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 = 0.033836, 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, Rules used = {4612, 4610, 3475} \[ \cot (a) \log (\cos (a-x))-\cot (a) \log (\cos (x))-x \]

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]]

Rule 4612

Int[Tan[(a_.) + (b_.)*(x_)]*Tan[(c_) + (d_.)*(x_)], x_Symbol] :> -Simp[(b*x)/d, x] + Dist[(b*Cos[(b*c - a*d)/d
])/d, Int[Sec[a + b*x]*Sec[c + d*x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0
]

Rule 4610

Int[Sec[(a_.) + (b_.)*(x_)]*Sec[(c_) + (d_.)*(x_)], x_Symbol] :> -Dist[Csc[(b*c - a*d)/d], Int[Tan[a + b*x], x
], x] + Dist[Csc[(b*c - a*d)/b], Int[Tan[c + d*x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && Ne
Q[b*c - a*d, 0]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int -\tan (a-x) \tan (x) \, dx &=-x+\cos (a) \int \sec (a-x) \sec (x) \, dx\\ &=-x+\cot (a) \int \tan (a-x) \, dx+\cot (a) \int \tan (x) \, dx\\ &=-x+\cot (a) \log (\cos (a-x))-\cot (a) \log (\cos (x))\\ \end{align*}

Mathematica [A]  time = 0.0772765, 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 [A]  time = 0.036, size = 18, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( 1+\tan \left ( x \right ) \tan \left ( a \right ) \right ) }{\tan \left ( a \right ) }}-x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-tan(x)*tan(a-x),x)

[Out]

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

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Maxima [B]  time = 1.47707, size = 251, normalized size = 11.95 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-tan(x)*tan(a-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)*arctan2(sin(2*a) + sin(2*x), co
s(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 [B]  time = 2.09668, size = 261, normalized size = 12.43 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-tan(x)*tan(a-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 [B]  time = 1.31194, size = 138, normalized size = 6.57 \begin{align*} - \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} \end{align*}

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 [A]  time = 1.0918, size = 24, normalized size = 1.14 \begin{align*} -x + \frac{\log \left ({\left | \tan \left (a\right ) \tan \left (x\right ) + 1 \right |}\right )}{\tan \left (a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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