3.36 \(\int \csc ^2(x) \sec ^2(x) \, dx\)

Optimal. Leaf size=7 \[ \tan (x)-\cot (x) \]

[Out]

-Cot[x] + Tan[x]

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Rubi [A]  time = 0.0449599, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \tan (x)-\cot (x) \]

Antiderivative was successfully verified.

[In]  Int[Csc[x]^2*Sec[x]^2,x]

[Out]

-Cot[x] + Tan[x]

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Rubi in Sympy [A]  time = 1.67775, size = 15, normalized size = 2.14 \[ \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )}} - \frac{1}{\sin{\left (x \right )} \cos{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sin(x)/cos(x) - 1/(sin(x)*cos(x))

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Mathematica [A]  time = 0.00525028, size = 6, normalized size = 0.86 \[ -2 \cot (2 x) \]

Antiderivative was successfully verified.

[In]  Integrate[Csc[x]^2*Sec[x]^2,x]

[Out]

-2*Cot[2*x]

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Maple [A]  time = 0., size = 15, normalized size = 2.1 \[{\frac{1}{\cos \left ( x \right ) \sin \left ( x \right ) }}-2\,\cot \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/cos(x)^2/sin(x)^2,x)

[Out]

1/sin(x)/cos(x)-2*cot(x)

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Maxima [A]  time = 1.36047, size = 12, normalized size = 1.71 \[ -\frac{1}{\tan \left (x\right )} + \tan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/tan(x) + tan(x)

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Fricas [A]  time = 0.225188, size = 24, normalized size = 3.43 \[ -\frac{2 \, \cos \left (x\right )^{2} - 1}{\cos \left (x\right ) \sin \left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*cos(x)^2 - 1)/(cos(x)*sin(x))

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Sympy [A]  time = 0.055305, size = 12, normalized size = 1.71 \[ - \frac{2 \cos{\left (2 x \right )}}{\sin{\left (2 x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/cos(x)**2/sin(x)**2,x)

[Out]

-2*cos(2*x)/sin(2*x)

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GIAC/XCAS [A]  time = 0.201517, size = 12, normalized size = 1.71 \[ -\frac{1}{\tan \left (x\right )} + \tan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/tan(x) + tan(x)