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

Optimal. Leaf size=38 \[ a x-\frac{b \log \left ((c+d x)^2+1\right )}{2 d}+\frac{b (c+d x) \tan ^{-1}(c+d x)}{d} \]

[Out]

a*x + (b*(c + d*x)*ArcTan[c + d*x])/d - (b*Log[1 + (c + d*x)^2])/(2*d)

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Rubi [A]  time = 0.0184687, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5039, 4846, 260} \[ a x-\frac{b \log \left ((c+d x)^2+1\right )}{2 d}+\frac{b (c+d x) \tan ^{-1}(c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[a + b*ArcTan[c + d*x],x]

[Out]

a*x + (b*(c + d*x)*ArcTan[c + d*x])/d - (b*Log[1 + (c + d*x)^2])/(2*d)

Rule 5039

Int[((a_.) + ArcTan[(c_) + (d_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcTan[x])^p, x],
x, c + d*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]

Rule 4846

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x])^p, x] - Dist[b*c*p, Int[
(x*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \left (a+b \tan ^{-1}(c+d x)\right ) \, dx &=a x+b \int \tan ^{-1}(c+d x) \, dx\\ &=a x+\frac{b \operatorname{Subst}\left (\int \tan ^{-1}(x) \, dx,x,c+d x\right )}{d}\\ &=a x+\frac{b (c+d x) \tan ^{-1}(c+d x)}{d}-\frac{b \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,c+d x\right )}{d}\\ &=a x+\frac{b (c+d x) \tan ^{-1}(c+d x)}{d}-\frac{b \log \left (1+(c+d x)^2\right )}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.0132084, size = 49, normalized size = 1.29 \[ a x-\frac{b \left (\log \left (c^2+2 c d x+d^2 x^2+1\right )-2 c \tan ^{-1}(c+d x)\right )}{2 d}+b x \tan ^{-1}(c+d x) \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*ArcTan[c + d*x],x]

[Out]

a*x + b*x*ArcTan[c + d*x] - (b*(-2*c*ArcTan[c + d*x] + Log[1 + c^2 + 2*c*d*x + d^2*x^2]))/(2*d)

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Maple [A]  time = 0.033, size = 42, normalized size = 1.1 \begin{align*} ax+b\arctan \left ( dx+c \right ) x+{\frac{b\arctan \left ( dx+c \right ) c}{d}}-{\frac{b\ln \left ( 1+ \left ( dx+c \right ) ^{2} \right ) }{2\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+b*arctan(d*x+c),x)

[Out]

a*x+b*arctan(d*x+c)*x+b/d*arctan(d*x+c)*c-1/2*b*ln(1+(d*x+c)^2)/d

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Maxima [A]  time = 0.973068, size = 49, normalized size = 1.29 \begin{align*} a x + \frac{{\left (2 \,{\left (d x + c\right )} \arctan \left (d x + c\right ) - \log \left ({\left (d x + c\right )}^{2} + 1\right )\right )} b}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + 1/2*(2*(d*x + c)*arctan(d*x + c) - log((d*x + c)^2 + 1))*b/d

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Fricas [A]  time = 1.51522, size = 119, normalized size = 3.13 \begin{align*} \frac{2 \, a d x + 2 \,{\left (b d x + b c\right )} \arctan \left (d x + c\right ) - b \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*a*d*x + 2*(b*d*x + b*c)*arctan(d*x + c) - b*log(d^2*x^2 + 2*c*d*x + c^2 + 1))/d

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Sympy [A]  time = 0.565511, size = 51, normalized size = 1.34 \begin{align*} a x + b \left (\begin{cases} \frac{c \operatorname{atan}{\left (c + d x \right )}}{d} + x \operatorname{atan}{\left (c + d x \right )} - \frac{\log{\left (c^{2} + 2 c d x + d^{2} x^{2} + 1 \right )}}{2 d} & \text{for}\: d \neq 0 \\x \operatorname{atan}{\left (c \right )} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*atan(d*x+c),x)

[Out]

a*x + b*Piecewise((c*atan(c + d*x)/d + x*atan(c + d*x) - log(c**2 + 2*c*d*x + d**2*x**2 + 1)/(2*d), Ne(d, 0)),
(x*atan(c), True))

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Giac [A]  time = 1.08183, size = 49, normalized size = 1.29 \begin{align*} a x + \frac{{\left (2 \,{\left (d x + c\right )} \arctan \left (d x + c\right ) - \log \left ({\left (d x + c\right )}^{2} + 1\right )\right )} b}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + 1/2*(2*(d*x + c)*arctan(d*x + c) - log((d*x + c)^2 + 1))*b/d