Optimal. Leaf size=364 \[ -\sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt {2}-\tan (x)}{2^{3/4} \sqrt {\tan (x)}}\right )+\frac {4}{\sqrt {\tan (x)}}-\sqrt [4]{2} \coth ^{-1}\left (\frac {\tan (x)+\sqrt {2}}{2^{3/4} \sqrt {\tan (x)}}\right )-2 \sqrt {2} \tan ^{-1}\left (\frac {\cos (x) (\cos (x)-\sin (x))}{\sqrt {2} \sqrt {\sin (x) \cos ^3(x)}}\right )+\sqrt [4]{2} \tan ^{-1}\left (\frac {\cos (x) \left (\sqrt {2} \cos (x)-\sin (x)\right )}{2^{3/4} \sqrt {\sin (x) \cos ^3(x)}}\right )-2 \sqrt {2} \coth ^{-1}\left (\frac {\cos (x) (\sin (x)+\cos (x))}{\sqrt {2} \sqrt {\sin (x) \cos ^3(x)}}\right )+\sqrt [4]{2} \coth ^{-1}\left (\frac {\cos (x) \left (\sin (x)+\sqrt {2} \cos (x)\right )}{2^{3/4} \sqrt {\sin (x) \cos ^3(x)}}\right )+4 \csc (x) \sec (x) \sqrt {\sin (x) \cos ^3(x)}+\frac {1}{2} \csc ^2(x) \sec ^2(x) \log (\sin (x)) \sqrt {\sin (x) \cos ^3(x)} \sqrt {\sin ^3(x) \cos (x)}-\frac {1}{4} \sqrt {\tan (x)} \csc ^2(x) \log \left (\cos ^2(x)+1\right ) \sqrt {\sin ^3(x) \cos (x)}+\frac {1}{4} \csc ^2(x) \sec ^2(x) \log \left (\cos ^2(x)+1\right ) \sqrt {\sin (x) \cos ^3(x)} \sqrt {\sin ^3(x) \cos (x)}+\frac {1}{2} \sqrt {\tan (x)} \csc ^2(x) \log (\sin (x)) \sqrt {\sin ^3(x) \cos (x)} \]
________________________________________________________________________________________
Rubi [A] time = 4.73, antiderivative size = 665, normalized size of antiderivative = 1.83, number of steps used = 66, number of rules used = 21, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.512, Rules used = {6725, 6742, 6719, 325, 329, 297, 1162, 617, 204, 1165, 628, 466, 482, 6733, 15, 29, 266, 36, 31, 260, 444} \[ -\sqrt [4]{2} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt {\tan (x)}\right )+\sqrt [4]{2} \tan ^{-1}\left (\sqrt [4]{2} \sqrt {\tan (x)}+1\right )+\frac {4}{\sqrt {\tan (x)}}+\frac {\log \left (\tan (x)-2^{3/4} \sqrt {\tan (x)}+\sqrt {2}\right )}{2^{3/4}}-\frac {\log \left (\tan (x)+2^{3/4} \sqrt {\tan (x)}+\sqrt {2}\right )}{2^{3/4}}+4 \csc (x) \sec (x) \sqrt {\sin (x) \cos ^3(x)}+\frac {\sqrt [4]{2} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt {\tan (x)}\right ) \sec ^2(x) \sqrt {\sin (x) \cos ^3(x)}}{\sqrt {\tan (x)}}-\frac {\sqrt [4]{2} \tan ^{-1}\left (\sqrt [4]{2} \sqrt {\tan (x)}+1\right ) \sec ^2(x) \sqrt {\sin (x) \cos ^3(x)}}{\sqrt {\tan (x)}}-\frac {2 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (x)}\right ) \sec ^2(x) \sqrt {\sin (x) \cos ^3(x)}}{\sqrt {\tan (x)}}+\frac {2 \sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (x)}+1\right ) \sec ^2(x) \sqrt {\sin (x) \cos ^3(x)}}{\sqrt {\tan (x)}}+\frac {\sqrt {2} \sec ^2(x) \log \left (\tan (x)-\sqrt {2} \sqrt {\tan (x)}+1\right ) \sqrt {\sin (x) \cos ^3(x)}}{\sqrt {\tan (x)}}-\frac {\sqrt {2} \sec ^2(x) \log \left (\tan (x)+\sqrt {2} \sqrt {\tan (x)}+1\right ) \sqrt {\sin (x) \cos ^3(x)}}{\sqrt {\tan (x)}}-\frac {\sec ^2(x) \log \left (\tan (x)-2^{3/4} \sqrt {\tan (x)}+\sqrt {2}\right ) \sqrt {\sin (x) \cos ^3(x)}}{2^{3/4} \sqrt {\tan (x)}}+\frac {\sec ^2(x) \log \left (\tan (x)+2^{3/4} \sqrt {\tan (x)}+\sqrt {2}\right ) \sqrt {\sin (x) \cos ^3(x)}}{2^{3/4} \sqrt {\tan (x)}}-\frac {1}{2} \csc ^2(x) \sec ^2(x) \log \left (\sec ^2(x)\right ) \sqrt {\sin (x) \cos ^3(x)} \sqrt {\sin ^3(x) \cos (x)}+\frac {\sec ^2(x) \log (\tan (x)) \sqrt {\sin ^3(x) \cos (x)}}{2 \tan ^{\frac {3}{2}}(x)}-\frac {\sec ^2(x) \log \left (\tan ^2(x)+2\right ) \sqrt {\sin ^3(x) \cos (x)}}{4 \tan ^{\frac {3}{2}}(x)}+\frac {1}{4} \csc ^2(x) \sec ^2(x) \log \left (\tan ^2(x)+2\right ) \sqrt {\sin (x) \cos ^3(x)} \sqrt {\sin ^3(x) \cos (x)}+\csc ^2(x) \sec ^2(x) \log \left (\sqrt {\tan (x)}\right ) \sqrt {\sin (x) \cos ^3(x)} \sqrt {\sin ^3(x) \cos (x)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 15
Rule 29
Rule 31
Rule 36
Rule 204
Rule 260
Rule 266
Rule 297
Rule 325
Rule 329
Rule 444
Rule 466
Rule 482
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 6719
Rule 6725
Rule 6733
Rule 6742
Rubi steps
\begin {align*} \int \frac {\sqrt {\cos (x) \sin ^3(x)}-2 \sin (2 x)}{-\sqrt {\cos ^3(x) \sin (x)}+\sqrt {\tan (x)}} \, dx &=\operatorname {Subst}\left (\int \frac {\sqrt {\frac {x^3}{\left (1+x^2\right )^2}}-\frac {4 x}{1+x^2}}{\left (1+x^2\right ) \left (\sqrt {x}-\sqrt {\frac {x}{\left (1+x^2\right )^2}}\right )} \, dx,x,\tan (x)\right )\\ &=\operatorname {Subst}\left (\int \left (\frac {4 x}{\left (1+x^2\right )^2 \left (-\sqrt {x}+\sqrt {\frac {x}{\left (1+x^2\right )^2}}\right )}-\frac {\sqrt {\frac {x^3}{\left (1+x^2\right )^2}}}{\left (1+x^2\right ) \left (-\sqrt {x}+\sqrt {\frac {x}{\left (1+x^2\right )^2}}\right )}\right ) \, dx,x,\tan (x)\right )\\ &=4 \operatorname {Subst}\left (\int \frac {x}{\left (1+x^2\right )^2 \left (-\sqrt {x}+\sqrt {\frac {x}{\left (1+x^2\right )^2}}\right )} \, dx,x,\tan (x)\right )-\operatorname {Subst}\left (\int \frac {\sqrt {\frac {x^3}{\left (1+x^2\right )^2}}}{\left (1+x^2\right ) \left (-\sqrt {x}+\sqrt {\frac {x}{\left (1+x^2\right )^2}}\right )} \, dx,x,\tan (x)\right )\\ &=4 \operatorname {Subst}\left (\int \left (-\frac {1}{2 x^{3/2}}-\frac {\sqrt {\frac {x}{\left (1+x^2\right )^2}}}{2 x^2}+\frac {\sqrt {x}}{2 \left (2+x^2\right )}+\frac {\sqrt {\frac {x}{\left (1+x^2\right )^2}}}{2 \left (2+x^2\right )}\right ) \, dx,x,\tan (x)\right )-\frac {\left (\sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}\right ) \operatorname {Subst}\left (\int \frac {x^{3/2}}{\left (1+x^2\right )^2 \left (-\sqrt {x}+\sqrt {\frac {x}{\left (1+x^2\right )^2}}\right )} \, dx,x,\tan (x)\right )}{\tan ^{\frac {3}{2}}(x)}\\ &=\frac {4}{\sqrt {\tan (x)}}-2 \operatorname {Subst}\left (\int \frac {\sqrt {\frac {x}{\left (1+x^2\right )^2}}}{x^2} \, dx,x,\tan (x)\right )+2 \operatorname {Subst}\left (\int \frac {\sqrt {x}}{2+x^2} \, dx,x,\tan (x)\right )+2 \operatorname {Subst}\left (\int \frac {\sqrt {\frac {x}{\left (1+x^2\right )^2}}}{2+x^2} \, dx,x,\tan (x)\right )-\frac {\left (2 \sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (1+x^4\right )^2 \left (-\sqrt {x^2}+\sqrt {\frac {x^2}{\left (1+x^4\right )^2}}\right )} \, dx,x,\sqrt {\tan (x)}\right )}{\tan ^{\frac {3}{2}}(x)}\\ &=\frac {4}{\sqrt {\tan (x)}}+4 \operatorname {Subst}\left (\int \frac {x^2}{2+x^4} \, dx,x,\sqrt {\tan (x)}\right )-\frac {\left (2 \sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{2 \sqrt {x^2}}-\frac {\sqrt {\frac {x^2}{\left (1+x^4\right )^2}}}{2 x^2}+\frac {\left (x^2\right )^{3/2}}{2 \left (2+x^4\right )}+\frac {x^2 \sqrt {\frac {x^2}{\left (1+x^4\right )^2}}}{2 \left (2+x^4\right )}\right ) \, dx,x,\sqrt {\tan (x)}\right )}{\tan ^{\frac {3}{2}}(x)}-\frac {\left (2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \operatorname {Subst}\left (\int \frac {1}{x^{3/2} \left (1+x^2\right )} \, dx,x,\tan (x)\right )}{\sqrt {\tan (x)}}+\frac {\left (2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {x}}{\left (1+x^2\right ) \left (2+x^2\right )} \, dx,x,\tan (x)\right )}{\sqrt {\tan (x)}}\\ &=4 \csc (x) \sec (x) \sqrt {\cos ^3(x) \sin (x)}+\frac {4}{\sqrt {\tan (x)}}-2 \operatorname {Subst}\left (\int \frac {\sqrt {2}-x^2}{2+x^4} \, dx,x,\sqrt {\tan (x)}\right )+2 \operatorname {Subst}\left (\int \frac {\sqrt {2}+x^2}{2+x^4} \, dx,x,\sqrt {\tan (x)}\right )+\frac {\left (\sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x^2}} \, dx,x,\sqrt {\tan (x)}\right )}{\tan ^{\frac {3}{2}}(x)}+\frac {\left (\sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {\frac {x^2}{\left (1+x^4\right )^2}}}{x^2} \, dx,x,\sqrt {\tan (x)}\right )}{\tan ^{\frac {3}{2}}(x)}-\frac {\left (\sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}\right ) \operatorname {Subst}\left (\int \frac {\left (x^2\right )^{3/2}}{2+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{\tan ^{\frac {3}{2}}(x)}-\frac {\left (\sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {\frac {x^2}{\left (1+x^4\right )^2}}}{2+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{\tan ^{\frac {3}{2}}(x)}+\frac {\left (2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\tan (x)\right )}{\sqrt {\tan (x)}}+\frac {\left (4 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^4\right ) \left (2+x^4\right )} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}\\ &=4 \csc (x) \sec (x) \sqrt {\cos ^3(x) \sin (x)}+\frac {4}{\sqrt {\tan (x)}}+\frac {\operatorname {Subst}\left (\int \frac {2^{3/4}+2 x}{-\sqrt {2}-2^{3/4} x-x^2} \, dx,x,\sqrt {\tan (x)}\right )}{2^{3/4}}+\frac {\operatorname {Subst}\left (\int \frac {2^{3/4}-2 x}{-\sqrt {2}+2^{3/4} x-x^2} \, dx,x,\sqrt {\tan (x)}\right )}{2^{3/4}}+\left (\csc ^2(x) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (1+x^4\right )} \, dx,x,\sqrt {\tan (x)}\right )-\left (\csc ^2(x) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (1+x^4\right ) \left (2+x^4\right )} \, dx,x,\sqrt {\tan (x)}\right )+\frac {\left (\sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\sqrt {\tan (x)}\right )}{\tan ^{\frac {3}{2}}(x)}-\frac {\left (\sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}\right ) \operatorname {Subst}\left (\int \frac {x^3}{2+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{\tan ^{\frac {3}{2}}(x)}+2 \frac {\left (4 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}-\frac {\left (4 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \operatorname {Subst}\left (\int \frac {x^2}{2+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}+\operatorname {Subst}\left (\int \frac {1}{\sqrt {2}-2^{3/4} x+x^2} \, dx,x,\sqrt {\tan (x)}\right )+\operatorname {Subst}\left (\int \frac {1}{\sqrt {2}+2^{3/4} x+x^2} \, dx,x,\sqrt {\tan (x)}\right )\\ &=\frac {\log \left (\sqrt {2}-2^{3/4} \sqrt {\tan (x)}+\tan (x)\right )}{2^{3/4}}-\frac {\log \left (\sqrt {2}+2^{3/4} \sqrt {\tan (x)}+\tan (x)\right )}{2^{3/4}}+4 \csc (x) \sec (x) \sqrt {\cos ^3(x) \sin (x)}+\frac {\log (\tan (x)) \sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}}{2 \tan ^{\frac {3}{2}}(x)}-\frac {\log \left (2+\tan ^2(x)\right ) \sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}}{4 \tan ^{\frac {3}{2}}(x)}+\frac {4}{\sqrt {\tan (x)}}+\sqrt [4]{2} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt [4]{2} \sqrt {\tan (x)}\right )-\sqrt [4]{2} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt [4]{2} \sqrt {\tan (x)}\right )+\frac {1}{4} \left (\csc ^2(x) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}\right ) \operatorname {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,\tan ^2(x)\right )-\frac {1}{4} \left (\csc ^2(x) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}\right ) \operatorname {Subst}\left (\int \frac {1}{(1+x) (2+x)} \, dx,x,\tan ^2(x)\right )+2 \left (-\frac {\left (2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}+\frac {\left (2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}\right )+\frac {\left (2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}-x^2}{2+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}-\frac {\left (2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}+x^2}{2+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}\\ &=-\sqrt [4]{2} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt {\tan (x)}\right )+\sqrt [4]{2} \tan ^{-1}\left (1+\sqrt [4]{2} \sqrt {\tan (x)}\right )+\frac {\log \left (\sqrt {2}-2^{3/4} \sqrt {\tan (x)}+\tan (x)\right )}{2^{3/4}}-\frac {\log \left (\sqrt {2}+2^{3/4} \sqrt {\tan (x)}+\tan (x)\right )}{2^{3/4}}+4 \csc (x) \sec (x) \sqrt {\cos ^3(x) \sin (x)}+\frac {\log (\tan (x)) \sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}}{2 \tan ^{\frac {3}{2}}(x)}-\frac {\log \left (2+\tan ^2(x)\right ) \sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}}{4 \tan ^{\frac {3}{2}}(x)}+\frac {4}{\sqrt {\tan (x)}}+\frac {1}{4} \left (\csc ^2(x) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\tan ^2(x)\right )-2 \left (\frac {1}{4} \left (\csc ^2(x) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\tan ^2(x)\right )\right )+\frac {1}{4} \left (\csc ^2(x) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}\right ) \operatorname {Subst}\left (\int \frac {1}{2+x} \, dx,x,\tan ^2(x)\right )-\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}-2^{3/4} x+x^2} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}-\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}+2^{3/4} x+x^2} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}-\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \operatorname {Subst}\left (\int \frac {2^{3/4}+2 x}{-\sqrt {2}-2^{3/4} x-x^2} \, dx,x,\sqrt {\tan (x)}\right )}{2^{3/4} \sqrt {\tan (x)}}-\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \operatorname {Subst}\left (\int \frac {2^{3/4}-2 x}{-\sqrt {2}+2^{3/4} x-x^2} \, dx,x,\sqrt {\tan (x)}\right )}{2^{3/4} \sqrt {\tan (x)}}+2 \left (\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}+\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}+\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {2} \sqrt {\tan (x)}}+\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {2} \sqrt {\tan (x)}}\right )\\ &=-\sqrt [4]{2} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt {\tan (x)}\right )+\sqrt [4]{2} \tan ^{-1}\left (1+\sqrt [4]{2} \sqrt {\tan (x)}\right )+\frac {\log \left (\sqrt {2}-2^{3/4} \sqrt {\tan (x)}+\tan (x)\right )}{2^{3/4}}-\frac {\log \left (\sqrt {2}+2^{3/4} \sqrt {\tan (x)}+\tan (x)\right )}{2^{3/4}}+4 \csc (x) \sec (x) \sqrt {\cos ^3(x) \sin (x)}+\csc ^2(x) \log (\cos (x)) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}+\frac {1}{2} \csc ^2(x) \log (\tan (x)) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}+\frac {1}{4} \csc ^2(x) \log \left (2+\tan ^2(x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}+\frac {\log (\tan (x)) \sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}}{2 \tan ^{\frac {3}{2}}(x)}-\frac {\log \left (2+\tan ^2(x)\right ) \sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}}{4 \tan ^{\frac {3}{2}}(x)}+\frac {4}{\sqrt {\tan (x)}}-\frac {\log \left (\sqrt {2}-2^{3/4} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{2^{3/4} \sqrt {\tan (x)}}+\frac {\log \left (\sqrt {2}+2^{3/4} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{2^{3/4} \sqrt {\tan (x)}}-\frac {\left (\sqrt [4]{2} \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt [4]{2} \sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}+\frac {\left (\sqrt [4]{2} \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt [4]{2} \sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}+2 \left (\frac {\log \left (1-\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {2} \sqrt {\tan (x)}}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {2} \sqrt {\tan (x)}}+\frac {\left (\sqrt {2} \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}-\frac {\left (\sqrt {2} \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}\right )\\ &=-\sqrt [4]{2} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt {\tan (x)}\right )+\sqrt [4]{2} \tan ^{-1}\left (1+\sqrt [4]{2} \sqrt {\tan (x)}\right )+\frac {\log \left (\sqrt {2}-2^{3/4} \sqrt {\tan (x)}+\tan (x)\right )}{2^{3/4}}-\frac {\log \left (\sqrt {2}+2^{3/4} \sqrt {\tan (x)}+\tan (x)\right )}{2^{3/4}}+4 \csc (x) \sec (x) \sqrt {\cos ^3(x) \sin (x)}+\csc ^2(x) \log (\cos (x)) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}+\frac {1}{2} \csc ^2(x) \log (\tan (x)) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}+\frac {1}{4} \csc ^2(x) \log \left (2+\tan ^2(x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}+2 \left (-\frac {\sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (x)}\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {\tan (x)}}+\frac {\sqrt {2} \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (x)}\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {\tan (x)}}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {2} \sqrt {\tan (x)}}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {2} \sqrt {\tan (x)}}\right )+\frac {\log (\tan (x)) \sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}}{2 \tan ^{\frac {3}{2}}(x)}-\frac {\log \left (2+\tan ^2(x)\right ) \sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}}{4 \tan ^{\frac {3}{2}}(x)}+\frac {4}{\sqrt {\tan (x)}}+\frac {\sqrt [4]{2} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt {\tan (x)}\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {\tan (x)}}-\frac {\sqrt [4]{2} \tan ^{-1}\left (1+\sqrt [4]{2} \sqrt {\tan (x)}\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {\tan (x)}}-\frac {\log \left (\sqrt {2}-2^{3/4} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{2^{3/4} \sqrt {\tan (x)}}+\frac {\log \left (\sqrt {2}+2^{3/4} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{2^{3/4} \sqrt {\tan (x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 11.37, size = 385, normalized size = 1.06 \[ \frac {4 \sqrt {2} \cos ^2(x)^{3/4} \tan ^{\frac {3}{2}}(x) \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};\frac {2 \sin ^2(x)}{\cos (2 x)+3}\right )}{3 (\cos (2 x)+3)^{3/4}}+\frac {4}{\sqrt {\tan (x)}}+4 \csc (x) \sec (x) \sqrt {\sin (x) \cos ^3(x)}-\frac {(1+i) \sec ^4\left (\frac {x}{2}\right ) \sqrt {\sin (x) \cos ^3(x)} \left ((4+4 i) \Pi \left (-i;\left .\sin ^{-1}\left (\sqrt {\tan \left (\frac {x}{2}\right )}\right )\right |-1\right )-(4+4 i) \Pi \left (i;\left .\sin ^{-1}\left (\sqrt {\tan \left (\frac {x}{2}\right )}\right )\right |-1\right )+\sqrt [4]{-1} \left (-\Pi \left (-\sqrt [4]{-1};\left .\sin ^{-1}\left (\sqrt {\tan \left (\frac {x}{2}\right )}\right )\right |-1\right )+\Pi \left (\sqrt [4]{-1};\left .\sin ^{-1}\left (\sqrt {\tan \left (\frac {x}{2}\right )}\right )\right |-1\right )-\Pi \left (-(-1)^{3/4};\left .\sin ^{-1}\left (\sqrt {\tan \left (\frac {x}{2}\right )}\right )\right |-1\right )+\Pi \left ((-1)^{3/4};\left .\sin ^{-1}\left (\sqrt {\tan \left (\frac {x}{2}\right )}\right )\right |-1\right )\right )\right )}{\sqrt {\tan \left (\frac {x}{2}\right )} \left (\tan ^2\left (\frac {x}{2}\right )-1\right ) \sqrt {\cos (x) \sec ^2\left (\frac {x}{2}\right )}}+\frac {1}{4} \sqrt {\tan (x)} \csc ^2(x) \left (2 \log (\tan (x))-\log \left (\tan ^2(x)+2\right )\right ) \sqrt {\sin ^3(x) \cos (x)}-\frac {\cos (x) \csc \left (\frac {x}{2}\right ) \sec \left (\frac {x}{2}\right ) \sqrt {\sin ^3(x) \cos (x)} \left (-\log \left (\tan ^4\left (\frac {x}{2}\right )+1\right )-2 \log \left (\tan \left (\frac {x}{2}\right )\right )+4 \log \left (\sec ^2\left (\frac {x}{2}\right )\right )\right )}{8 \sqrt {\sin (x) \cos ^3(x)}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\cos (x) \sin ^3(x)}-2 \sin (2 x)}{-\sqrt {\cos ^3(x) \sin (x)}+\sqrt {\tan (x)}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {\sqrt {\cos \relax (x) \sin \relax (x)^{3}} - 2 \, \sin \left (2 \, x\right )}{\sqrt {\cos \relax (x)^{3} \sin \relax (x)} - \sqrt {\tan \relax (x)}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 4.60, size = 22964, normalized size = 63.09
method | result | size |
default | \(\text {Expression too large to display}\) | \(22964\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ -\int \frac {2\,\sin \left (2\,x\right )-\sqrt {\cos \relax (x)\,{\sin \relax (x)}^3}}{\sqrt {\mathrm {tan}\relax (x)}-\sqrt {{\cos \relax (x)}^3\,\sin \relax (x)}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________