3.410 \(\int \frac{\csc (x)}{\sin ^{\frac{3}{2}}(2 x)} \, dx\)

Optimal. Leaf size=29 \[ \frac{4 \sin (x)}{3 \sqrt{\sin (2 x)}}-\frac{2 \cos (x)}{3 \sin ^{\frac{3}{2}}(2 x)} \]

[Out]

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

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Rubi [A]  time = 0.0694859, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{4 \sin (x)}{3 \sqrt{\sin (2 x)}}-\frac{2 \cos (x)}{3 \sin ^{\frac{3}{2}}(2 x)} \]

Antiderivative was successfully verified.

[In]  Int[Csc[x]/Sin[2*x]^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 3.44304, size = 29, normalized size = 1. \[ \frac{4 \sin{\left (x \right )}}{3 \sqrt{\sin{\left (2 x \right )}}} - \frac{2 \cos{\left (x \right )}}{3 \sin ^{\frac{3}{2}}{\left (2 x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/sin(x)/sin(2*x)**(3/2),x)

[Out]

4*sin(x)/(3*sqrt(sin(2*x))) - 2*cos(x)/(3*sin(2*x)**(3/2))

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Mathematica [A]  time = 0.0385378, size = 24, normalized size = 0.83 \[ \sqrt{\sin (2 x)} \left (\frac{\sec (x)}{2}-\frac{1}{6} \cot (x) \csc (x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Csc[x]/Sin[2*x]^(3/2),x]

[Out]

(-(Cot[x]*Csc[x])/6 + Sec[x]/2)*Sqrt[Sin[2*x]]

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Maple [C]  time = 0.098, size = 121, normalized size = 4.2 \[ -{\frac{1}{12}\sqrt{-{1\tan \left ({\frac{x}{2}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}-1 \right ) ^{-1}}} \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}-1 \right ) \left ( 2\,\sqrt{1+\tan \left ( x/2 \right ) }\sqrt{-2\,\tan \left ( x/2 \right ) +2}\sqrt{-\tan \left ( x/2 \right ) }{\it EllipticF} \left ( \sqrt{1+\tan \left ( x/2 \right ) },1/2\,\sqrt{2} \right ) \tan \left ( x/2 \right ) - \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{4}+1 \right ) \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{\tan \left ({\frac{x}{2}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}-1 \right ) }}}{\frac{1}{\sqrt{ \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}-\tan \left ({\frac{x}{2}} \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/sin(x)/sin(2*x)^(3/2),x)

[Out]

-1/12*(-tan(1/2*x)/(tan(1/2*x)^2-1))^(1/2)*(tan(1/2*x)^2-1)/tan(1/2*x)*(2*(1+tan
(1/2*x))^(1/2)*(-2*tan(1/2*x)+2)^(1/2)*(-tan(1/2*x))^(1/2)*EllipticF((1+tan(1/2*
x))^(1/2),1/2*2^(1/2))*tan(1/2*x)-tan(1/2*x)^4+1)/(tan(1/2*x)*(tan(1/2*x)^2-1))^
(1/2)/(tan(1/2*x)^3-tan(1/2*x))^(1/2)

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Maxima [A]  time = 2.03308, size = 1446, normalized size = 49.86 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(((((cos(2*x) + sin(2*x) - 1)*cos(4*x) - cos(2*x)^2 - (cos(2*x) - sin(2*x)
- 1)*sin(4*x) - sin(2*x)^2 + 2*cos(2*x) - 1)*cos(1/2*arctan2(sin(x), -cos(x) + 1
)) - ((cos(2*x) - sin(2*x) - 1)*cos(4*x) - cos(2*x)^2 + (cos(2*x) + sin(2*x) - 1
)*sin(4*x) - sin(2*x)^2 + 2*cos(2*x) - 1)*sin(1/2*arctan2(sin(x), -cos(x) + 1)))
*cos(1/2*arctan2(sin(x), cos(x) + 1)) + (((cos(2*x) - sin(2*x) - 1)*cos(4*x) - c
os(2*x)^2 + (cos(2*x) + sin(2*x) - 1)*sin(4*x) - sin(2*x)^2 + 2*cos(2*x) - 1)*co
s(1/2*arctan2(sin(x), -cos(x) + 1)) + ((cos(2*x) + sin(2*x) - 1)*cos(4*x) - cos(
2*x)^2 - (cos(2*x) - sin(2*x) - 1)*sin(4*x) - sin(2*x)^2 + 2*cos(2*x) - 1)*sin(1
/2*arctan2(sin(x), -cos(x) + 1)))*sin(1/2*arctan2(sin(x), cos(x) + 1)))*cos(1/2*
arctan2(sin(2*x), cos(2*x) + 1)) + ((((cos(2*x) - sin(2*x) - 1)*cos(4*x) - cos(2
*x)^2 + (cos(2*x) + sin(2*x) - 1)*sin(4*x) - sin(2*x)^2 + 2*cos(2*x) - 1)*cos(1/
2*arctan2(sin(x), -cos(x) + 1)) + ((cos(2*x) + sin(2*x) - 1)*cos(4*x) - cos(2*x)
^2 - (cos(2*x) - sin(2*x) - 1)*sin(4*x) - sin(2*x)^2 + 2*cos(2*x) - 1)*sin(1/2*a
rctan2(sin(x), -cos(x) + 1)))*cos(1/2*arctan2(sin(x), cos(x) + 1)) - (((cos(2*x)
+ sin(2*x) - 1)*cos(4*x) - cos(2*x)^2 - (cos(2*x) - sin(2*x) - 1)*sin(4*x) - si
n(2*x)^2 + 2*cos(2*x) - 1)*cos(1/2*arctan2(sin(x), -cos(x) + 1)) - ((cos(2*x) -
sin(2*x) - 1)*cos(4*x) - cos(2*x)^2 + (cos(2*x) + sin(2*x) - 1)*sin(4*x) - sin(2
*x)^2 + 2*cos(2*x) - 1)*sin(1/2*arctan2(sin(x), -cos(x) + 1)))*sin(1/2*arctan2(s
in(x), cos(x) + 1)))*sin(1/2*arctan2(sin(2*x), cos(2*x) + 1)))/(((((cos(2*x)^2 +
sin(2*x)^2 - 2*cos(2*x) + 1)*cos(1/2*arctan2(sin(x), -cos(x) + 1))^2 + (cos(2*x
)^2 + sin(2*x)^2 - 2*cos(2*x) + 1)*sin(1/2*arctan2(sin(x), -cos(x) + 1))^2)*cos(
1/2*arctan2(sin(x), cos(x) + 1))^2 + ((cos(2*x)^2 + sin(2*x)^2 - 2*cos(2*x) + 1)
*cos(1/2*arctan2(sin(x), -cos(x) + 1))^2 + (cos(2*x)^2 + sin(2*x)^2 - 2*cos(2*x)
+ 1)*sin(1/2*arctan2(sin(x), -cos(x) + 1))^2)*sin(1/2*arctan2(sin(x), cos(x) +
1))^2)*cos(1/2*arctan2(sin(2*x), cos(2*x) + 1))^2 + (((cos(2*x)^2 + sin(2*x)^2 -
2*cos(2*x) + 1)*cos(1/2*arctan2(sin(x), -cos(x) + 1))^2 + (cos(2*x)^2 + sin(2*x
)^2 - 2*cos(2*x) + 1)*sin(1/2*arctan2(sin(x), -cos(x) + 1))^2)*cos(1/2*arctan2(s
in(x), cos(x) + 1))^2 + ((cos(2*x)^2 + sin(2*x)^2 - 2*cos(2*x) + 1)*cos(1/2*arct
an2(sin(x), -cos(x) + 1))^2 + (cos(2*x)^2 + sin(2*x)^2 - 2*cos(2*x) + 1)*sin(1/2
*arctan2(sin(x), -cos(x) + 1))^2)*sin(1/2*arctan2(sin(x), cos(x) + 1))^2)*sin(1/
2*arctan2(sin(2*x), cos(2*x) + 1))^2)*(cos(2*x)^2 + sin(2*x)^2 + 2*cos(2*x) + 1)
^(1/4)*(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1)^(1/4)*(cos(x)^2 + sin(x)^2 - 2*cos(x
) + 1)^(1/4))

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Fricas [A]  time = 0.223991, size = 42, normalized size = 1.45 \[ \frac{\sqrt{2}{\left (4 \, \cos \left (x\right )^{2} - 3\right )} \sqrt{\cos \left (x\right ) \sin \left (x\right )}}{6 \,{\left (\cos \left (x\right )^{3} - \cos \left (x\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*sqrt(2)*(4*cos(x)^2 - 3)*sqrt(cos(x)*sin(x))/(cos(x)^3 - cos(x))

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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(1/sin(x)/sin(2*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sin(2*x)^(3/2)*sin(x)), x)