3.396 \(\int \frac{\cos (x) \left (-\cos ^2(x)+2 \sqrt [4]{1+2 \sin (x)}\right )}{(1+2 \sin (x))^{3/2}} \, dx\)

Optimal. Leaf size=55 \[ \frac{1}{12} (2 \sin (x)+1)^{3/2}-\frac{1}{2} \sqrt{2 \sin (x)+1}-\frac{4}{\sqrt [4]{2 \sin (x)+1}}+\frac{3}{4 \sqrt{2 \sin (x)+1}} \]

[Out]

3/(4*Sqrt[1 + 2*Sin[x]]) - 4/(1 + 2*Sin[x])^(1/4) - Sqrt[1 + 2*Sin[x]]/2 + (1 +
2*Sin[x])^(3/2)/12

_______________________________________________________________________________________

Rubi [A]  time = 0.241763, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{1}{12} (2 \sin (x)+1)^{3/2}-\frac{1}{2} \sqrt{2 \sin (x)+1}-\frac{4}{\sqrt [4]{2 \sin (x)+1}}+\frac{3}{4 \sqrt{2 \sin (x)+1}} \]

Antiderivative was successfully verified.

[In]  Int[(Cos[x]*(-Cos[x]^2 + 2*(1 + 2*Sin[x])^(1/4)))/(1 + 2*Sin[x])^(3/2),x]

[Out]

3/(4*Sqrt[1 + 2*Sin[x]]) - 4/(1 + 2*Sin[x])^(1/4) - Sqrt[1 + 2*Sin[x]]/2 + (1 +
2*Sin[x])^(3/2)/12

_______________________________________________________________________________________

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(cos(x)*(-cos(x)**2+2*(1+2*sin(x))**(1/4))/(1+2*sin(x))**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.206393, size = 50, normalized size = 0.91 \[ \frac{4 \sin ^2(x)-8 \sin (x)+5 \sqrt{2 \sin (x)+1}-48 \sqrt [4]{2 \sin (x)+1}+4}{12 \sqrt{2 \sin (x)+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Cos[x]*(-Cos[x]^2 + 2*(1 + 2*Sin[x])^(1/4)))/(1 + 2*Sin[x])^(3/2),x]

[Out]

(4 - 8*Sin[x] + 4*Sin[x]^2 - 48*(1 + 2*Sin[x])^(1/4) + 5*Sqrt[1 + 2*Sin[x]])/(12
*Sqrt[1 + 2*Sin[x]])

_______________________________________________________________________________________

Maple [F]  time = 0.631, size = 0, normalized size = 0. \[ \int{\cos \left ( x \right ) \left ( - \left ( \cos \left ( x \right ) \right ) ^{2}+2\,\sqrt [4]{1+2\,\sin \left ( x \right ) } \right ) \left ( 1+2\,\sin \left ( x \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)*(-cos(x)^2+2*(1+2*sin(x))^(1/4))/(1+2*sin(x))^(3/2),x)

[Out]

int(cos(x)*(-cos(x)^2+2*(1+2*sin(x))^(1/4))/(1+2*sin(x))^(3/2),x)

_______________________________________________________________________________________

Maxima [A]  time = 6.59508, size = 58, normalized size = 1.05 \[ \frac{1}{12} \,{\left (2 \, \sin \left (x\right ) + 1\right )}^{\frac{3}{2}} - \frac{16 \,{\left (2 \, \sin \left (x\right ) + 1\right )}^{\frac{1}{4}} - 3}{4 \, \sqrt{2 \, \sin \left (x\right ) + 1}} - \frac{1}{2} \, \sqrt{2 \, \sin \left (x\right ) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(cos(x)^2 - 2*(2*sin(x) + 1)^(1/4))*cos(x)/(2*sin(x) + 1)^(3/2),x, algorithm="maxima")

[Out]

1/12*(2*sin(x) + 1)^(3/2) - 1/4*(16*(2*sin(x) + 1)^(1/4) - 3)/sqrt(2*sin(x) + 1)
- 1/2*sqrt(2*sin(x) + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.223371, size = 54, normalized size = 0.98 \[ -\frac{{\left (\cos \left (x\right )^{2} + 2 \, \sin \left (x\right ) - 2\right )} \sqrt{2 \, \sin \left (x\right ) + 1} + 12 \,{\left (2 \, \sin \left (x\right ) + 1\right )}^{\frac{3}{4}}}{3 \,{\left (2 \, \sin \left (x\right ) + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(cos(x)^2 - 2*(2*sin(x) + 1)^(1/4))*cos(x)/(2*sin(x) + 1)^(3/2),x, algorithm="fricas")

[Out]

-1/3*((cos(x)^2 + 2*sin(x) - 2)*sqrt(2*sin(x) + 1) + 12*(2*sin(x) + 1)^(3/4))/(2
*sin(x) + 1)

_______________________________________________________________________________________

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(cos(x)*(-cos(x)**2+2*(1+2*sin(x))**(1/4))/(1+2*sin(x))**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{{\left (\cos \left (x\right )^{2} - 2 \,{\left (2 \, \sin \left (x\right ) + 1\right )}^{\frac{1}{4}}\right )} \cos \left (x\right )}{{\left (2 \, \sin \left (x\right ) + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(cos(x)^2 - 2*(2*sin(x) + 1)^(1/4))*cos(x)/(2*sin(x) + 1)^(3/2),x, algorithm="giac")

[Out]

integrate(-(cos(x)^2 - 2*(2*sin(x) + 1)^(1/4))*cos(x)/(2*sin(x) + 1)^(3/2), x)