Integrand size = 8, antiderivative size = 28 \[ \int \tan ^4(a+b x) \, dx=x-\frac {\tan (a+b x)}{b}+\frac {\tan ^3(a+b x)}{3 b} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 28, 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} \[ \int \tan ^4(a+b x) \, dx=\frac {\tan ^3(a+b x)}{3 b}-\frac {\tan (a+b x)}{b}+x \]
[In]
[Out]
Rule 8
Rule 3554
Rubi steps \begin{align*} \text {integral}& = \frac {\tan ^3(a+b x)}{3 b}-\int \tan ^2(a+b x) \, dx \\ & = -\frac {\tan (a+b x)}{b}+\frac {\tan ^3(a+b x)}{3 b}+\int 1 \, dx \\ & = x-\frac {\tan (a+b x)}{b}+\frac {\tan ^3(a+b x)}{3 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \tan ^4(a+b x) \, dx=\frac {\arctan (\tan (a+b x))}{b}-\frac {\tan (a+b x)}{b}+\frac {\tan ^3(a+b x)}{3 b} \]
[In]
[Out]
Time = 0.10 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {\frac {\left (\tan ^{3}\left (b x +a \right )\right )}{3}-\tan \left (b x +a \right )+b x +a}{b}\) | \(28\) |
default | \(\frac {\frac {\left (\tan ^{3}\left (b x +a \right )\right )}{3}-\tan \left (b x +a \right )+b x +a}{b}\) | \(28\) |
risch | \(x -\frac {4 i \left (3 \,{\mathrm e}^{4 i \left (b x +a \right )}+3 \,{\mathrm e}^{2 i \left (b x +a \right )}+2\right )}{3 b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{3}}\) | \(46\) |
norman | \(\frac {x \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-x +\frac {2 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}-\frac {20 \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}+\frac {2 \left (\tan ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}+3 x \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-3 x \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{\left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{3}}\) | \(108\) |
parallelrisch | \(\frac {3 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) x b -9 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) x b +6 \left (\tan ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+9 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) x b -20 \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-3 b x +6 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{3 b \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{3} \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )^{3}}\) | \(119\) |
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \tan ^4(a+b x) \, dx=\frac {3 \, b x \cos \left (b x + a\right )^{3} - {\left (4 \, \cos \left (b x + a\right )^{2} - 1\right )} \sin \left (b x + a\right )}{3 \, b \cos \left (b x + a\right )^{3}} \]
[In]
[Out]
\[ \int \tan ^4(a+b x) \, dx=\int \sin ^{4}{\left (a + b x \right )} \sec ^{4}{\left (a + b x \right )}\, dx \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \tan ^4(a+b x) \, dx=\frac {\tan \left (b x + a\right )^{3} + 3 \, b x + 3 \, a - 3 \, \tan \left (b x + a\right )}{3 \, b} \]
[In]
[Out]
none
Time = 0.34 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \tan ^4(a+b x) \, dx=\frac {\tan \left (b x + a\right )^{3} + 3 \, b x + 3 \, a - 3 \, \tan \left (b x + a\right )}{3 \, b} \]
[In]
[Out]
Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \tan ^4(a+b x) \, dx=x-\frac {\mathrm {tan}\left (a+b\,x\right )-\frac {{\mathrm {tan}\left (a+b\,x\right )}^3}{3}}{b} \]
[In]
[Out]