3.31 \(\int -\tan (a-x) \tan (x) \, dx\)

Optimal. Leaf size=21 \[ \cot (a) \log (\cos (a-x))-\cot (a) \log (\cos (x))-x \]

[Out]

-x + Cot[a]*Log[Cos[a - x]] - Cot[a]*Log[Cos[x]]

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Rubi [A]  time = 0.0604058, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \cot (a) \log (\cos (a-x))-\cot (a) \log (\cos (x))-x \]

Antiderivative was successfully verified.

[In]  Int[-(Tan[a - x]*Tan[x]),x]

[Out]

-x + Cot[a]*Log[Cos[a - x]] - Cot[a]*Log[Cos[x]]

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Rubi in Sympy [A]  time = 5.88019, size = 19, normalized size = 0.9 \[ - x - \frac{\log{\left (\cos{\left (x \right )} \right )}}{\tan{\left (a \right )}} + \frac{\log{\left (\cos{\left (a - x \right )} \right )}}{\tan{\left (a \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(-tan(x)*tan(a-x),x)

[Out]

-x - log(cos(x))/tan(a) + log(cos(a - x))/tan(a)

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Mathematica [A]  time = 0.114657, size = 21, normalized size = 1. \[ \cot (a) \log (\cos (a-x))-\cot (a) \log (\cos (x))-x \]

Antiderivative was successfully verified.

[In]  Integrate[-(Tan[a - x]*Tan[x]),x]

[Out]

-x + Cot[a]*Log[Cos[a - x]] - Cot[a]*Log[Cos[x]]

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Maple [B]  time = 0.132, size = 96, normalized size = 4.6 \[ -{\frac{ \left ( \cos \left ( a \right ) \right ) ^{2}\arctan \left ( \tan \left ( x \right ) \right ) }{ \left ( \cos \left ( a \right ) \right ) ^{2}+ \left ( \sin \left ( a \right ) \right ) ^{2}}}+{\frac{ \left ( \cos \left ( a \right ) \right ) ^{3}\ln \left ( \sin \left ( a \right ) \tan \left ( x \right ) +\cos \left ( a \right ) \right ) }{ \left ( \left ( \cos \left ( a \right ) \right ) ^{2}+ \left ( \sin \left ( a \right ) \right ) ^{2} \right ) \sin \left ( a \right ) }}-{\frac{ \left ( \sin \left ( a \right ) \right ) ^{2}\arctan \left ( \tan \left ( x \right ) \right ) }{ \left ( \cos \left ( a \right ) \right ) ^{2}+ \left ( \sin \left ( a \right ) \right ) ^{2}}}+{\frac{\cos \left ( a \right ) \sin \left ( a \right ) \ln \left ( \sin \left ( a \right ) \tan \left ( x \right ) +\cos \left ( a \right ) \right ) }{ \left ( \cos \left ( a \right ) \right ) ^{2}+ \left ( \sin \left ( a \right ) \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(-tan(x)*tan(a-x),x)

[Out]

-1/(cos(a)^2+sin(a)^2)*cos(a)^2*arctan(tan(x))+cos(a)^3/(cos(a)^2+sin(a)^2)/sin(
a)*ln(sin(a)*tan(x)+cos(a))-1/(cos(a)^2+sin(a)^2)*sin(a)^2*arctan(tan(x))+sin(a)
*cos(a)/(cos(a)^2+sin(a)^2)*ln(sin(a)*tan(x)+cos(a))

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Maxima [A]  time = 1.50116, size = 251, normalized size = 11.95 \[ -\frac{{\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2} - 2 \, \cos \left (2 \, a\right ) + 1\right )} x +{\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2} - 1\right )} \arctan \left (\sin \left (2 \, a\right ) + \sin \left (2 \, x\right ), \cos \left (2 \, a\right ) + \cos \left (2 \, x\right )\right ) -{\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2} - 1\right )} \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right ) + 1\right ) - \log \left (\cos \left (2 \, a\right )^{2} + 2 \, \cos \left (2 \, a\right ) \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + \sin \left (2 \, a\right )^{2} + 2 \, \sin \left (2 \, a\right ) \sin \left (2 \, x\right ) + \sin \left (2 \, x\right )^{2}\right ) \sin \left (2 \, a\right ) + \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right ) \sin \left (2 \, a\right )}{\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2} - 2 \, \cos \left (2 \, a\right ) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-tan(a - x)*tan(x),x, algorithm="maxima")

[Out]

-((cos(2*a)^2 + sin(2*a)^2 - 2*cos(2*a) + 1)*x + (cos(2*a)^2 + sin(2*a)^2 - 1)*a
rctan2(sin(2*a) + sin(2*x), cos(2*a) + cos(2*x)) - (cos(2*a)^2 + sin(2*a)^2 - 1)
*arctan2(sin(2*x), cos(2*x) + 1) - log(cos(2*a)^2 + 2*cos(2*a)*cos(2*x) + cos(2*
x)^2 + sin(2*a)^2 + 2*sin(2*a)*sin(2*x) + sin(2*x)^2)*sin(2*a) + log(cos(2*x)^2
+ sin(2*x)^2 + 2*cos(2*x) + 1)*sin(2*a))/(cos(2*a)^2 + sin(2*a)^2 - 2*cos(2*a) +
1)

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Fricas [A]  time = 0.238456, size = 120, normalized size = 5.71 \[ \frac{{\left (\cos \left (2 \, a\right ) + 1\right )} \log \left (-\frac{{\left (\cos \left (2 \, a\right ) - 1\right )} \tan \left (x\right )^{2} - 2 \, \sin \left (2 \, a\right ) \tan \left (x\right ) - \cos \left (2 \, a\right ) - 1}{{\left (\cos \left (2 \, a\right ) + 1\right )} \tan \left (x\right )^{2} + \cos \left (2 \, a\right ) + 1}\right ) -{\left (\cos \left (2 \, a\right ) + 1\right )} \log \left (\frac{1}{\tan \left (x\right )^{2} + 1}\right ) - 2 \, x \sin \left (2 \, a\right )}{2 \, \sin \left (2 \, a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-tan(a - x)*tan(x),x, algorithm="fricas")

[Out]

1/2*((cos(2*a) + 1)*log(-((cos(2*a) - 1)*tan(x)^2 - 2*sin(2*a)*tan(x) - cos(2*a)
- 1)/((cos(2*a) + 1)*tan(x)^2 + cos(2*a) + 1)) - (cos(2*a) + 1)*log(1/(tan(x)^2
+ 1)) - 2*x*sin(2*a))/sin(2*a)

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Sympy [A]  time = 1.85757, size = 138, normalized size = 6.57 \[ - \left (\begin{cases} \frac{2 x \tan{\left (a \right )}}{2 \tan ^{2}{\left (a \right )} + 2} - \frac{2 \log{\left (\tan{\left (x \right )} + \frac{1}{\tan{\left (a \right )}} \right )}}{2 \tan ^{2}{\left (a \right )} + 2} + \frac{\log{\left (\tan ^{2}{\left (x \right )} + 1 \right )}}{2 \tan ^{2}{\left (a \right )} + 2} & \text{for}\: a \neq 0 \\\frac{\log{\left (\tan ^{2}{\left (x \right )} + 1 \right )}}{2} & \text{otherwise} \end{cases}\right ) \tan{\left (a \right )} + \begin{cases} - \frac{2 x \tan{\left (a \right )}}{2 \tan ^{3}{\left (a \right )} + 2 \tan{\left (a \right )}} + \frac{2 \log{\left (\tan{\left (x \right )} + \frac{1}{\tan{\left (a \right )}} \right )}}{2 \tan ^{3}{\left (a \right )} + 2 \tan{\left (a \right )}} + \frac{\log{\left (\tan ^{2}{\left (x \right )} + 1 \right )} \tan ^{2}{\left (a \right )}}{2 \tan ^{3}{\left (a \right )} + 2 \tan{\left (a \right )}} & \text{for}\: a \neq 0 \\- x + \tan{\left (x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-tan(x)*tan(a-x),x)

[Out]

-Piecewise((2*x*tan(a)/(2*tan(a)**2 + 2) - 2*log(tan(x) + 1/tan(a))/(2*tan(a)**2
+ 2) + log(tan(x)**2 + 1)/(2*tan(a)**2 + 2), Ne(a, 0)), (log(tan(x)**2 + 1)/2,
True))*tan(a) + Piecewise((-2*x*tan(a)/(2*tan(a)**3 + 2*tan(a)) + 2*log(tan(x) +
1/tan(a))/(2*tan(a)**3 + 2*tan(a)) + log(tan(x)**2 + 1)*tan(a)**2/(2*tan(a)**3
+ 2*tan(a)), Ne(a, 0)), (-x + tan(x), True))

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GIAC/XCAS [A]  time = 0.217077, size = 24, normalized size = 1.14 \[ -x + \frac{{\rm ln}\left ({\left | \tan \left (a\right ) \tan \left (x\right ) + 1 \right |}\right )}{\tan \left (a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-tan(a - x)*tan(x),x, algorithm="giac")

[Out]

-x + ln(abs(tan(a)*tan(x) + 1))/tan(a)