### 3.1260 $$\int \frac{1}{(a+b x) \sqrt{1+a^2+2 a b x+b^2 x^2}} \, dx$$

Optimal. Leaf size=27 $-\frac{\tanh ^{-1}\left (\sqrt{a^2+2 a b x+b^2 x^2+1}\right )}{b}$

[Out]

-(ArcTanh[Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]]/b)

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Rubi [A]  time = 0.0206604, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.069, Rules used = {688, 208} $-\frac{\tanh ^{-1}\left (\sqrt{a^2+2 a b x+b^2 x^2+1}\right )}{b}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

-(ArcTanh[Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]]/b)

Rule 688

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[4*c, Subst[Int[1/(b^2*e
- 4*a*c*e + 4*c*e*x^2), x], x, Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0]
&& EqQ[2*c*d - b*e, 0]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{1}{(a+b x) \sqrt{1+a^2+2 a b x+b^2 x^2}} \, dx &=\left (4 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{4 a^2 b^3-4 \left (1+a^2\right ) b^3+4 b^3 x^2} \, dx,x,\sqrt{1+a^2+2 a b x+b^2 x^2}\right )\\ &=-\frac{\tanh ^{-1}\left (\sqrt{1+a^2+2 a b x+b^2 x^2}\right )}{b}\\ \end{align*}

Mathematica [A]  time = 0.0081073, size = 19, normalized size = 0.7 $-\frac{\tanh ^{-1}\left (\sqrt{(a+b x)^2+1}\right )}{b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

-(ArcTanh[Sqrt[1 + (a + b*x)^2]]/b)

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Maple [A]  time = 0.045, size = 24, normalized size = 0.9 \begin{align*} -{\frac{1}{b}{\it Artanh} \left ({\frac{1}{\sqrt{ \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+1}}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/b*arctanh(1/((x+1/b*a)^2*b^2+1)^(1/2))

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Maxima [A]  time = 2.5732, size = 19, normalized size = 0.7 \begin{align*} -\frac{\operatorname{arsinh}\left (\frac{1}{{\left | b x + a \right |}}\right )}{b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arcsinh(1/abs(b*x + a))/b

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Fricas [B]  time = 2.05115, size = 157, normalized size = 5.81 \begin{align*} -\frac{\log \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 1\right ) - \log \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 1\right )}{b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((a + b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x)

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Giac [B]  time = 1.23572, size = 120, normalized size = 4.44 \begin{align*} \frac{\log \left (\frac{{\left | -2 \,{\left (x{\left | b \right |} - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} b - 2 \, a{\left | b \right |} - 2 \,{\left | b \right |} \right |}}{{\left | -2 \,{\left (x{\left | b \right |} - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} b - 2 \, a{\left | b \right |} + 2 \,{\left | b \right |} \right |}}\right )}{{\left | b \right |}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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