Optimal. Leaf size=111 \[ 2 \tan ^{-1}\left (\frac {\cos (x)}{\sqrt {-5+\sin ^2(x)}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {5} \cos (x)}{\sqrt {-5+\sin ^2(x)}}\right )}{\sqrt {5}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {-5+\sin ^2(x)}}{\sqrt {5}}\right )}{\sqrt {5}}-2 \tanh ^{-1}\left (\frac {\sin (x)}{\sqrt {-5+\sin ^2(x)}}\right )+2 \sqrt {-5+\sin ^2(x)}+\frac {2}{5} \csc (x) \sqrt {-5+\sin ^2(x)} \]
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Rubi [A]
time = 0.41, 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 = {4486, 4441,
462, 223, 212, 4451, 6857, 209, 267, 1024, 385, 455, 65} \begin {gather*} 2 \text {ArcTan}\left (\frac {\cos (x)}{\sqrt {-\cos ^2(x)-4}}\right )-\frac {\text {ArcTan}\left (\frac {\sqrt {5} \cos (x)}{\sqrt {-\cos ^2(x)-4}}\right )}{\sqrt {5}}-\frac {2 \text {ArcTan}\left (\frac {\sqrt {-\cos ^2(x)-4}}{\sqrt {5}}\right )}{\sqrt {5}}+2 \sqrt {-\cos ^2(x)-4}-2 \tanh ^{-1}\left (\frac {\sin (x)}{\sqrt {\sin ^2(x)-5}}\right )+\frac {2}{5} \sqrt {\sin ^2(x)-5} \csc (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 209
Rule 212
Rule 223
Rule 267
Rule 385
Rule 455
Rule 462
Rule 1024
Rule 4441
Rule 4451
Rule 4486
Rule 6857
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 \text {Subst}\left (\int \frac {1-x^2}{x^2 \sqrt {-5+x^2}} \, dx,x,\sin (x)\right )-\text {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 \text {Subst}\left (\int \frac {1}{\sqrt {-5+x^2}} \, dx,x,\sin (x)\right )-\text {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 \text {Subst}\left (\int \frac {1}{\sqrt {-4-x^2}} \, dx,x,\cos (x)\right )-2 \text {Subst}\left (\int \frac {x}{\sqrt {-4-x^2}} \, dx,x,\cos (x)\right )-2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sin (x)}{\sqrt {-5+\sin ^2(x)}}\right )-\text {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 \text {Subst}\left (\int \frac {x}{\sqrt {-4-x^2} \left (1-x^2\right )} \, dx,x,\cos (x)\right )+2 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\cos (x)}{\sqrt {-4-\cos ^2(x)}}\right )-\text {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)}+\text {Subst}\left (\int \frac {1}{\sqrt {-4-x} (1-x)} \, dx,x,\cos ^2(x)\right )-\text {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 \text {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*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.18, size = 295, normalized size = 2.66 \begin {gather*} \frac {2 \sqrt {2} \left (-2 \cos ^3(x)+\cos (2 x)+2 \cos ^2(x) \cot (x)\right ) \left (18+2 \cos (2 x)+20 \sqrt {2} \tanh ^{-1}\left (\frac {2 \sqrt {2} \tan \left (\frac {x}{2}\right )}{\sqrt {-\left ((9+\cos (2 x)) \sec ^4\left (\frac {x}{2}\right )\right )}}\right ) \cos ^3\left (\frac {x}{2}\right ) \sqrt {-\left ((9+\cos (2 x)) \sec ^4\left (\frac {x}{2}\right )\right )} \sin \left (\frac {x}{2}\right )+85 \sin (x)+\sqrt {10} \tan ^{-1}\left (\frac {\sqrt {10} \cos (x)}{\sqrt {-9-\cos (2 x)}}\right ) \sqrt {-9-\cos (2 x)} \sin (x)+2 \sqrt {10} \tan ^{-1}\left (\frac {\sqrt {-9-\cos (2 x)}}{\sqrt {10}}\right ) \sqrt {-9-\cos (2 x)} \sin (x)+10 i \sqrt {2} \sqrt {-9-\cos (2 x)} \log \left (i \sqrt {2} \cos (x)+\sqrt {-9-\cos (2 x)}\right ) \sin (x)+5 \sin (3 x)\right )}{5 \sqrt {-9-\cos (2 x)} (-6 \cos (x)-2 \cos (3 x)+2 \sin (x)+2 \sin (2 x)-2 \sin (3 x)+\sin (4 x))} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.30, size = 131, normalized size = 1.18
method | result | size |
default | \(-2 \ln \left (\sin \left (x \right )+\sqrt {-5+\sin ^{2}\left (x \right )}\right )+2 \sqrt {-5+\sin ^{2}\left (x \right )}+\frac {2 \sqrt {5}\, \arctan \left (\frac {\sqrt {5}}{\sqrt {-5+\sin ^{2}\left (x \right )}}\right )}{5}+\frac {2 \sqrt {-5+\sin ^{2}\left (x \right )}}{5 \sin \left (x \right )}-\frac {\sqrt {\left (-5+\sin ^{2}\left (x \right )\right ) \left (\cos ^{2}\left (x \right )\right )}\, \left (-\sqrt {5}\, \arctan \left (\frac {\left (3 \left (\sin ^{2}\left (x \right )\right )-5\right ) \sqrt {5}}{5 \sqrt {-\left (\cos ^{4}\left (x \right )\right )-4 \left (\cos ^{2}\left (x \right )\right )}}\right )-10 \arcsin \left (1+\frac {\left (\cos ^{2}\left (x \right )\right )}{2}\right )\right )}{10 \cos \left (x \right ) \sqrt {-5+\sin ^{2}\left (x \right )}}\) | \(131\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 1.50, size = 115, normalized size = 1.04 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\cos \left (2\,x\right )\,\sin \left (x\right )-2\,{\cos \left (x\right )}^3\,\left (\sin \left (x\right )-1\right )}{{\sin \left (x\right )}^2\,\sqrt {{\sin \left (x\right )}^2-5}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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