3.426 \(\int \frac{\csc ^2(x) \left (-2 \cos ^3(x) (-1+\sin (x))+\cos (2 x) \sin (x)\right )}{\sqrt{-5+\sin ^2(x)}} \, dx\)

Optimal. Leaf size=111 \[ 2 \sqrt{\sin ^2(x)-5}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{\sin ^2(x)-5}}{\sqrt{5}}\right )}{\sqrt{5}}-2 \tanh ^{-1}\left (\frac{\sin (x)}{\sqrt{\sin ^2(x)-5}}\right )+\frac{2}{5} \sqrt{\sin ^2(x)-5} \csc (x)+2 \tan ^{-1}\left (\frac{\cos (x)}{\sqrt{\sin ^2(x)-5}}\right )-\frac{\tan ^{-1}\left (\frac{\sqrt{5} \cos (x)}{\sqrt{\sin ^2(x)-5}}\right )}{\sqrt{5}} \]

[Out]

2*ArcTan[Cos[x]/Sqrt[-5 + Sin[x]^2]] - ArcTan[(Sqrt[5]*Cos[x])/Sqrt[-5 + Sin[x]^
2]]/Sqrt[5] - (2*ArcTan[Sqrt[-5 + Sin[x]^2]/Sqrt[5]])/Sqrt[5] - 2*ArcTanh[Sin[x]
/Sqrt[-5 + Sin[x]^2]] + 2*Sqrt[-5 + Sin[x]^2] + (2*Csc[x]*Sqrt[-5 + Sin[x]^2])/5

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Rubi [A]  time = 1.00023, antiderivative size = 119, normalized size of antiderivative = 1.07, number of steps used = 18, number of rules used = 14, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.424 \[ 2 \sqrt{-\cos ^2(x)-4}-2 \tanh ^{-1}\left (\frac{\sin (x)}{\sqrt{\sin ^2(x)-5}}\right )+2 \tan ^{-1}\left (\frac{\cos (x)}{\sqrt{-\cos ^2(x)-4}}\right )-\frac{\tan ^{-1}\left (\frac{\sqrt{5} \cos (x)}{\sqrt{-\cos ^2(x)-4}}\right )}{\sqrt{5}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{-\cos ^2(x)-4}}{\sqrt{5}}\right )}{\sqrt{5}}+\frac{2}{5} \sqrt{\sin ^2(x)-5} \csc (x) \]

Antiderivative was successfully verified.

[In]  Int[(Csc[x]^2*(-2*Cos[x]^3*(-1 + Sin[x]) + Cos[2*x]*Sin[x]))/Sqrt[-5 + Sin[x]^2],x]

[Out]

2*ArcTan[Cos[x]/Sqrt[-4 - Cos[x]^2]] - ArcTan[(Sqrt[5]*Cos[x])/Sqrt[-4 - Cos[x]^
2]]/Sqrt[5] - (2*ArcTan[Sqrt[-4 - Cos[x]^2]/Sqrt[5]])/Sqrt[5] - 2*ArcTanh[Sin[x]
/Sqrt[-5 + Sin[x]^2]] + 2*Sqrt[-4 - Cos[x]^2] + (2*Csc[x]*Sqrt[-5 + Sin[x]^2])/5

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-2*cos(x)**3*(-1+sin(x))+cos(2*x)*sin(x))/sin(x)**2/(-5+sin(x)**2)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 3.77235, size = 338, normalized size = 3.05 \[ \frac{(16-32 i) \sqrt{5} \sqrt{\frac{(1+2 i) (\cos (x)-2 i)}{\cos (x)+1}} \sqrt{\frac{(1-2 i) (\cos (x)+2 i)}{\cos (x)+1}} \cos ^2\left (\frac{x}{2}\right ) F\left (\sin ^{-1}\left (\frac{(1+2 i) \tan \left (\frac{x}{2}\right )}{\sqrt{5}}\right )|-\frac{7}{25}+\frac{24 i}{25}\right )-(32-64 i) \sqrt{5} \sqrt{\frac{(1+2 i) (\cos (x)-2 i)}{\cos (x)+1}} \sqrt{\frac{(1-2 i) (\cos (x)+2 i)}{\cos (x)+1}} \cos ^2\left (\frac{x}{2}\right ) \Pi \left (\frac{3}{5}+\frac{4 i}{5};\sin ^{-1}\left (\frac{(1+2 i) \tan \left (\frac{x}{2}\right )}{\sqrt{5}}\right )|-\frac{7}{25}+\frac{24 i}{25}\right )-5 \left (18 \csc (x)+10 i \sqrt{2} \sqrt{-\cos (2 x)-9} \log \left (\sqrt{-\cos (2 x)-9}+i \sqrt{2} \cos (x)\right )+\sqrt{10} \sqrt{-\cos (2 x)-9} \tan ^{-1}\left (\frac{\sqrt{10} \cos (x)}{\sqrt{-\cos (2 x)-9}}\right )+2 \sqrt{10} \sqrt{-\cos (2 x)-9} \tan ^{-1}\left (\frac{\sqrt{-\cos (2 x)-9}}{\sqrt{10}}\right )+2 \cos (2 x) \csc (x)+5 \sin (3 x) \csc (x)+85\right )}{25 \sqrt{2} \sqrt{-\cos (2 x)-9}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Csc[x]^2*(-2*Cos[x]^3*(-1 + Sin[x]) + Cos[2*x]*Sin[x]))/Sqrt[-5 + Sin[x]^2],x]

[Out]

((16 - 32*I)*Sqrt[5]*Cos[x/2]^2*Sqrt[((1 + 2*I)*(-2*I + Cos[x]))/(1 + Cos[x])]*S
qrt[((1 - 2*I)*(2*I + Cos[x]))/(1 + Cos[x])]*EllipticF[ArcSin[((1 + 2*I)*Tan[x/2
])/Sqrt[5]], -7/25 + (24*I)/25] - (32 - 64*I)*Sqrt[5]*Cos[x/2]^2*Sqrt[((1 + 2*I)
*(-2*I + Cos[x]))/(1 + Cos[x])]*Sqrt[((1 - 2*I)*(2*I + Cos[x]))/(1 + Cos[x])]*El
lipticPi[3/5 + (4*I)/5, ArcSin[((1 + 2*I)*Tan[x/2])/Sqrt[5]], -7/25 + (24*I)/25]
- 5*(85 + Sqrt[10]*ArcTan[(Sqrt[10]*Cos[x])/Sqrt[-9 - Cos[2*x]]]*Sqrt[-9 - Cos[
2*x]] + 2*Sqrt[10]*ArcTan[Sqrt[-9 - Cos[2*x]]/Sqrt[10]]*Sqrt[-9 - Cos[2*x]] + 18
*Csc[x] + 2*Cos[2*x]*Csc[x] + (10*I)*Sqrt[2]*Sqrt[-9 - Cos[2*x]]*Log[I*Sqrt[2]*C
os[x] + Sqrt[-9 - Cos[2*x]]] + 5*Csc[x]*Sin[3*x]))/(25*Sqrt[2]*Sqrt[-9 - Cos[2*x
]])

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Maple [A]  time = 0.266, size = 130, normalized size = 1.2 \[{\frac{2}{5\,\sin \left ( x \right ) }\sqrt{-5+ \left ( \sin \left ( x \right ) \right ) ^{2}}}+2\,\sqrt{-5+ \left ( \sin \left ( x \right ) \right ) ^{2}}-2\,\ln \left ( \sin \left ( x \right ) +\sqrt{-5+ \left ( \sin \left ( x \right ) \right ) ^{2}} \right ) +{\frac{2\,\sqrt{5}}{5}\arctan \left ({\sqrt{5}{\frac{1}{\sqrt{-5+ \left ( \sin \left ( x \right ) \right ) ^{2}}}}} \right ) }+{\frac{1}{10\,\cos \left ( x \right ) }\sqrt{ \left ( -5+ \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \left ( \cos \left ( x \right ) \right ) ^{2}} \left ( \sqrt{5}\arctan \left ({\frac{\sqrt{5} \left ( 3\, \left ( \sin \left ( x \right ) \right ) ^{2}-5 \right ) }{5}{\frac{1}{\sqrt{- \left ( \cos \left ( x \right ) \right ) ^{4}-4\, \left ( \cos \left ( x \right ) \right ) ^{2}}}}} \right ) +10\,\arcsin \left ( 1+1/2\, \left ( \cos \left ( x \right ) \right ) ^{2} \right ) \right ){\frac{1}{\sqrt{-5+ \left ( \sin \left ( x \right ) \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-2*cos(x)^3*(-1+sin(x))+cos(2*x)*sin(x))/sin(x)^2/(-5+sin(x)^2)^(1/2),x)

[Out]

2/5*(-5+sin(x)^2)^(1/2)/sin(x)+2*(-5+sin(x)^2)^(1/2)-2*ln(sin(x)+(-5+sin(x)^2)^(
1/2))+2/5*5^(1/2)*arctan(5^(1/2)/(-5+sin(x)^2)^(1/2))+1/10*((-5+sin(x)^2)*cos(x)
^2)^(1/2)*(5^(1/2)*arctan(1/5*5^(1/2)*(3*sin(x)^2-5)/(-cos(x)^4-4*cos(x)^2)^(1/2
))+10*arcsin(1+1/2*cos(x)^2))/cos(x)/(-5+sin(x)^2)^(1/2)

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Maxima [A]  time = 1.55068, size = 155, normalized size = 1.4 \[ \frac{2}{5} \, \sqrt{5} \arcsin \left (\frac{\sqrt{5}}{{\left | \sin \left (x\right ) \right |}}\right ) - \frac{1}{10} i \, \sqrt{5} \operatorname{arsinh}\left (\frac{\cos \left (x\right )}{2 \,{\left (\cos \left (x\right ) + 1\right )}} - \frac{2}{\cos \left (x\right ) + 1}\right ) - \frac{1}{10} i \, \sqrt{5} \operatorname{arsinh}\left (-\frac{\cos \left (x\right )}{2 \,{\left (\cos \left (x\right ) - 1\right )}} - \frac{2}{\cos \left (x\right ) - 1}\right ) + 2 \, \sqrt{\sin \left (x\right )^{2} - 5} + \frac{2 \, \sqrt{\sin \left (x\right )^{2} - 5}}{5 \, \sin \left (x\right )} - 2 i \, \operatorname{arsinh}\left (\frac{1}{2} \, \cos \left (x\right )\right ) - 2 \, \log \left (2 \, \sqrt{\sin \left (x\right )^{2} - 5} + 2 \, \sin \left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*(sin(x) - 1)*cos(x)^3 - cos(2*x)*sin(x))/(sqrt(sin(x)^2 - 5)*sin(x)^2),x, algorithm="maxima")

[Out]

2/5*sqrt(5)*arcsin(sqrt(5)/abs(sin(x))) - 1/10*I*sqrt(5)*arcsinh(1/2*cos(x)/(cos
(x) + 1) - 2/(cos(x) + 1)) - 1/10*I*sqrt(5)*arcsinh(-1/2*cos(x)/(cos(x) - 1) - 2
/(cos(x) - 1)) + 2*sqrt(sin(x)^2 - 5) + 2/5*sqrt(sin(x)^2 - 5)/sin(x) - 2*I*arcs
inh(1/2*cos(x)) - 2*log(2*sqrt(sin(x)^2 - 5) + 2*sin(x))

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Fricas [A]  time = 1.4217, size = 1, normalized size = 0.01 \[ 0 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*(sin(x) - 1)*cos(x)^3 - cos(2*x)*sin(x))/(sqrt(sin(x)^2 - 5)*sin(x)^2),x, algorithm="fricas")

[Out]

0

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*cos(x)**3*(-1+sin(x))+cos(2*x)*sin(x))/sin(x)**2/(-5+sin(x)**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.435009, size = 356, normalized size = 3.21 \[ \pi{\rm sign}\left (-2 i \, \sqrt{\cos \left (x\right )^{2} + 4} - 4 i\right ){\rm sign}\left (\cos \left (x\right )\right ) - \frac{1}{5} \, \sqrt{5}{\left (\pi{\rm sign}\left (-2 i \, \sqrt{\cos \left (x\right )^{2} + 4} - 4 i\right ){\rm sign}\left (\cos \left (x\right )\right ) + 2 \, \arctan \left (\frac{\sqrt{5}{\left (\frac{{\left (i \, \sqrt{\cos \left (x\right )^{2} + 4} + 2 i\right )}^{2}}{\cos \left (x\right )^{2}} - 1\right )} \cos \left (x\right )}{5 \,{\left (-2 i \, \sqrt{\cos \left (x\right )^{2} + 4} - 4 i\right )}}\right )\right )} + \frac{1}{10} \, \sqrt{5}{\left (\pi{\rm sign}\left (-2 i \, \sqrt{\cos \left (x\right )^{2} + 4} - 4 i\right ){\rm sign}\left (\cos \left (x\right )\right ) + 2 \, \arctan \left (\frac{\sqrt{5}{\left (\frac{{\left (-i \, \sqrt{\cos \left (x\right )^{2} + 4} - 2 i\right )}^{2}}{\cos \left (x\right )^{2}} - 1\right )} \cos \left (x\right )}{5 \,{\left (-2 i \, \sqrt{\cos \left (x\right )^{2} + 4} - 4 i\right )}}\right )\right )} - \frac{2}{5} \, \sqrt{5} \arctan \left (\frac{1}{5} \, \sqrt{5} \sqrt{\sin \left (x\right )^{2} - 5}\right ) + 2 \, \sqrt{\sin \left (x\right )^{2} - 5} + \frac{4}{{\left (\sqrt{\sin \left (x\right )^{2} - 5} - \sin \left (x\right )\right )}^{2} + 5} + 2 \, \arctan \left (\frac{{\left (\frac{{\left (i \, \sqrt{\cos \left (x\right )^{2} + 4} + 2 i\right )}^{2}}{\cos \left (x\right )^{2}} - 1\right )} \cos \left (x\right )}{-2 i \, \sqrt{\cos \left (x\right )^{2} + 4} - 4 i}\right ) +{\rm ln}\left ({\left (\sqrt{\sin \left (x\right )^{2} - 5} - \sin \left (x\right )\right )}^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*(sin(x) - 1)*cos(x)^3 - cos(2*x)*sin(x))/(sqrt(sin(x)^2 - 5)*sin(x)^2),x, algorithm="giac")

[Out]

pi*sign(-2*I*sqrt(cos(x)^2 + 4) - 4*I)*sign(cos(x)) - 1/5*sqrt(5)*(pi*sign(-2*I*
sqrt(cos(x)^2 + 4) - 4*I)*sign(cos(x)) + 2*arctan(1/5*sqrt(5)*((I*sqrt(cos(x)^2
+ 4) + 2*I)^2/cos(x)^2 - 1)*cos(x)/(-2*I*sqrt(cos(x)^2 + 4) - 4*I))) + 1/10*sqrt
(5)*(pi*sign(-2*I*sqrt(cos(x)^2 + 4) - 4*I)*sign(cos(x)) + 2*arctan(1/5*sqrt(5)*
((-I*sqrt(cos(x)^2 + 4) - 2*I)^2/cos(x)^2 - 1)*cos(x)/(-2*I*sqrt(cos(x)^2 + 4) -
4*I))) - 2/5*sqrt(5)*arctan(1/5*sqrt(5)*sqrt(sin(x)^2 - 5)) + 2*sqrt(sin(x)^2 -
5) + 4/((sqrt(sin(x)^2 - 5) - sin(x))^2 + 5) + 2*arctan(((I*sqrt(cos(x)^2 + 4)
+ 2*I)^2/cos(x)^2 - 1)*cos(x)/(-2*I*sqrt(cos(x)^2 + 4) - 4*I)) + ln((sqrt(sin(x)
^2 - 5) - sin(x))^2)