3.39 \(\int x \tan ^{-1}(\tan (a+b x)) \, dx\)

Optimal. Leaf size=23 \[ \frac {1}{2} x^2 \tan ^{-1}(\tan (a+b x))-\frac {b x^3}{6} \]

[Out]

-1/6*b*x^3+1/2*x^2*arctan(tan(b*x+a))

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5171, 30} \[ \frac {1}{2} x^2 \tan ^{-1}(\tan (a+b x))-\frac {b x^3}{6} \]

Antiderivative was successfully verified.

[In]

Int[x*ArcTan[Tan[a + b*x]],x]

[Out]

-(b*x^3)/6 + (x^2*ArcTan[Tan[a + b*x]])/2

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 5171

Int[ArcTan[(c_.) + (d_.)*Tan[(a_.) + (b_.)*(x_)]]*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^(m
+ 1)*ArcTan[c + d*Tan[a + b*x]])/(f*(m + 1)), x] - Dist[(I*b)/(f*(m + 1)), Int[(e + f*x)^(m + 1)/(c + I*d + c*
E^(2*I*a + 2*I*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c + I*d)^2, -1]

Rubi steps

\begin {align*} \int x \tan ^{-1}(\tan (a+b x)) \, dx &=\frac {1}{2} x^2 \tan ^{-1}(\tan (a+b x))-\frac {1}{2} b \int x^2 \, dx\\ &=-\frac {b x^3}{6}+\frac {1}{2} x^2 \tan ^{-1}(\tan (a+b x))\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 20, normalized size = 0.87 \[ -\frac {1}{6} x^2 \left (b x-3 \tan ^{-1}(\tan (a+b x))\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*ArcTan[Tan[a + b*x]],x]

[Out]

-1/6*(x^2*(b*x - 3*ArcTan[Tan[a + b*x]]))

________________________________________________________________________________________

fricas [A]  time = 0.49, size = 13, normalized size = 0.57 \[ \frac {1}{3} \, b x^{3} + \frac {1}{2} \, a x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arctan(tan(b*x+a)),x, algorithm="fricas")

[Out]

1/3*b*x^3 + 1/2*a*x^2

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arctan(tan(b*x+a)),x, algorithm="giac")

[Out]

sage0*x

________________________________________________________________________________________

maple [A]  time = 0.18, size = 20, normalized size = 0.87 \[ -\frac {b \,x^{3}}{6}+\frac {x^{2} \arctan \left (\tan \left (b x +a \right )\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arctan(tan(b*x+a)),x)

[Out]

-1/6*b*x^3+1/2*x^2*arctan(tan(b*x+a))

________________________________________________________________________________________

maxima [B]  time = 0.32, size = 57, normalized size = 2.48 \[ \frac {\frac {3 \, {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a\right )} \arctan \left (\tan \left (b x + a\right )\right )}{b} - \frac {{\left (b x + a\right )}^{3} - 3 \, {\left (b x + a\right )}^{2} a}{b}}{6 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arctan(tan(b*x+a)),x, algorithm="maxima")

[Out]

1/6*(3*((b*x + a)^2 - 2*(b*x + a)*a)*arctan(tan(b*x + a))/b - ((b*x + a)^3 - 3*(b*x + a)^2*a)/b)/b

________________________________________________________________________________________

mupad [B]  time = 0.06, size = 19, normalized size = 0.83 \[ \frac {x^2\,\mathrm {atan}\left (\mathrm {tan}\left (a+b\,x\right )\right )}{2}-\frac {b\,x^3}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*atan(tan(a + b*x)),x)

[Out]

(x^2*atan(tan(a + b*x)))/2 - (b*x^3)/6

________________________________________________________________________________________

sympy [A]  time = 0.20, size = 32, normalized size = 1.39 \[ - \frac {b x^{3}}{6} + \frac {x^{2} \left (\operatorname {atan}{\left (\tan {\left (a + b x \right )} \right )} + \pi \left \lfloor {\frac {a + b x - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*atan(tan(b*x+a)),x)

[Out]

-b*x**3/6 + x**2*(atan(tan(a + b*x)) + pi*floor((a + b*x - pi/2)/pi))/2

________________________________________________________________________________________