Optimal. Leaf size=111 \[ 2 \sqrt{\sin ^2(x)-5}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{\sin ^2(x)-5}}{\sqrt{5}}\right )}{\sqrt{5}}-2 \tanh ^{-1}\left (\frac{\sin (x)}{\sqrt{\sin ^2(x)-5}}\right )+\frac{2}{5} \sqrt{\sin ^2(x)-5} \csc (x)+2 \tan ^{-1}\left (\frac{\cos (x)}{\sqrt{\sin ^2(x)-5}}\right )-\frac{\tan ^{-1}\left (\frac{\sqrt{5} \cos (x)}{\sqrt{\sin ^2(x)-5}}\right )}{\sqrt{5}} \]
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Rubi [A] time = 0.574979, antiderivative size = 119, normalized size of antiderivative = 1.07, number of steps used = 18, number of rules used = 13, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.394, Rules used = {4401, 4356, 451, 217, 206, 4366, 6725, 203, 261, 1010, 377, 444, 63} \[ 2 \sqrt{-\cos ^2(x)-4}-2 \tanh ^{-1}\left (\frac{\sin (x)}{\sqrt{\sin ^2(x)-5}}\right )+2 \tan ^{-1}\left (\frac{\cos (x)}{\sqrt{-\cos ^2(x)-4}}\right )-\frac{\tan ^{-1}\left (\frac{\sqrt{5} \cos (x)}{\sqrt{-\cos ^2(x)-4}}\right )}{\sqrt{5}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{-\cos ^2(x)-4}}{\sqrt{5}}\right )}{\sqrt{5}}+\frac{2}{5} \sqrt{\sin ^2(x)-5} \csc (x) \]
Antiderivative was successfully verified.
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Rule 4401
Rule 4356
Rule 451
Rule 217
Rule 206
Rule 4366
Rule 6725
Rule 203
Rule 261
Rule 1010
Rule 377
Rule 444
Rule 63
Rubi steps
\begin{align*} \int \frac{\csc ^2(x) \left (-2 \cos ^3(x) (-1+\sin (x))+\cos (2 x) \sin (x)\right )}{\sqrt{-5+\sin ^2(x)}} \, dx &=\int \left (\frac{2 \cos (x) \cot ^2(x)}{\sqrt{-5+\sin ^2(x)}}+\frac{\left (-2 \cos ^3(x)+\cos (2 x)\right ) \csc (x)}{\sqrt{-5+\sin ^2(x)}}\right ) \, dx\\ &=2 \int \frac{\cos (x) \cot ^2(x)}{\sqrt{-5+\sin ^2(x)}} \, dx+\int \frac{\left (-2 \cos ^3(x)+\cos (2 x)\right ) \csc (x)}{\sqrt{-5+\sin ^2(x)}} \, dx\\ &=2 \operatorname{Subst}\left (\int \frac{1-x^2}{x^2 \sqrt{-5+x^2}} \, dx,x,\sin (x)\right )-\operatorname{Subst}\left (\int \frac{-1+2 x^2-2 x^3}{\sqrt{-4-x^2} \left (1-x^2\right )} \, dx,x,\cos (x)\right )\\ &=\frac{2}{5} \csc (x) \sqrt{-5+\sin ^2(x)}-2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{-5+x^2}} \, dx,x,\sin (x)\right )-\operatorname{Subst}\left (\int \left (-\frac{2}{\sqrt{-4-x^2}}+\frac{2 x}{\sqrt{-4-x^2}}+\frac{1-2 x}{\sqrt{-4-x^2} \left (1-x^2\right )}\right ) \, dx,x,\cos (x)\right )\\ &=\frac{2}{5} \csc (x) \sqrt{-5+\sin ^2(x)}+2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{-4-x^2}} \, dx,x,\cos (x)\right )-2 \operatorname{Subst}\left (\int \frac{x}{\sqrt{-4-x^2}} \, dx,x,\cos (x)\right )-2 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sin (x)}{\sqrt{-5+\sin ^2(x)}}\right )-\operatorname{Subst}\left (\int \frac{1-2 x}{\sqrt{-4-x^2} \left (1-x^2\right )} \, dx,x,\cos (x)\right )\\ &=-2 \tanh ^{-1}\left (\frac{\sin (x)}{\sqrt{-5+\sin ^2(x)}}\right )+2 \sqrt{-4-\cos ^2(x)}+\frac{2}{5} \csc (x) \sqrt{-5+\sin ^2(x)}+2 \operatorname{Subst}\left (\int \frac{x}{\sqrt{-4-x^2} \left (1-x^2\right )} \, dx,x,\cos (x)\right )+2 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\cos (x)}{\sqrt{-4-\cos ^2(x)}}\right )-\operatorname{Subst}\left (\int \frac{1}{\sqrt{-4-x^2} \left (1-x^2\right )} \, dx,x,\cos (x)\right )\\ &=2 \tan ^{-1}\left (\frac{\cos (x)}{\sqrt{-4-\cos ^2(x)}}\right )-2 \tanh ^{-1}\left (\frac{\sin (x)}{\sqrt{-5+\sin ^2(x)}}\right )+2 \sqrt{-4-\cos ^2(x)}+\frac{2}{5} \csc (x) \sqrt{-5+\sin ^2(x)}+\operatorname{Subst}\left (\int \frac{1}{\sqrt{-4-x} (1-x)} \, dx,x,\cos ^2(x)\right )-\operatorname{Subst}\left (\int \frac{1}{1+5 x^2} \, dx,x,\frac{\cos (x)}{\sqrt{-4-\cos ^2(x)}}\right )\\ &=2 \tan ^{-1}\left (\frac{\cos (x)}{\sqrt{-4-\cos ^2(x)}}\right )-\frac{\tan ^{-1}\left (\frac{\sqrt{5} \cos (x)}{\sqrt{-4-\cos ^2(x)}}\right )}{\sqrt{5}}-2 \tanh ^{-1}\left (\frac{\sin (x)}{\sqrt{-5+\sin ^2(x)}}\right )+2 \sqrt{-4-\cos ^2(x)}+\frac{2}{5} \csc (x) \sqrt{-5+\sin ^2(x)}-2 \operatorname{Subst}\left (\int \frac{1}{5+x^2} \, dx,x,\sqrt{-4-\cos ^2(x)}\right )\\ &=2 \tan ^{-1}\left (\frac{\cos (x)}{\sqrt{-4-\cos ^2(x)}}\right )-\frac{\tan ^{-1}\left (\frac{\sqrt{5} \cos (x)}{\sqrt{-4-\cos ^2(x)}}\right )}{\sqrt{5}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{-4-\cos ^2(x)}}{\sqrt{5}}\right )}{\sqrt{5}}-2 \tanh ^{-1}\left (\frac{\sin (x)}{\sqrt{-5+\sin ^2(x)}}\right )+2 \sqrt{-4-\cos ^2(x)}+\frac{2}{5} \csc (x) \sqrt{-5+\sin ^2(x)}\\ \end{align*}
Mathematica [C] time = 2.32738, size = 338, normalized size = 3.05 \[ \frac{(16-32 i) \sqrt{5} \sqrt{\frac{(1+2 i) (\cos (x)-2 i)}{\cos (x)+1}} \sqrt{\frac{(1-2 i) (\cos (x)+2 i)}{\cos (x)+1}} \cos ^2\left (\frac{x}{2}\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{(1+2 i) \tan \left (\frac{x}{2}\right )}{\sqrt{5}}\right ),-\frac{7}{25}+\frac{24 i}{25}\right )-(32-64 i) \sqrt{5} \sqrt{\frac{(1+2 i) (\cos (x)-2 i)}{\cos (x)+1}} \sqrt{\frac{(1-2 i) (\cos (x)+2 i)}{\cos (x)+1}} \cos ^2\left (\frac{x}{2}\right ) \Pi \left (\frac{3}{5}+\frac{4 i}{5};\sin ^{-1}\left (\frac{(1+2 i) \tan \left (\frac{x}{2}\right )}{\sqrt{5}}\right )|-\frac{7}{25}+\frac{24 i}{25}\right )-5 \left (18 \csc (x)+10 i \sqrt{2} \sqrt{-\cos (2 x)-9} \log \left (\sqrt{-\cos (2 x)-9}+i \sqrt{2} \cos (x)\right )+\sqrt{10} \sqrt{-\cos (2 x)-9} \tan ^{-1}\left (\frac{\sqrt{10} \cos (x)}{\sqrt{-\cos (2 x)-9}}\right )+2 \sqrt{10} \sqrt{-\cos (2 x)-9} \tan ^{-1}\left (\frac{\sqrt{-\cos (2 x)-9}}{\sqrt{10}}\right )+2 \cos (2 x) \csc (x)+5 \sin (3 x) \csc (x)+85\right )}{25 \sqrt{2} \sqrt{-\cos (2 x)-9}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.184, size = 130, normalized size = 1.2 \begin{align*} -2\,\ln \left ( \sin \left ( x \right ) +\sqrt{-5+ \left ( \sin \left ( x \right ) \right ) ^{2}} \right ) +2\,\sqrt{-5+ \left ( \sin \left ( x \right ) \right ) ^{2}}+{\frac{2\,\sqrt{5}}{5}\arctan \left ({\sqrt{5}{\frac{1}{\sqrt{-5+ \left ( \sin \left ( x \right ) \right ) ^{2}}}}} \right ) }+{\frac{2}{5\,\sin \left ( x \right ) }\sqrt{-5+ \left ( \sin \left ( x \right ) \right ) ^{2}}}+{\frac{1}{10\,\cos \left ( x \right ) }\sqrt{ \left ( -5+ \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \left ( \cos \left ( x \right ) \right ) ^{2}} \left ( \sqrt{5}\arctan \left ({\frac{ \left ( 3\, \left ( \sin \left ( x \right ) \right ) ^{2}-5 \right ) \sqrt{5}}{5}{\frac{1}{\sqrt{- \left ( \cos \left ( x \right ) \right ) ^{4}-4\, \left ( \cos \left ( x \right ) \right ) ^{2}}}}} \right ) +10\,\arcsin \left ( 1+1/2\, \left ( \cos \left ( x \right ) \right ) ^{2} \right ) \right ){\frac{1}{\sqrt{-5+ \left ( \sin \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.51108, size = 155, normalized size = 1.4 \begin{align*} \frac{2}{5} \, \sqrt{5} \arcsin \left (\frac{\sqrt{5}}{{\left | \sin \left (x\right ) \right |}}\right ) - \frac{1}{10} i \, \sqrt{5} \operatorname{arsinh}\left (\frac{\cos \left (x\right )}{2 \,{\left (\cos \left (x\right ) + 1\right )}} - \frac{2}{\cos \left (x\right ) + 1}\right ) - \frac{1}{10} i \, \sqrt{5} \operatorname{arsinh}\left (-\frac{\cos \left (x\right )}{2 \,{\left (\cos \left (x\right ) - 1\right )}} - \frac{2}{\cos \left (x\right ) - 1}\right ) + 2 \, \sqrt{\sin \left (x\right )^{2} - 5} + \frac{2 \, \sqrt{\sin \left (x\right )^{2} - 5}}{5 \, \sin \left (x\right )} - 2 i \, \operatorname{arsinh}\left (\frac{1}{2} \, \cos \left (x\right )\right ) - 2 \, \log \left (2 \, \sqrt{\sin \left (x\right )^{2} - 5} + 2 \, \sin \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 [C] time = 1.78554, size = 356, normalized size = 3.21 \begin{align*} \pi \mathrm{sgn}\left (-2 i \, \sqrt{\cos \left (x\right )^{2} + 4} - 4 i\right ) \mathrm{sgn}\left (\cos \left (x\right )\right ) - \frac{1}{5} \, \sqrt{5}{\left (\pi \mathrm{sgn}\left (-2 i \, \sqrt{\cos \left (x\right )^{2} + 4} - 4 i\right ) \mathrm{sgn}\left (\cos \left (x\right )\right ) + 2 \, \arctan \left (\frac{\sqrt{5}{\left (\frac{{\left (i \, \sqrt{\cos \left (x\right )^{2} + 4} + 2 i\right )}^{2}}{\cos \left (x\right )^{2}} - 1\right )} \cos \left (x\right )}{5 \,{\left (-2 i \, \sqrt{\cos \left (x\right )^{2} + 4} - 4 i\right )}}\right )\right )} + \frac{1}{10} \, \sqrt{5}{\left (\pi \mathrm{sgn}\left (-2 i \, \sqrt{\cos \left (x\right )^{2} + 4} - 4 i\right ) \mathrm{sgn}\left (\cos \left (x\right )\right ) + 2 \, \arctan \left (\frac{\sqrt{5}{\left (\frac{{\left (-i \, \sqrt{\cos \left (x\right )^{2} + 4} - 2 i\right )}^{2}}{\cos \left (x\right )^{2}} - 1\right )} \cos \left (x\right )}{5 \,{\left (-2 i \, \sqrt{\cos \left (x\right )^{2} + 4} - 4 i\right )}}\right )\right )} - \frac{2}{5} \, \sqrt{5} \arctan \left (\frac{1}{5} \, \sqrt{5} \sqrt{\sin \left (x\right )^{2} - 5}\right ) + 2 \, \sqrt{\sin \left (x\right )^{2} - 5} + \frac{4}{{\left (\sqrt{\sin \left (x\right )^{2} - 5} - \sin \left (x\right )\right )}^{2} + 5} + 2 \, \arctan \left (\frac{{\left (\frac{{\left (i \, \sqrt{\cos \left (x\right )^{2} + 4} + 2 i\right )}^{2}}{\cos \left (x\right )^{2}} - 1\right )} \cos \left (x\right )}{-2 i \, \sqrt{\cos \left (x\right )^{2} + 4} - 4 i}\right ) + \log \left ({\left (\sqrt{\sin \left (x\right )^{2} - 5} - \sin \left (x\right )\right )}^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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