Integrand size = 13, antiderivative size = 42 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^{3/2}} \, dx=\frac {\text {arctanh}(\sin (x)) \cos (x)}{2 a \sqrt {a \cos ^2(x)}}+\frac {\tan (x)}{2 a \sqrt {a \cos ^2(x)}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3255, 3283, 3286, 3855} \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^{3/2}} \, dx=\frac {\cos (x) \text {arctanh}(\sin (x))}{2 a \sqrt {a \cos ^2(x)}}+\frac {\tan (x)}{2 a \sqrt {a \cos ^2(x)}} \]
[In]
[Out]
Rule 3255
Rule 3283
Rule 3286
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (a \cos ^2(x)\right )^{3/2}} \, dx \\ & = \frac {\tan (x)}{2 a \sqrt {a \cos ^2(x)}}+\frac {\int \frac {1}{\sqrt {a \cos ^2(x)}} \, dx}{2 a} \\ & = \frac {\tan (x)}{2 a \sqrt {a \cos ^2(x)}}+\frac {\cos (x) \int \sec (x) \, dx}{2 a \sqrt {a \cos ^2(x)}} \\ & = \frac {\text {arctanh}(\sin (x)) \cos (x)}{2 a \sqrt {a \cos ^2(x)}}+\frac {\tan (x)}{2 a \sqrt {a \cos ^2(x)}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.62 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^{3/2}} \, dx=\frac {\text {arctanh}(\sin (x)) \cos (x)+\tan (x)}{2 a \sqrt {a \cos ^2(x)}} \]
[In]
[Out]
Time = 0.72 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.98
method | result | size |
default | \(-\frac {\left (-\ln \left (1+\sin \left (x \right )\right )+\ln \left (\sin \left (x \right )-1\right )\right ) \left (\cos ^{2}\left (x \right )\right )-2 \sin \left (x \right )}{4 a \cos \left (x \right ) \sqrt {a \left (\cos ^{2}\left (x \right )\right )}}\) | \(41\) |
risch | \(-\frac {i \left ({\mathrm e}^{2 i x}-1\right )}{a \left ({\mathrm e}^{2 i x}+1\right ) \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}-\frac {\ln \left ({\mathrm e}^{i x}-i\right ) \cos \left (x \right )}{a \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}+\frac {\ln \left ({\mathrm e}^{i x}+i\right ) \cos \left (x \right )}{a \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}\) | \(109\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^{3/2}} \, dx=-\frac {\sqrt {a \cos \left (x\right )^{2}} {\left (\cos \left (x\right )^{2} \log \left (-\frac {\sin \left (x\right ) - 1}{\sin \left (x\right ) + 1}\right ) - 2 \, \sin \left (x\right )\right )}}{4 \, a^{2} \cos \left (x\right )^{3}} \]
[In]
[Out]
\[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^{3/2}} \, dx=\int \frac {1}{\left (- a \sin ^{2}{\left (x \right )} + a\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (34) = 68\).
Time = 0.42 (sec) , antiderivative size = 304, normalized size of antiderivative = 7.24 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^{3/2}} \, dx=\frac {4 \, {\left (\sin \left (3 \, x\right ) - \sin \left (x\right )\right )} \cos \left (4 \, x\right ) + {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1\right ) - {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1\right ) - 4 \, {\left (\cos \left (3 \, x\right ) - \cos \left (x\right )\right )} \sin \left (4 \, x\right ) + 4 \, {\left (2 \, \cos \left (2 \, x\right ) + 1\right )} \sin \left (3 \, x\right ) - 8 \, \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) + 8 \, \cos \left (x\right ) \sin \left (2 \, x\right ) - 8 \, \cos \left (2 \, x\right ) \sin \left (x\right ) - 4 \, \sin \left (x\right )}{4 \, {\left (a \cos \left (4 \, x\right )^{2} + 4 \, a \cos \left (2 \, x\right )^{2} + a \sin \left (4 \, x\right )^{2} + 4 \, a \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a \sin \left (2 \, x\right )^{2} + 2 \, {\left (2 \, a \cos \left (2 \, x\right ) + a\right )} \cos \left (4 \, x\right ) + 4 \, a \cos \left (2 \, x\right ) + a\right )} \sqrt {a}} \]
[In]
[Out]
none
Time = 0.35 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^{3/2}} \, dx=-\frac {\frac {\log \left ({\left | -\sqrt {a} \tan \left (x\right ) + \sqrt {a \tan \left (x\right )^{2} + a} \right |}\right )}{\sqrt {a}} - \frac {\sqrt {a \tan \left (x\right )^{2} + a} \tan \left (x\right )}{a}}{2 \, a} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (a-a\,{\sin \left (x\right )}^2\right )}^{3/2}} \,d x \]
[In]
[Out]