Integrand size = 17, antiderivative size = 38 \[ \int \csc ^2(a+b x) \sec ^4(a+b x) \, dx=-\frac {\cot (a+b x)}{b}+\frac {2 \tan (a+b x)}{b}+\frac {\tan ^3(a+b x)}{3 b} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2700, 276} \[ \int \csc ^2(a+b x) \sec ^4(a+b x) \, dx=\frac {\tan ^3(a+b x)}{3 b}+\frac {2 \tan (a+b x)}{b}-\frac {\cot (a+b x)}{b} \]
[In]
[Out]
Rule 276
Rule 2700
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^2} \, dx,x,\tan (a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \left (2+\frac {1}{x^2}+x^2\right ) \, dx,x,\tan (a+b x)\right )}{b} \\ & = -\frac {\cot (a+b x)}{b}+\frac {2 \tan (a+b x)}{b}+\frac {\tan ^3(a+b x)}{3 b} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.21 \[ \int \csc ^2(a+b x) \sec ^4(a+b x) \, dx=-\frac {\cot (a+b x)}{b}+\frac {5 \tan (a+b x)}{3 b}+\frac {\sec ^2(a+b x) \tan (a+b x)}{3 b} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.21
method | result | size |
risch | \(-\frac {16 i \left (2 \,{\mathrm e}^{2 i \left (b x +a \right )}+1\right )}{3 b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{3} \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}\) | \(46\) |
derivativedivides | \(\frac {\frac {1}{3 \cos \left (b x +a \right )^{3} \sin \left (b x +a \right )}+\frac {4}{3 \sin \left (b x +a \right ) \cos \left (b x +a \right )}-\frac {8 \cot \left (b x +a \right )}{3}}{b}\) | \(50\) |
default | \(\frac {\frac {1}{3 \cos \left (b x +a \right )^{3} \sin \left (b x +a \right )}+\frac {4}{3 \sin \left (b x +a \right ) \cos \left (b x +a \right )}-\frac {8 \cot \left (b x +a \right )}{3}}{b}\) | \(50\) |
parallelrisch | \(\frac {3 \left (\tan ^{7}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-36 \left (\tan ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+50 \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-36 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )+3 \cot \left (\frac {b x}{2}+\frac {a}{2}\right )}{6 b \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{3} \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )^{3}}\) | \(94\) |
norman | \(\frac {\frac {1}{2 b}-\frac {6 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}+\frac {25 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}-\frac {6 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}+\frac {\tan ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )}{2 b}}{\left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{3} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}\) | \(98\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.13 \[ \int \csc ^2(a+b x) \sec ^4(a+b x) \, dx=-\frac {8 \, \cos \left (b x + a\right )^{4} - 4 \, \cos \left (b x + a\right )^{2} - 1}{3 \, b \cos \left (b x + a\right )^{3} \sin \left (b x + a\right )} \]
[In]
[Out]
\[ \int \csc ^2(a+b x) \sec ^4(a+b x) \, dx=\int \frac {\sec ^{4}{\left (a + b x \right )}}{\sin ^{2}{\left (a + b x \right )}}\, dx \]
[In]
[Out]
none
Time = 0.18 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.84 \[ \int \csc ^2(a+b x) \sec ^4(a+b x) \, dx=\frac {\tan \left (b x + a\right )^{3} - \frac {3}{\tan \left (b x + a\right )} + 6 \, \tan \left (b x + a\right )}{3 \, b} \]
[In]
[Out]
none
Time = 0.36 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.84 \[ \int \csc ^2(a+b x) \sec ^4(a+b x) \, dx=\frac {\tan \left (b x + a\right )^{3} - \frac {3}{\tan \left (b x + a\right )} + 6 \, \tan \left (b x + a\right )}{3 \, b} \]
[In]
[Out]
Time = 0.14 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.87 \[ \int \csc ^2(a+b x) \sec ^4(a+b x) \, dx=\frac {{\mathrm {tan}\left (a+b\,x\right )}^4+6\,{\mathrm {tan}\left (a+b\,x\right )}^2-3}{3\,b\,\mathrm {tan}\left (a+b\,x\right )} \]
[In]
[Out]