3.81 \(\int \frac{1}{c+(a+b x)^2} \, dx\)

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

[Out]

ArcTan[(a + b*x)/Sqrt[c]]/(b*Sqrt[c])

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Rubi [A]  time = 0.0081073, 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 = {247, 203} \[ \frac{\tan ^{-1}\left (\frac{a+b x}{\sqrt{c}}\right )}{b \sqrt{c}} \]

Antiderivative was successfully verified.

[In]

Int[(c + (a + b*x)^2)^(-1),x]

[Out]

ArcTan[(a + b*x)/Sqrt[c]]/(b*Sqrt[c])

Rule 247

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0033573, size = 21, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{a+b x}{\sqrt{c}}\right )}{b \sqrt{c}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + (a + b*x)^2)^(-1),x]

[Out]

ArcTan[(a + b*x)/Sqrt[c]]/(b*Sqrt[c])

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Maple [A]  time = 0.003, size = 28, normalized size = 1.3 \begin{align*}{\frac{1}{b}\arctan \left ({\frac{2\,{b}^{2}x+2\,ab}{2\,b}{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c+(b*x+a)^2),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+(b*x+a)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.74942, size = 205, normalized size = 9.76 \begin{align*} \left [-\frac{\sqrt{-c} \log \left (\frac{b^{2} x^{2} + 2 \, a b x + a^{2} - 2 \,{\left (b x + a\right )} \sqrt{-c} - c}{b^{2} x^{2} + 2 \, a b x + a^{2} + c}\right )}{2 \, b c}, \frac{\arctan \left (\frac{b x + a}{\sqrt{c}}\right )}{b \sqrt{c}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+(b*x+a)^2),x, algorithm="fricas")

[Out]

[-1/2*sqrt(-c)*log((b^2*x^2 + 2*a*b*x + a^2 - 2*(b*x + a)*sqrt(-c) - c)/(b^2*x^2 + 2*a*b*x + a^2 + c))/(b*c),
arctan((b*x + a)/sqrt(c))/(b*sqrt(c))]

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Sympy [B]  time = 0.175442, size = 54, normalized size = 2.57 \begin{align*} \frac{- \frac{\sqrt{- \frac{1}{c}} \log{\left (x + \frac{a - c \sqrt{- \frac{1}{c}}}{b} \right )}}{2} + \frac{\sqrt{- \frac{1}{c}} \log{\left (x + \frac{a + c \sqrt{- \frac{1}{c}}}{b} \right )}}{2}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+(b*x+a)**2),x)

[Out]

(-sqrt(-1/c)*log(x + (a - c*sqrt(-1/c))/b)/2 + sqrt(-1/c)*log(x + (a + c*sqrt(-1/c))/b)/2)/b

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Giac [A]  time = 1.12321, size = 23, normalized size = 1.1 \begin{align*} \frac{\arctan \left (\frac{b x + a}{\sqrt{c}}\right )}{b \sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+(b*x+a)^2),x, algorithm="giac")

[Out]

arctan((b*x + a)/sqrt(c))/(b*sqrt(c))