) \, dx,x,\cot (a+b x)\right )}{8 b} \\ & = \frac {35 \cot (a+b x)}{8 b}-\frac {35 \cot ^3(a+b x)}{24 b}+\frac {7 \cos ^2(a+b x) \cot ^3(a+b x)}{8 b}+\frac {\cos ^4(a+b x) \cot ^3(a+b x)}{4 b}-\frac {35 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\cot (a+b x)\right )}{8 b} \\ & = \frac {35 x}{8}+\frac {35 \cot (a+b x)}{8 b}-\frac {35 \cot ^3(a+b x)}{24 b}+\frac {7 \cos ^2(a+b x) \cot ^3(a+b x)}{8 b}+\frac {\cos ^4(a+b x) \cot ^3(a+b x)}{4 b} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.66 \[ \int \cos ^4(a+b x) \cot ^4(a+b x) \, dx=\frac {420 (a+b x)-32 \cot (a+b x) \left (-10+\csc ^2(a+b x)\right )+72 \sin (2 (a+b x))+3 \sin (4 (a+b x))}{96 b} \]
[In]
[Out]
Time = 0.31 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.16
method | result | size |
parallelrisch | \(\frac {\left (2520 b x \sin \left (b x +a \right )-840 b x \sin \left (3 b x +3 a \right )+525 \cos \left (b x +a \right )+63 \cos \left (5 b x +5 a \right )-847 \cos \left (3 b x +3 a \right )+3 \cos \left (7 b x +7 a \right )\right ) \left (\sec ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \left (\csc ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{6144 b}\) | \(93\) |
derivativedivides | \(\frac {-\frac {\cos ^{9}\left (b x +a \right )}{3 \sin \left (b x +a \right )^{3}}+\frac {2 \left (\cos ^{9}\left (b x +a \right )\right )}{\sin \left (b x +a \right )}+2 \left (\cos ^{7}\left (b x +a \right )+\frac {7 \left (\cos ^{5}\left (b x +a \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (b x +a \right )\right )}{24}+\frac {35 \cos \left (b x +a \right )}{16}\right ) \sin \left (b x +a \right )+\frac {35 b x}{8}+\frac {35 a}{8}}{b}\) | \(94\) |
default | \(\frac {-\frac {\cos ^{9}\left (b x +a \right )}{3 \sin \left (b x +a \right )^{3}}+\frac {2 \left (\cos ^{9}\left (b x +a \right )\right )}{\sin \left (b x +a \right )}+2 \left (\cos ^{7}\left (b x +a \right )+\frac {7 \left (\cos ^{5}\left (b x +a \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (b x +a \right )\right )}{24}+\frac {35 \cos \left (b x +a \right )}{16}\right ) \sin \left (b x +a \right )+\frac {35 b x}{8}+\frac {35 a}{8}}{b}\) | \(94\) |
risch | \(\frac {35 x}{8}-\frac {i {\mathrm e}^{4 i \left (b x +a \right )}}{64 b}-\frac {3 i {\mathrm e}^{2 i \left (b x +a \right )}}{8 b}+\frac {3 i {\mathrm e}^{-2 i \left (b x +a \right )}}{8 b}+\frac {i {\mathrm e}^{-4 i \left (b x +a \right )}}{64 b}+\frac {4 i \left (6 \,{\mathrm e}^{4 i \left (b x +a \right )}-9 \,{\mathrm e}^{2 i \left (b x +a \right )}+5\right )}{3 b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3}}\) | \(108\) |
norman | \(\frac {-\frac {1}{24 b}+\frac {35 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{24 b}+\frac {63 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8 b}+\frac {35 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8 b}-\frac {35 \left (\tan ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8 b}-\frac {63 \left (\tan ^{10}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8 b}-\frac {35 \left (\tan ^{12}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{24 b}+\frac {\tan ^{14}\left (\frac {b x}{2}+\frac {a}{2}\right )}{24 b}+\frac {35 x \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8}+\frac {35 x \left (\tan ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2}+\frac {105 x \left (\tan ^{7}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4}+\frac {35 x \left (\tan ^{9}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2}+\frac {35 x \left (\tan ^{11}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{4} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}\) | \(216\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.11 \[ \int \cos ^4(a+b x) \cot ^4(a+b x) \, dx=-\frac {6 \, \cos \left (b x + a\right )^{7} + 21 \, \cos \left (b x + a\right )^{5} - 140 \, \cos \left (b x + a\right )^{3} - 105 \, {\left (b x \cos \left (b x + a\right )^{2} - b x\right )} \sin \left (b x + a\right ) + 105 \, \cos \left (b x + a\right )}{24 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \]
[In]
[Out]
Time = 1.15 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.76 \[ \int \cos ^4(a+b x) \cot ^4(a+b x) \, dx=\begin {cases} \frac {35 x \sin ^{4}{\left (a + b x \right )}}{8} + \frac {35 x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{4} + \frac {35 x \cos ^{4}{\left (a + b x \right )}}{8} + \frac {35 \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b} + \frac {175 \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{24 b} + \frac {7 \cos ^{5}{\left (a + b x \right )}}{3 b \sin {\left (a + b x \right )}} - \frac {\cos ^{7}{\left (a + b x \right )}}{3 b \sin ^{3}{\left (a + b x \right )}} & \text {for}\: b \neq 0 \\\frac {x \cos ^{8}{\left (a \right )}}{\sin ^{4}{\left (a \right )}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.94 \[ \int \cos ^4(a+b x) \cot ^4(a+b x) \, dx=\frac {105 \, b x + 105 \, a + \frac {105 \, \tan \left (b x + a\right )^{6} + 175 \, \tan \left (b x + a\right )^{4} + 56 \, \tan \left (b x + a\right )^{2} - 8}{\tan \left (b x + a\right )^{7} + 2 \, \tan \left (b x + a\right )^{5} + \tan \left (b x + a\right )^{3}}}{24 \, b} \]
[In]
[Out]
none
Time = 0.50 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.85 \[ \int \cos ^4(a+b x) \cot ^4(a+b x) \, dx=\frac {105 \, b x + 105 \, a + \frac {3 \, {\left (11 \, \tan \left (b x + a\right )^{3} + 13 \, \tan \left (b x + a\right )\right )}}{{\left (\tan \left (b x + a\right )^{2} + 1\right )}^{2}} + \frac {8 \, {\left (9 \, \tan \left (b x + a\right )^{2} - 1\right )}}{\tan \left (b x + a\right )^{3}}}{24 \, b} \]
[In]
[Out]
Time = 1.46 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.70 \[ \int \cos ^4(a+b x) \cot ^4(a+b x) \, dx=\frac {35\,x}{8}+\frac {{\cos \left (a+b\,x\right )}^4\,\left (\frac {35\,{\mathrm {tan}\left (a+b\,x\right )}^6}{8}+\frac {175\,{\mathrm {tan}\left (a+b\,x\right )}^4}{24}+\frac {7\,{\mathrm {tan}\left (a+b\,x\right )}^2}{3}-\frac {1}{3}\right )}{b\,{\mathrm {tan}\left (a+b\,x\right )}^3} \]
[In]
[Out]