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

Optimal. Leaf size=19 \[ \log (x) \left (\tan ^{-1}(\cot (a+b x))+b x\right )-b x \]

[Out]

-b*x+(b*x+1/2*Pi-arccot(cot(b*x+a)))*ln(x)

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Rubi [A]  time = 0.03, antiderivative size = 19, normalized size of antiderivative = 1.00, 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} \[ \log (x) \left (\tan ^{-1}(\cot (a+b x))+b x\right )-b x \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[Cot[a + b*x]]/x,x]

[Out]

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

Rule 29

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

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 19, normalized size = 1.00 \[ \log (x) \left (\tan ^{-1}(\cot (a+b x))+b x\right )-b x \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTan[Cot[a + b*x]]/x,x]

[Out]

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

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fricas [A]  time = 0.47, size = 14, normalized size = 0.74 \[ -b x + \frac {1}{2} \, {\left (\pi - 2 \, a\right )} \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/2*pi-arccot(cot(b*x+a)))/x,x, algorithm="fricas")

[Out]

-b*x + 1/2*(pi - 2*a)*log(x)

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giac [A]  time = 0.11, size = 15, normalized size = 0.79 \[ -b x + \frac {1}{2} \, {\left (\pi - 2 \, a\right )} \log \left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/2*pi-arccot(cot(b*x+a)))/x,x, algorithm="giac")

[Out]

-b*x + 1/2*(pi - 2*a)*log(abs(x))

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maple [A]  time = 0.39, size = 35, normalized size = 1.84 \[ \frac {\pi \ln \relax (x )}{2}-b x -a \ln \relax (x )-\ln \relax (x ) \left (\mathrm {arccot}\left (\cot \left (b x +a \right )\right )-b x -a \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/2*Pi-arccot(cot(b*x+a)))/x,x)

[Out]

1/2*Pi*ln(x)-b*x-a*ln(x)-ln(x)*(arccot(cot(b*x+a))-b*x-a)

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maxima [A]  time = 0.31, size = 14, normalized size = 0.74 \[ -b x + \frac {1}{2} \, {\left (\pi - 2 \, a\right )} \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/2*pi-arccot(cot(b*x+a)))/x,x, algorithm="maxima")

[Out]

-b*x + 1/2*(pi - 2*a)*log(x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {\frac {\Pi }{2}-\mathrm {acot}\left (\mathrm {cot}\left (a+b\,x\right )\right )}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((Pi/2 - acot(cot(a + b*x)))/x,x)

[Out]

int((Pi/2 - acot(cot(a + b*x)))/x, x)

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sympy [A]  time = 4.57, size = 27, normalized size = 1.42 \[ b x \log {\relax (x )} - b x - \log {\relax (x )} \operatorname {acot}{\left (\cot {\left (a + b x \right )} \right )} + \frac {\pi \log {\relax (x )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/2*pi-acot(cot(b*x+a)))/x,x)

[Out]

b*x*log(x) - b*x - log(x)*acot(cot(a + b*x)) + pi*log(x)/2

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