Optimal. Leaf size=111 \[ 2 \sqrt {\sin ^2(x)-5}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {\sin ^2(x)-5}}{\sqrt {5}}\right )}{\sqrt {5}}-2 \tanh ^{-1}\left (\frac {\sin (x)}{\sqrt {\sin ^2(x)-5}}\right )+\frac {2}{5} \sqrt {\sin ^2(x)-5} \csc (x)+2 \tan ^{-1}\left (\frac {\cos (x)}{\sqrt {\sin ^2(x)-5}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {5} \cos (x)}{\sqrt {\sin ^2(x)-5}}\right )}{\sqrt {5}} \]
________________________________________________________________________________________
Rubi [A] time = 0.57, 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.
[In]
[Out]
Rule 63
Rule 203
Rule 206
Rule 217
Rule 261
Rule 377
Rule 444
Rule 451
Rule 1010
Rule 4356
Rule 4366
Rule 4401
Rule 6725
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 = 2.32, size = 338, normalized size = 3.05 \[ \frac {(16-32 i) \sqrt {5} \sqrt {\frac {(1+2 i) (\cos (x)-2 i)}{\cos (x)+1}} \sqrt {\frac {(1-2 i) (\cos (x)+2 i)}{\cos (x)+1}} \cos ^2\left (\frac {x}{2}\right ) \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {(1+2 i) \tan \left (\frac {x}{2}\right )}{\sqrt {5}}\right ),-\frac {7}{25}+\frac {24 i}{25}\right )-(32-64 i) \sqrt {5} \sqrt {\frac {(1+2 i) (\cos (x)-2 i)}{\cos (x)+1}} \sqrt {\frac {(1-2 i) (\cos (x)+2 i)}{\cos (x)+1}} \cos ^2\left (\frac {x}{2}\right ) \Pi \left (\frac {3}{5}+\frac {4 i}{5};\sin ^{-1}\left (\frac {(1+2 i) \tan \left (\frac {x}{2}\right )}{\sqrt {5}}\right )|-\frac {7}{25}+\frac {24 i}{25}\right )-5 \left (18 \csc (x)+10 i \sqrt {2} \sqrt {-\cos (2 x)-9} \log \left (\sqrt {-\cos (2 x)-9}+i \sqrt {2} \cos (x)\right )+\sqrt {10} \sqrt {-\cos (2 x)-9} \tan ^{-1}\left (\frac {\sqrt {10} \cos (x)}{\sqrt {-\cos (2 x)-9}}\right )+2 \sqrt {10} \sqrt {-\cos (2 x)-9} \tan ^{-1}\left (\frac {\sqrt {-\cos (2 x)-9}}{\sqrt {10}}\right )+2 \cos (2 x) \csc (x)+5 \sin (3 x) \csc (x)+85\right )}{25 \sqrt {2} \sqrt {-\cos (2 x)-9}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^2(x) \left (-2 \cos ^3(x) (-1+\sin (x))+\cos (2 x) \sin (x)\right )}{\sqrt {-5+\sin ^2(x)}} \, dx \]
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 \[ \int -\frac {2 \, {\left (\sin \relax (x) - 1\right )} \cos \relax (x)^{3} - \cos \left (2 \, x\right ) \sin \relax (x)}{\sqrt {\sin \relax (x)^{2} - 5} \sin \relax (x)^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.44, size = 131, normalized size = 1.18
method | result | size |
default | \(-2 \ln \left (\sin \relax (x )+\sqrt {-5+\sin ^{2}\relax (x )}\right )+2 \sqrt {-5+\sin ^{2}\relax (x )}+\frac {2 \sqrt {5}\, \arctan \left (\frac {\sqrt {5}}{\sqrt {-5+\sin ^{2}\relax (x )}}\right )}{5}+\frac {2 \sqrt {-5+\sin ^{2}\relax (x )}}{5 \sin \relax (x )}-\frac {\sqrt {\left (-5+\sin ^{2}\relax (x )\right ) \left (\cos ^{2}\relax (x )\right )}\, \left (-\sqrt {5}\, \arctan \left (\frac {\left (3 \left (\sin ^{2}\relax (x )\right )-5\right ) \sqrt {5}}{5 \sqrt {-\left (\cos ^{4}\relax (x )\right )-4 \left (\cos ^{2}\relax (x )\right )}}\right )-10 \arcsin \left (1+\frac {\left (\cos ^{2}\relax (x )\right )}{2}\right )\right )}{10 \cos \relax (x ) \sqrt {-5+\sin ^{2}\relax (x )}}\) | \(131\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [C] time = 1.02, size = 115, normalized size = 1.04 \[ \frac {2}{5} \, \sqrt {5} \arcsin \left (\frac {\sqrt {5}}{{\left | \sin \relax (x) \right |}}\right ) - \frac {1}{10} i \, \sqrt {5} \operatorname {arsinh}\left (\frac {\cos \relax (x)}{2 \, {\left (\cos \relax (x) + 1\right )}} - \frac {2}{\cos \relax (x) + 1}\right ) - \frac {1}{10} i \, \sqrt {5} \operatorname {arsinh}\left (-\frac {\cos \relax (x)}{2 \, {\left (\cos \relax (x) - 1\right )}} - \frac {2}{\cos \relax (x) - 1}\right ) + 2 \, \sqrt {\sin \relax (x)^{2} - 5} + \frac {2 \, \sqrt {\sin \relax (x)^{2} - 5}}{5 \, \sin \relax (x)} - 2 i \, \operatorname {arsinh}\left (\frac {1}{2} \, \cos \relax (x)\right ) - 2 \, \log \left (2 \, \sqrt {\sin \relax (x)^{2} - 5} + 2 \, \sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\cos \left (2\,x\right )\,\sin \relax (x)-2\,{\cos \relax (x)}^3\,\left (\sin \relax (x)-1\right )}{{\sin \relax (x)}^2\,\sqrt {{\sin \relax (x)}^2-5}} \,d x \]
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]
________________________________________________________________________________________