3.13.60 \(\int \frac {1}{(a+b x) \sqrt {1+a^2+2 a b x+b^2 x^2}} \, dx\) [1260]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 = {702, 214} \begin {gather*} -\frac {\tanh ^{-1}\left (\sqrt {a^2+2 a b x+b^2 x^2+1}\right )}{b} \end {gather*}

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 214

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

Rule 702

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]

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 ) \text {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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(103\) vs. \(2(27)=54\).
time = 0.34, size = 103, normalized size = 3.81 \begin {gather*} \frac {\left (b+\sqrt {b^2}\right ) \tanh ^{-1}\left (a+\sqrt {b^2} x-\sqrt {1+a^2+2 a b x+b^2 x^2}\right )+\left (b-\sqrt {b^2}\right ) \tanh ^{-1}\left (a-\sqrt {b^2} x+\sqrt {1+a^2+2 a b x+b^2 x^2}\right )}{b \sqrt {b^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.75, size = 24, normalized size = 0.89

method	result	size
default	\(-\frac {\arctanh \left (\frac {1}{\sqrt {b^{2} \left (x +\frac {a}{b}\right )^{2}+1}}\right )}{b}\)	\(24\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.48, size = 14, normalized size = 0.52 \begin {gather*} -\frac {\operatorname {arsinh}\left (\frac {1}{{\left | b x + a \right |}}\right )}{b} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (25) = 50\).
time = 3.31, size = 66, normalized size = 2.44 \begin {gather*} -\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x\right ) \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (25) = 50\).
time = 1.92, size = 89, normalized size = 3.30 \begin {gather*} \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 {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{\left (a+b\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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