Optimal. Leaf size=108 \[ -\frac{\sqrt{\sin (2 x)} \cos (x) \sin ^{-1}(\cos (x)-\sin (x))}{\sqrt{\sin (x) \cos ^3(x)}}-\frac{\sin (2 x)}{\sqrt{\sin (x) \cos ^3(x)}}-\frac{\sqrt{\sin (2 x)} \cos (x) \tanh ^{-1}(\sin (x))}{\sqrt{\sin (x) \cos ^3(x)}}-\sqrt{2} \log \left (\sin (x)+\cos (x)-\sqrt{2} \sec (x) \sqrt{\sin (x) \cos ^3(x)}\right ) \]
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Rubi [B] time = 1.53986, antiderivative size = 234, normalized size of antiderivative = 2.17, number of steps used = 27, number of rules used = 11, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.407, Rules used = {6719, 6725, 215, 329, 211, 1165, 628, 1162, 617, 204, 321} \[ -2 \sec ^2(x) \sqrt{\sin (x) \cos ^3(x)}-\frac{\sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (x)}\right ) \sec ^2(x) \sqrt{\sin (x) \cos ^3(x)}}{\sqrt{\tan (x)}}+\frac{\sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (x)}+1\right ) \sec ^2(x) \sqrt{\sin (x) \cos ^3(x)}}{\sqrt{\tan (x)}}-\frac{\sec ^2(x) \log \left (\tan (x)-\sqrt{2} \sqrt{\tan (x)}+1\right ) \sqrt{\sin (x) \cos ^3(x)}}{\sqrt{2} \sqrt{\tan (x)}}+\frac{\sec ^2(x) \log \left (\tan (x)+\sqrt{2} \sqrt{\tan (x)}+1\right ) \sqrt{\sin (x) \cos ^3(x)}}{\sqrt{2} \sqrt{\tan (x)}}-\sqrt{2} \cot (x) \sec ^2(x)^{3/2} \sqrt{\sin (x) \cos (x)} \sqrt{\sin (x) \cos ^3(x)} \sinh ^{-1}(\tan (x)) \]
Antiderivative was successfully verified.
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Rule 6719
Rule 6725
Rule 215
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rule 321
Rubi steps
\begin{align*} \int \frac{\cos (2 x)-\sqrt{\sin (2 x)}}{\sqrt{\cos ^3(x) \sin (x)}} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt{\frac{x}{\left (1+x^2\right )^2}} \left (1-x^2-\frac{x}{\sqrt{\frac{x}{2+2 x^2}}}\right )}{x} \, dx,x,\tan (x)\right )\\ &=\frac{\left (\sec ^2(x) \sqrt{\cos ^3(x) \sin (x)}\right ) \operatorname{Subst}\left (\int \frac{1-x^2-\frac{x}{\sqrt{\frac{x}{2+2 x^2}}}}{\sqrt{x} \left (1+x^2\right )} \, dx,x,\tan (x)\right )}{\sqrt{\tan (x)}}\\ &=\frac{\left (\sec ^2(x) \sqrt{\cos ^3(x) \sin (x)}\right ) \operatorname{Subst}\left (\int \left (-\frac{\sqrt{2} \sqrt{\frac{x}{1+x^2}}}{\sqrt{x}}+\frac{1}{\sqrt{x} \left (1+x^2\right )}-\frac{x^{3/2}}{1+x^2}\right ) \, dx,x,\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{x} \left (1+x^2\right )} \, dx,x,\tan (x)\right )}{\sqrt{\tan (x)}}-\frac{\left (\sec ^2(x) \sqrt{\cos ^3(x) \sin (x)}\right ) \operatorname{Subst}\left (\int \frac{x^{3/2}}{1+x^2} \, dx,x,\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{\sqrt{\frac{x}{1+x^2}}}{\sqrt{x}} \, dx,x,\tan (x)\right )}{\sqrt{\tan (x)}}\\ &=-2 \sec ^2(x) \sqrt{\cos ^3(x) \sin (x)}-\left (\sqrt{2} \cot (x) \sec ^2(x)^{3/2} \sqrt{\cos (x) \sin (x)} \sqrt{\cos ^3(x) \sin (x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,\tan (x)\right )+\frac{\left (\sec ^2(x) \sqrt{\cos ^3(x) \sin (x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \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{1}{1+x^4} \, dx,x,\sqrt{\tan (x)}\right )}{\sqrt{\tan (x)}}\\ &=-2 \sec ^2(x) \sqrt{\cos ^3(x) \sin (x)}-\sqrt{2} \sinh ^{-1}(\tan (x)) \cot (x) \sec ^2(x)^{3/2} \sqrt{\cos (x) \sin (x)} \sqrt{\cos ^3(x) \sin (x)}+\frac{\left (\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 (\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}{1+x^4} \, dx,x,\sqrt{\tan (x)}\right )}{\sqrt{\tan (x)}}\\ &=-2 \sec ^2(x) \sqrt{\cos ^3(x) \sin (x)}-\sqrt{2} \sinh ^{-1}(\tan (x)) \cot (x) \sec ^2(x)^{3/2} \sqrt{\cos (x) \sin (x)} \sqrt{\cos ^3(x) \sin (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 )}{2 \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 )}{2 \sqrt{\tan (x)}}+\frac{\left (\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 (\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 (\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 )}{2 \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 )}{2 \sqrt{2} \sqrt{\tan (x)}}\\ &=-2 \sec ^2(x) \sqrt{\cos ^3(x) \sin (x)}-\sqrt{2} \sinh ^{-1}(\tan (x)) \cot (x) \sec ^2(x)^{3/2} \sqrt{\cos (x) \sin (x)} \sqrt{\cos ^3(x) \sin (x)}-\frac{\log \left (1-\sqrt{2} \sqrt{\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt{\cos ^3(x) \sin (x)}}{2 \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)}}{2 \sqrt{2} \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 )}{2 \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 )}{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 )}{2 \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 )}{2 \sqrt{2} \sqrt{\tan (x)}}+\frac{\left (\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{2} \sqrt{\tan (x)}}-\frac{\left (\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{2} \sqrt{\tan (x)}}\\ &=-2 \sec ^2(x) \sqrt{\cos ^3(x) \sin (x)}-\sqrt{2} \sinh ^{-1}(\tan (x)) \cot (x) \sec ^2(x)^{3/2} \sqrt{\cos (x) \sin (x)} \sqrt{\cos ^3(x) \sin (x)}-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (x)}\right ) \sec ^2(x) \sqrt{\cos ^3(x) \sin (x)}}{\sqrt{2} \sqrt{\tan (x)}}+\frac{\tan ^{-1}\left (1+\sqrt{2} \sqrt{\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{\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 (\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{2} \sqrt{\tan (x)}}-\frac{\left (\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{2} \sqrt{\tan (x)}}\\ &=-2 \sec ^2(x) \sqrt{\cos ^3(x) \sin (x)}-\sqrt{2} \sinh ^{-1}(\tan (x)) \cot (x) \sec ^2(x)^{3/2} \sqrt{\cos (x) \sin (x)} \sqrt{\cos ^3(x) \sin (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{\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)}}\\ \end{align*}
Mathematica [C] time = 0.261313, size = 105, normalized size = 0.97 \[ \frac{-4 \sin (x) \cos ^3(x) \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\cos ^2(x)\right )-3 \sqrt [4]{\sin ^2(x)} \cos (x) \left (2 \sin (x)+\sqrt{\sin (2 x)} \left (\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )\right )\right )}{3 \sqrt [4]{\sin ^2(x)} \sqrt{\sin (x) \cos ^3(x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.317, size = 247, normalized size = 2.3 \begin{align*} -2\,{\frac{\cos \left ( x \right ) \sin \left ( x \right ) }{\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{3}\sin \left ( x \right ) }}}-{\frac{\cos \left ( x \right ) \sqrt{2} \left ( \sin \left ( x \right ) \right ) ^{2}}{\cos \left ( x \right ) -1} \left ( i{\it EllipticPi} \left ( \sqrt{-{\frac{-\sin \left ( x \right ) -1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}},{\frac{1}{2}}-{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) -i{\it EllipticPi} \left ( \sqrt{-{\frac{-\sin \left ( x \right ) -1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}},{\frac{1}{2}}+{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) +{\it EllipticPi} \left ( \sqrt{-{\frac{-\sin \left ( x \right ) -1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}},{\frac{1}{2}}-{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) +{\it EllipticPi} \left ( \sqrt{-{\frac{-\sin \left ( x \right ) -1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}},{\frac{1}{2}}+{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) -2\,{\it EllipticF} \left ( \sqrt{-{\frac{-\sin \left ( x \right ) -1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}},1/2\,\sqrt{2} \right ) \right ) \sqrt{-{\frac{-\sin \left ( x \right ) -1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}\sqrt{{\frac{\sin \left ( x \right ) -1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}\sqrt{{\frac{\cos \left ( x \right ) -1}{\sin \left ( x \right ) }}}{\frac{1}{\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{3}\sin \left ( x \right ) }}}}+2\,{\frac{\cos \left ( x \right ) \sqrt{2}\sqrt{\cos \left ( x \right ) \sin \left ( x \right ) }}{\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{3}\sin \left ( x \right ) }}{\it Artanh} \left ({\frac{\cos \left ( x \right ) -1}{\sin \left ( x \right ) }} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 76.0529, size = 2007, normalized size = 18.58 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{\sqrt{\sin \left (2 \, x\right )} - \cos \left (2 \, x\right )}{\sqrt{\cos \left (x\right )^{3} \sin \left (x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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