Integrand size = 13, antiderivative size = 25 \[ \int \frac {\cos ^3(x)}{\sqrt {\sin ^3(x)}} \, dx=-\frac {2 \sin (x)}{\sqrt {\sin ^3(x)}}-\frac {2}{3} \sqrt {\sin ^3(x)} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3286, 2644, 14} \[ \int \frac {\cos ^3(x)}{\sqrt {\sin ^3(x)}} \, dx=-\frac {2 \sin (x)}{\sqrt {\sin ^3(x)}}-\frac {2}{3} \sqrt {\sin ^3(x)} \]
[In]
[Out]
Rule 14
Rule 2644
Rule 3286
Rubi steps \begin{align*} \text {integral}& = \frac {\sin ^{\frac {3}{2}}(x) \int \frac {\cos ^3(x)}{\sin ^{\frac {3}{2}}(x)} \, dx}{\sqrt {\sin ^3(x)}} \\ & = \frac {\sin ^{\frac {3}{2}}(x) \text {Subst}\left (\int \frac {1-x^2}{x^{3/2}} \, dx,x,\sin (x)\right )}{\sqrt {\sin ^3(x)}} \\ & = \frac {\sin ^{\frac {3}{2}}(x) \text {Subst}\left (\int \left (\frac {1}{x^{3/2}}-\sqrt {x}\right ) \, dx,x,\sin (x)\right )}{\sqrt {\sin ^3(x)}} \\ & = -\frac {2 \sin (x)}{\sqrt {\sin ^3(x)}}-\frac {2}{3} \sqrt {\sin ^3(x)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {\cos ^3(x)}{\sqrt {\sin ^3(x)}} \, dx=\frac {(-7+\cos (2 x)) \sin (x)}{3 \sqrt {\sin ^3(x)}} \]
[In]
[Out]
Time = 0.67 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68
method | result | size |
derivativedivides | \(-\frac {4 \sin \left (x \right ) \left (\sin \left (x \right )^{2}+3\right )}{3 \sqrt {-\sin \left (3 x \right )+3 \sin \left (x \right )}}\) | \(17\) |
default | \(-\frac {4 \sin \left (x \right ) \left (\sin \left (x \right )^{2}+3\right )}{3 \sqrt {-\sin \left (3 x \right )+3 \sin \left (x \right )}}\) | \(17\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {\cos ^3(x)}{\sqrt {\sin ^3(x)}} \, dx=-\frac {2 \, {\left (\cos \left (x\right )^{2} - 4\right )} \sqrt {-{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}}{3 \, {\left (\cos \left (x\right )^{2} - 1\right )}} \]
[In]
[Out]
Time = 0.43 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {\cos ^3(x)}{\sqrt {\sin ^3(x)}} \, dx=- \frac {8 \sin ^{3}{\left (x \right )}}{3 \sqrt {\sin ^{3}{\left (x \right )}}} - \frac {2 \sin {\left (x \right )} \cos ^{2}{\left (x \right )}}{\sqrt {\sin ^{3}{\left (x \right )}}} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {\cos ^3(x)}{\sqrt {\sin ^3(x)}} \, dx=-\frac {2}{3} \, \sqrt {\sin \left (x\right )^{3}} - \frac {2 \, \sin \left (x\right )}{\sqrt {\sin \left (x\right )^{3}}} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.52 \[ \int \frac {\cos ^3(x)}{\sqrt {\sin ^3(x)}} \, dx=-\frac {2}{3} \, \sin \left (x\right )^{\frac {3}{2}} - \frac {2}{\sqrt {\sin \left (x\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {\cos ^3(x)}{\sqrt {\sin ^3(x)}} \, dx=\int \frac {{\cos \left (x\right )}^3}{\sqrt {{\sin \left (x\right )}^3}} \,d x \]
[In]
[Out]