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

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

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Rubi [A]  time = 0.03, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 3475

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

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 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
]

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*}

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Mathematica [A]  time = 0.08, size = 21, normalized size = 1.00 \[ \cot (a) \log (\cos (a-x))-\cot (a) \log (\cos (x))-x \]

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\tan (a-x) \tan (x) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[-(Tan[a - x]*Tan[x]),x]

[Out]

Could not integrate

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fricas [B]  time = 1.12, size = 89, normalized size = 4.24 \[ \frac {{\left (\cos \left (2 \, a\right ) + 1\right )} \log \left (-\frac {{\left (\cos \left (2 \, a\right ) - 1\right )} \tan \relax (x)^{2} - 2 \, \sin \left (2 \, a\right ) \tan \relax (x) - \cos \left (2 \, a\right ) - 1}{{\left (\cos \left (2 \, a\right ) + 1\right )} \tan \relax (x)^{2} + \cos \left (2 \, a\right ) + 1}\right ) - {\left (\cos \left (2 \, a\right ) + 1\right )} \log \left (\frac {1}{\tan \relax (x)^{2} + 1}\right ) - 2 \, x \sin \left (2 \, a\right )}{2 \, \sin \left (2 \, a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-tan(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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giac [A]  time = 0.95, size = 18, normalized size = 0.86 \[ -x + \frac {\log \left ({\left | \tan \relax (a) \tan \relax (x) + 1 \right |}\right )}{\tan \relax (a)} \]

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)

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maple [A]  time = 0.12, size = 20, normalized size = 0.95

method	result	size
derivativedivides	\(-\arctan \left (\tan \relax (x )\right )+\frac {\ln \left (1+\tan \relax (x ) \tan \relax (a )\right )}{\tan \relax (a )}\)	\(20\)
default	\(-\arctan \left (\tan \relax (x )\right )+\frac {\ln \left (1+\tan \relax (x ) \tan \relax (a )\right )}{\tan \relax (a )}\)	\(20\)
risch	\(-x +\frac {i \ln \left ({\mathrm e}^{2 i a}+{\mathrm e}^{2 i x}\right ) {\mathrm e}^{2 i a}}{{\mathrm e}^{2 i a}-1}+\frac {i \ln \left ({\mathrm e}^{2 i a}+{\mathrm e}^{2 i x}\right )}{{\mathrm e}^{2 i a}-1}-\frac {i \ln \left (1+{\mathrm e}^{2 i x}\right ) {\mathrm e}^{2 i a}}{{\mathrm e}^{2 i a}-1}-\frac {i \ln \left (1+{\mathrm e}^{2 i x}\right )}{{\mathrm e}^{2 i a}-1}\)	\(103\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-tan(x)*tan(a-x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 1.03, size = 186, normalized size = 8.86 \[ -\frac {{\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2} - 2 \, \cos \left (2 \, a\right ) + 1\right )} x + {\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2} - 1\right )} \arctan \left (\sin \left (2 \, a\right ) + \sin \left (2 \, x\right ), \cos \left (2 \, a\right ) + \cos \left (2 \, x\right )\right ) - {\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2} - 1\right )} \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right ) + 1\right ) - \log \left (\cos \left (2 \, a\right )^{2} + 2 \, \cos \left (2 \, a\right ) \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + \sin \left (2 \, a\right )^{2} + 2 \, \sin \left (2 \, a\right ) \sin \left (2 \, x\right ) + \sin \left (2 \, x\right )^{2}\right ) \sin \left (2 \, a\right ) + \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right ) \sin \left (2 \, a\right )}{\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2} - 2 \, \cos \left (2 \, a\right ) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-tan(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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mupad [B]  time = 1.33, size = 118, normalized size = 5.62 \[ -x-\frac {\frac {\sin \left (2\,a\right )\,\ln \left ({\sin \left (2\,a+x\right )}^2\,2{}\mathrm {i}+{\sin \left (2\,a\right )}^2\,2{}\mathrm {i}-{\sin \relax (x)}^2\,2{}\mathrm {i}+\sin \left (4\,a\right )-\sin \left (2\,x\right )+\sin \left (4\,a+2\,x\right )\right )}{2}-\frac {\sin \left (2\,a\right )\,\ln \left (\sin \left (2\,a\right )\,\left (2\,{\sin \relax (a)}^2-1\right )-{\sin \left (2\,a\right )}^2\,1{}\mathrm {i}+\sin \left (2\,a\right )\,\left (2\,{\sin \relax (x)}^2-1\right )-\sin \left (2\,a\right )\,\sin \left (2\,x\right )\,1{}\mathrm {i}\right )}{2}}{{\sin \relax (a)}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-tan(a - x)*tan(x),x)

[Out]

- x - ((sin(2*a)*log(sin(4*a) - sin(2*x) + sin(4*a + 2*x) - sin(x)^2*2i + sin(2*a + x)^2*2i + sin(2*a)^2*2i))/
2 - (sin(2*a)*log(sin(2*a)*(2*sin(a)^2 - 1) - sin(2*a)^2*1i + sin(2*a)*(2*sin(x)^2 - 1) - sin(2*a)*sin(2*x)*1i
))/2)/sin(a)^2

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sympy [B]  time = 1.54, size = 138, normalized size = 6.57 \[ - \left (\begin {cases} \frac {2 x \tan {\relax (a )}}{2 \tan ^{2}{\relax (a )} + 2} - \frac {2 \log {\left (\tan {\relax (x )} + \frac {1}{\tan {\relax (a )}} \right )}}{2 \tan ^{2}{\relax (a )} + 2} + \frac {\log {\left (\tan ^{2}{\relax (x )} + 1 \right )}}{2 \tan ^{2}{\relax (a )} + 2} & \text {for}\: a \neq 0 \\\frac {\log {\left (\tan ^{2}{\relax (x )} + 1 \right )}}{2} & \text {otherwise} \end {cases}\right ) \tan {\relax (a )} + \begin {cases} - \frac {2 x \tan {\relax (a )}}{2 \tan ^{3}{\relax (a )} + 2 \tan {\relax (a )}} + \frac {2 \log {\left (\tan {\relax (x )} + \frac {1}{\tan {\relax (a )}} \right )}}{2 \tan ^{3}{\relax (a )} + 2 \tan {\relax (a )}} + \frac {\log {\left (\tan ^{2}{\relax (x )} + 1 \right )} \tan ^{2}{\relax (a )}}{2 \tan ^{3}{\relax (a )} + 2 \tan {\relax (a )}} & \text {for}\: a \neq 0 \\- x + \tan {\relax (x )} & \text {otherwise} \end {cases} \]

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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