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

Optimal. Leaf size=21 \[ b x-\log (x) \left (b x-\tan ^{-1}(\tan (a+b x))\right ) \]

[Out]

b*x - (b*x - ArcTan[Tan[a + b*x]])*Log[x]

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Rubi [A]  time = 0.0313861, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2158, 29} \[ b x-\log (x) \left (b x-\tan ^{-1}(\tan (a+b x))\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[Tan[a + b*x]]/x,x]

[Out]

b*x - (b*x - ArcTan[Tan[a + b*x]])*Log[x]

Rule 2158

Int[(v_)/(u_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(b*x)/a, x] - Dist[(b*u
- a*v)/a, Int[1/u, x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

\begin{align*} \int \frac{\tan ^{-1}(\tan (a+b x))}{x} \, dx &=b x-\left (b x-\tan ^{-1}(\tan (a+b x))\right ) \int \frac{1}{x} \, dx\\ &=b x-\left (b x-\tan ^{-1}(\tan (a+b x))\right ) \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0135539, size = 19, normalized size = 0.9 \[ \log (x) \left (\tan ^{-1}(\tan (a+b x))-b x\right )+b x \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTan[Tan[a + b*x]]/x,x]

[Out]

b*x + (-(b*x) + ArcTan[Tan[a + b*x]])*Log[x]

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Maple [A]  time = 0.045, size = 21, normalized size = 1. \begin{align*} \ln \left ( x \right ) \arctan \left ( \tan \left ( bx+a \right ) \right ) -\ln \left ( x \right ) xb+bx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x)*arctan(tan(b*x+a))-ln(x)*x*b+b*x

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Maxima [B]  time = 1.46779, size = 57, normalized size = 2.71 \begin{align*} \frac{b \arctan \left (\tan \left (b x + a\right )\right ) \log \left (b x\right ) +{\left (b x -{\left (b x + a\right )} \log \left (b x\right ) + a \log \left (b x\right ) + a\right )} b}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*arctan(tan(b*x + a))*log(b*x) + (b*x - (b*x + a)*log(b*x) + a*log(b*x) + a)*b)/b

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Fricas [A]  time = 1.80399, size = 22, normalized size = 1.05 \begin{align*} b x + a \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x + a*log(x)

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Sympy [A]  time = 0.893141, size = 34, normalized size = 1.62 \begin{align*} - b x \log{\left (x \right )} + b x + \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 ) \log{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.10772, size = 31, normalized size = 1.48 \begin{align*} b x -{\left (\pi \left \lfloor \frac{a}{\pi } + \frac{1}{2} \right \rfloor - a\right )} \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x - (pi*floor(a/pi + 1/2) - a)*log(abs(x))