Integrand size = 13, antiderivative size = 40 \[ \int \frac {\sin ^2(x)}{a+a \csc (x)} \, dx=\frac {3 x}{2 a}+\frac {2 \cos (x)}{a}-\frac {3 \cos (x) \sin (x)}{2 a}+\frac {\cos (x) \sin (x)}{a+a \csc (x)} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3904, 3872, 2715, 8, 2718} \[ \int \frac {\sin ^2(x)}{a+a \csc (x)} \, dx=\frac {3 x}{2 a}+\frac {2 \cos (x)}{a}-\frac {3 \sin (x) \cos (x)}{2 a}+\frac {\sin (x) \cos (x)}{a \csc (x)+a} \]
[In]
[Out]
Rule 8
Rule 2715
Rule 2718
Rule 3872
Rule 3904
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (x) \sin (x)}{a+a \csc (x)}-\frac {\int (-3 a+2 a \csc (x)) \sin ^2(x) \, dx}{a^2} \\ & = \frac {\cos (x) \sin (x)}{a+a \csc (x)}-\frac {2 \int \sin (x) \, dx}{a}+\frac {3 \int \sin ^2(x) \, dx}{a} \\ & = \frac {2 \cos (x)}{a}-\frac {3 \cos (x) \sin (x)}{2 a}+\frac {\cos (x) \sin (x)}{a+a \csc (x)}+\frac {3 \int 1 \, dx}{2 a} \\ & = \frac {3 x}{2 a}+\frac {2 \cos (x)}{a}-\frac {3 \cos (x) \sin (x)}{2 a}+\frac {\cos (x) \sin (x)}{a+a \csc (x)} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \frac {\sin ^2(x)}{a+a \csc (x)} \, dx=-\frac {-6 x-4 \cos (x)+\frac {8 \sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )}+\sin (2 x)}{4 a} \]
[In]
[Out]
Time = 0.40 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.72
method | result | size |
parallelrisch | \(\frac {6 x +8-\sin \left (2 x \right )+4 \cos \left (x \right )+4 \sec \left (x \right )-4 \tan \left (x \right )}{4 a}\) | \(29\) |
risch | \(\frac {3 x}{2 a}+\frac {{\mathrm e}^{i x}}{2 a}+\frac {{\mathrm e}^{-i x}}{2 a}+\frac {2}{\left (i+{\mathrm e}^{i x}\right ) a}-\frac {\sin \left (2 x \right )}{4 a}\) | \(52\) |
default | \(\frac {\frac {2 \left (\frac {\tan \left (\frac {x}{2}\right )^{3}}{2}+\tan \left (\frac {x}{2}\right )^{2}-\frac {\tan \left (\frac {x}{2}\right )}{2}+1\right )}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{2}}+3 \arctan \left (\tan \left (\frac {x}{2}\right )\right )+\frac {16}{8 \tan \left (\frac {x}{2}\right )+8}}{a}\) | \(58\) |
norman | \(\frac {\frac {3}{a}-\frac {\tan \left (\frac {x}{2}\right )^{5}}{a}+\frac {2 \tan \left (\frac {x}{2}\right )^{4}}{a}+\frac {\tan \left (\frac {x}{2}\right )^{3}}{a}+\frac {3 \tan \left (\frac {x}{2}\right )^{2}}{a}+\frac {3 x}{2 a}+\frac {3 x \tan \left (\frac {x}{2}\right )}{2 a}+\frac {3 x \tan \left (\frac {x}{2}\right )^{2}}{a}+\frac {3 x \tan \left (\frac {x}{2}\right )^{3}}{a}+\frac {3 x \tan \left (\frac {x}{2}\right )^{4}}{2 a}+\frac {3 x \tan \left (\frac {x}{2}\right )^{5}}{2 a}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )}\) | \(133\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.32 \[ \int \frac {\sin ^2(x)}{a+a \csc (x)} \, dx=\frac {\cos \left (x\right )^{3} + 3 \, {\left (x + 1\right )} \cos \left (x\right ) + 2 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{2} - 3 \, x - \cos \left (x\right ) + 2\right )} \sin \left (x\right ) + 3 \, x + 2}{2 \, {\left (a \cos \left (x\right ) + a \sin \left (x\right ) + a\right )}} \]
[In]
[Out]
\[ \int \frac {\sin ^2(x)}{a+a \csc (x)} \, dx=\frac {\int \frac {\sin ^{2}{\left (x \right )}}{\csc {\left (x \right )} + 1}\, dx}{a} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (36) = 72\).
Time = 0.34 (sec) , antiderivative size = 128, normalized size of antiderivative = 3.20 \[ \int \frac {\sin ^2(x)}{a+a \csc (x)} \, dx=\frac {\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {5 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {3 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + 4}{a + \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {2 \, a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {2 \, a \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {a \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {a \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}} + \frac {3 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.40 \[ \int \frac {\sin ^2(x)}{a+a \csc (x)} \, dx=\frac {3 \, x}{2 \, a} + \frac {\tan \left (\frac {1}{2} \, x\right )^{3} + 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} - \tan \left (\frac {1}{2} \, x\right ) + 2}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{2} a} + \frac {2}{a {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}} \]
[In]
[Out]
Time = 18.30 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.48 \[ \int \frac {\sin ^2(x)}{a+a \csc (x)} \, dx=\frac {3\,x}{2\,a}+\frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+5\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\mathrm {tan}\left (\frac {x}{2}\right )+4}{a\,{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}^2\,\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )} \]
[In]
[Out]