3.1.33 \(\int \tan ^{-1}(2 \tan (x)) \, dx\) [33]

Optimal. Leaf size=80 \[ x \tan ^{-1}(2 \tan (x))+\frac {1}{2} i x \log \left (1-3 e^{2 i x}\right )-\frac {1}{2} i x \log \left (1-\frac {1}{3} e^{2 i x}\right )-\frac {1}{4} \text {Li}_2\left (\frac {1}{3} e^{2 i x}\right )+\frac {1}{4} \text {Li}_2\left (3 e^{2 i x}\right ) \]

[Out]

x*arctan(2*tan(x))+1/2*I*x*ln(1-3*exp(2*I*x))-1/2*I*x*ln(1-1/3*exp(2*I*x))-1/4*polylog(2,1/3*exp(2*I*x))+1/4*p
olylog(2,3*exp(2*I*x))

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Rubi [A]
time = 0.05, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5275, 2221, 2317, 2438} \begin {gather*} -\frac {1}{4} \text {Li}_2\left (\frac {1}{3} e^{2 i x}\right )+\frac {1}{4} \text {Li}_2\left (3 e^{2 i x}\right )+\frac {1}{2} i x \log \left (1-3 e^{2 i x}\right )-\frac {1}{2} i x \log \left (1-\frac {1}{3} e^{2 i x}\right )+x \tan ^{-1}(2 \tan (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTan[2*Tan[x]],x]

[Out]

x*ArcTan[2*Tan[x]] + (I/2)*x*Log[1 - 3*E^((2*I)*x)] - (I/2)*x*Log[1 - E^((2*I)*x)/3] - PolyLog[2, E^((2*I)*x)/
3]/4 + PolyLog[2, 3*E^((2*I)*x)]/4

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 5275

Int[ArcTan[(c_.) + (d_.)*Tan[(a_.) + (b_.)*(x_)]], x_Symbol] :> Simp[x*ArcTan[c + d*Tan[a + b*x]], x] + (Dist[
b*(1 - I*c - d), Int[x*(E^(2*I*a + 2*I*b*x)/(1 - I*c + d + (1 - I*c - d)*E^(2*I*a + 2*I*b*x))), x], x] - Dist[
b*(1 + I*c + d), Int[x*(E^(2*I*a + 2*I*b*x)/(1 + I*c - d + (1 + I*c + d)*E^(2*I*a + 2*I*b*x))), x], x]) /; Fre
eQ[{a, b, c, d}, x] && NeQ[(c + I*d)^2, -1]

Rubi steps

\begin {align*} \int \tan ^{-1}(2 \tan (x)) \, dx &=x \tan ^{-1}(2 \tan (x))-3 \int \frac {e^{2 i x} x}{-1+3 e^{2 i x}} \, dx-\int \frac {e^{2 i x} x}{3-e^{2 i x}} \, dx\\ &=x \tan ^{-1}(2 \tan (x))+\frac {1}{2} i x \log \left (1-3 e^{2 i x}\right )-\frac {1}{2} i x \log \left (1-\frac {1}{3} e^{2 i x}\right )-\frac {1}{2} i \int \log \left (1-3 e^{2 i x}\right ) \, dx+\frac {1}{2} i \int \log \left (1-\frac {1}{3} e^{2 i x}\right ) \, dx\\ &=x \tan ^{-1}(2 \tan (x))+\frac {1}{2} i x \log \left (1-3 e^{2 i x}\right )-\frac {1}{2} i x \log \left (1-\frac {1}{3} e^{2 i x}\right )-\frac {1}{4} \text {Subst}\left (\int \frac {\log (1-3 x)}{x} \, dx,x,e^{2 i x}\right )+\frac {1}{4} \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{3}\right )}{x} \, dx,x,e^{2 i x}\right )\\ &=x \tan ^{-1}(2 \tan (x))+\frac {1}{2} i x \log \left (1-3 e^{2 i x}\right )-\frac {1}{2} i x \log \left (1-\frac {1}{3} e^{2 i x}\right )-\frac {1}{4} \text {Li}_2\left (\frac {1}{3} e^{2 i x}\right )+\frac {1}{4} \text {Li}_2\left (3 e^{2 i x}\right )\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(262\) vs. \(2(80)=160\).
time = 0.16, size = 262, normalized size = 3.28 \begin {gather*} x \tan ^{-1}(2 \tan (x))-\frac {1}{4} i \left (4 i x \tan ^{-1}\left (\frac {\cot (x)}{2}\right )+2 i \cos ^{-1}\left (\frac {5}{3}\right ) \tan ^{-1}(2 \tan (x))+\left (\cos ^{-1}\left (\frac {5}{3}\right )+2 \tan ^{-1}\left (\frac {\cot (x)}{2}\right )+2 \tan ^{-1}(2 \tan (x))\right ) \log \left (\frac {2 i \sqrt {\frac {2}{3}} e^{-i x}}{\sqrt {-5+3 \cos (2 x)}}\right )+\left (\cos ^{-1}\left (\frac {5}{3}\right )-2 \tan ^{-1}\left (\frac {\cot (x)}{2}\right )-2 \tan ^{-1}(2 \tan (x))\right ) \log \left (\frac {2 i \sqrt {\frac {2}{3}} e^{i x}}{\sqrt {-5+3 \cos (2 x)}}\right )-\left (\cos ^{-1}\left (\frac {5}{3}\right )-2 \tan ^{-1}(2 \tan (x))\right ) \log \left (\frac {4 i-4 \tan (x)}{i+2 \tan (x)}\right )-\left (\cos ^{-1}\left (\frac {5}{3}\right )+2 \tan ^{-1}(2 \tan (x))\right ) \log \left (\frac {4 (i+\tan (x))}{3 i+6 \tan (x)}\right )+i \left (-\text {Li}_2\left (\frac {-3 i+6 \tan (x)}{i+2 \tan (x)}\right )+\text {Li}_2\left (\frac {-i+2 \tan (x)}{3 i+6 \tan (x)}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcTan[2*Tan[x]],x]

[Out]

x*ArcTan[2*Tan[x]] - (I/4)*((4*I)*x*ArcTan[Cot[x]/2] + (2*I)*ArcCos[5/3]*ArcTan[2*Tan[x]] + (ArcCos[5/3] + 2*A
rcTan[Cot[x]/2] + 2*ArcTan[2*Tan[x]])*Log[((2*I)*Sqrt[2/3])/(E^(I*x)*Sqrt[-5 + 3*Cos[2*x]])] + (ArcCos[5/3] -
2*ArcTan[Cot[x]/2] - 2*ArcTan[2*Tan[x]])*Log[((2*I)*Sqrt[2/3]*E^(I*x))/Sqrt[-5 + 3*Cos[2*x]]] - (ArcCos[5/3] -
2*ArcTan[2*Tan[x]])*Log[(4*I - 4*Tan[x])/(I + 2*Tan[x])] - (ArcCos[5/3] + 2*ArcTan[2*Tan[x]])*Log[(4*(I + Tan
[x]))/(3*I + 6*Tan[x])] + I*(-PolyLog[2, (-3*I + 6*Tan[x])/(I + 2*Tan[x])] + PolyLog[2, (-I + 2*Tan[x])/(3*I +
6*Tan[x])]))

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[ArcTan[2*Tan[x]],x]')

[Out]

cought exception: maximum recursion depth exceeded

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Maple [A]
time = 0.25, size = 113, normalized size = 1.41

method	result	size
derivativedivides	\(\arctan \left (2 \tan \left (x \right )\right ) \arctan \left (\tan \left (x \right )\right )+\frac {i \arctan \left (\tan \left (x \right )\right ) \ln \left (1-\frac {3 \left (1+i \tan \left (x \right )\right )^{2}}{1+\tan ^{2}\left (x \right )}\right )}{2}+\frac {\polylog \left (2, \frac {3 \left (1+i \tan \left (x \right )\right )^{2}}{1+\tan ^{2}\left (x \right )}\right )}{4}-\frac {i \arctan \left (\tan \left (x \right )\right ) \ln \left (1-\frac {\left (1+i \tan \left (x \right )\right )^{2}}{3 \left (1+\tan ^{2}\left (x \right )\right )}\right )}{2}-\frac {\polylog \left (2, \frac {\left (1+i \tan \left (x \right )\right )^{2}}{3+3 \left (\tan ^{2}\left (x \right )\right )}\right )}{4}\)	\(113\)
default	\(\arctan \left (2 \tan \left (x \right )\right ) \arctan \left (\tan \left (x \right )\right )+\frac {i \arctan \left (\tan \left (x \right )\right ) \ln \left (1-\frac {3 \left (1+i \tan \left (x \right )\right )^{2}}{1+\tan ^{2}\left (x \right )}\right )}{2}+\frac {\polylog \left (2, \frac {3 \left (1+i \tan \left (x \right )\right )^{2}}{1+\tan ^{2}\left (x \right )}\right )}{4}-\frac {i \arctan \left (\tan \left (x \right )\right ) \ln \left (1-\frac {\left (1+i \tan \left (x \right )\right )^{2}}{3 \left (1+\tan ^{2}\left (x \right )\right )}\right )}{2}-\frac {\polylog \left (2, \frac {\left (1+i \tan \left (x \right )\right )^{2}}{3+3 \left (\tan ^{2}\left (x \right )\right )}\right )}{4}\)	\(113\)
risch	\(\frac {x \pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 i x}+1}\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 i x}-\frac {1}{3}\right )}{{\mathrm e}^{2 i x}+1}\right )^{2}}{4}+\frac {x \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-\frac {1}{3}\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 i x}-\frac {1}{3}\right )}{{\mathrm e}^{2 i x}+1}\right )^{2}}{4}-\frac {x \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-3\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 i x}-3\right )}{{\mathrm e}^{2 i x}+1}\right )^{2}}{4}-\frac {x \pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 i x}+1}\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 i x}-3\right )}{{\mathrm e}^{2 i x}+1}\right )^{2}}{4}+\frac {\dilog \left (1+\sqrt {3}\, {\mathrm e}^{i x}\right )}{2}+\frac {\dilog \left (1-\sqrt {3}\, {\mathrm e}^{i x}\right )}{2}-\frac {\dilog \left (1-\frac {\sqrt {3}\, {\mathrm e}^{i x}}{3}\right )}{2}-\frac {\dilog \left (1+\frac {\sqrt {3}\, {\mathrm e}^{i x}}{3}\right )}{2}-\frac {\pi x}{2}-\frac {x \pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 i x}+1}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-\frac {1}{3}\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 i x}-\frac {1}{3}\right )}{{\mathrm e}^{2 i x}+1}\right )}{4}+\frac {x \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-3\right )\right ) \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 i x}+1}\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 i x}-3\right )}{{\mathrm e}^{2 i x}+1}\right )}{4}+\frac {x \pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 i x}-3\right )}{{\mathrm e}^{2 i x}+1}\right )^{2}}{2}-\frac {\ln \left ({\mathrm e}^{i x}\right ) \ln \left (1+\frac {\sqrt {3}\, {\mathrm e}^{i x}}{3}\right )}{2}+\frac {\ln \left ({\mathrm e}^{i x}\right ) \ln \left ({\mathrm e}^{2 i x}-3\right )}{2}-\frac {\ln \left ({\mathrm e}^{i x}\right ) \ln \left (1-\frac {\sqrt {3}\, {\mathrm e}^{i x}}{3}\right )}{2}-\frac {i x \ln \left ({\mathrm e}^{2 i x}-\frac {1}{3}\right )}{2}-\frac {i x \ln \left (3\right )}{2}+\frac {i x \ln \left (1+\sqrt {3}\, {\mathrm e}^{i x}\right )}{2}+\frac {i x \ln \left (1-\sqrt {3}\, {\mathrm e}^{i x}\right )}{2}-\frac {x \pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 i x}-3\right )}{{\mathrm e}^{2 i x}+1}\right )^{3}}{4}-\frac {x \pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 i x}-\frac {1}{3}\right )}{{\mathrm e}^{2 i x}+1}\right )^{3}}{4}\)	\(500\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(2*tan(x)),x,method=_RETURNVERBOSE)

[Out]

arctan(2*tan(x))*arctan(tan(x))+1/2*I*arctan(tan(x))*ln(1-3*(1+I*tan(x))^2/(1+tan(x)^2))+1/4*polylog(2,3*(1+I*
tan(x))^2/(1+tan(x)^2))-1/2*I*arctan(tan(x))*ln(1-1/3*(1+I*tan(x))^2/(1+tan(x)^2))-1/4*polylog(2,1/3*(1+I*tan(
x))^2/(1+tan(x)^2))

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Maxima [A]
time = 0.38, size = 84, normalized size = 1.05 \begin {gather*} x \arctan \left (2 \, \tan \left (x\right )\right ) - \frac {1}{8} \, \log \left (4 \, \tan \left (x\right )^{2} + 4\right ) \log \left (4 \, \tan \left (x\right )^{2} + 1\right ) + \frac {1}{8} \, \log \left (4 \, \tan \left (x\right )^{2} + 1\right ) \log \left (\frac {4}{9} \, \tan \left (x\right )^{2} + \frac {4}{9}\right ) - \frac {1}{4} \, {\rm Li}_2\left (2 i \, \tan \left (x\right ) - 1\right ) + \frac {1}{4} \, {\rm Li}_2\left (\frac {2}{3} i \, \tan \left (x\right ) + \frac {1}{3}\right ) + \frac {1}{4} \, {\rm Li}_2\left (-\frac {2}{3} i \, \tan \left (x\right ) + \frac {1}{3}\right ) - \frac {1}{4} \, {\rm Li}_2\left (-2 i \, \tan \left (x\right ) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(2*tan(x)),x, algorithm="maxima")

[Out]

x*arctan(2*tan(x)) - 1/8*log(4*tan(x)^2 + 4)*log(4*tan(x)^2 + 1) + 1/8*log(4*tan(x)^2 + 1)*log(4/9*tan(x)^2 +
4/9) - 1/4*dilog(2*I*tan(x) - 1) + 1/4*dilog(2/3*I*tan(x) + 1/3) + 1/4*dilog(-2/3*I*tan(x) + 1/3) - 1/4*dilog(
-2*I*tan(x) - 1)

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (50) = 100\).
time = 0.35, size = 220, normalized size = 2.75 \begin {gather*} x \arctan \left (2 \, \tan \left (x\right )\right ) - \frac {1}{4} i \, x \log \left (\frac {2 \, {\left (2 \, \tan \left (x\right )^{2} + 3 i \, \tan \left (x\right ) - 1\right )}}{\tan \left (x\right )^{2} + 1}\right ) + \frac {1}{4} i \, x \log \left (\frac {2 \, {\left (2 \, \tan \left (x\right )^{2} + i \, \tan \left (x\right ) + 1\right )}}{3 \, {\left (\tan \left (x\right )^{2} + 1\right )}}\right ) - \frac {1}{4} i \, x \log \left (\frac {2 \, {\left (2 \, \tan \left (x\right )^{2} - i \, \tan \left (x\right ) + 1\right )}}{3 \, {\left (\tan \left (x\right )^{2} + 1\right )}}\right ) + \frac {1}{4} i \, x \log \left (\frac {2 \, {\left (2 \, \tan \left (x\right )^{2} - 3 i \, \tan \left (x\right ) - 1\right )}}{\tan \left (x\right )^{2} + 1}\right ) + \frac {1}{8} \, {\rm Li}_2\left (-\frac {2 \, {\left (2 \, \tan \left (x\right )^{2} + 3 i \, \tan \left (x\right ) - 1\right )}}{\tan \left (x\right )^{2} + 1} + 1\right ) - \frac {1}{8} \, {\rm Li}_2\left (-\frac {2 \, {\left (2 \, \tan \left (x\right )^{2} + i \, \tan \left (x\right ) + 1\right )}}{3 \, {\left (\tan \left (x\right )^{2} + 1\right )}} + 1\right ) - \frac {1}{8} \, {\rm Li}_2\left (-\frac {2 \, {\left (2 \, \tan \left (x\right )^{2} - i \, \tan \left (x\right ) + 1\right )}}{3 \, {\left (\tan \left (x\right )^{2} + 1\right )}} + 1\right ) + \frac {1}{8} \, {\rm Li}_2\left (-\frac {2 \, {\left (2 \, \tan \left (x\right )^{2} - 3 i \, \tan \left (x\right ) - 1\right )}}{\tan \left (x\right )^{2} + 1} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(2*tan(x)),x, algorithm="fricas")

[Out]

x*arctan(2*tan(x)) - 1/4*I*x*log(2*(2*tan(x)^2 + 3*I*tan(x) - 1)/(tan(x)^2 + 1)) + 1/4*I*x*log(2/3*(2*tan(x)^2
+ I*tan(x) + 1)/(tan(x)^2 + 1)) - 1/4*I*x*log(2/3*(2*tan(x)^2 - I*tan(x) + 1)/(tan(x)^2 + 1)) + 1/4*I*x*log(2
*(2*tan(x)^2 - 3*I*tan(x) - 1)/(tan(x)^2 + 1)) + 1/8*dilog(-2*(2*tan(x)^2 + 3*I*tan(x) - 1)/(tan(x)^2 + 1) + 1
) - 1/8*dilog(-2/3*(2*tan(x)^2 + I*tan(x) + 1)/(tan(x)^2 + 1) + 1) - 1/8*dilog(-2/3*(2*tan(x)^2 - I*tan(x) + 1
)/(tan(x)^2 + 1) + 1) + 1/8*dilog(-2*(2*tan(x)^2 - 3*I*tan(x) - 1)/(tan(x)^2 + 1) + 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \operatorname {atan}{\left (2 \tan {\left (x \right )} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atan(2*tan(x)),x)

[Out]

Integral(atan(2*tan(x)), x)

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Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(2*tan(x)),x)

[Out]

Could not integrate

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \mathrm {atan}\left (2\,\mathrm {tan}\left (x\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atan(2*tan(x)),x)

[Out]

int(atan(2*tan(x)), x)

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