3.851 \(\int \frac{1}{\sqrt{-1+a+b x} \sqrt{1+a+b x}} \, dx\)

Optimal. Leaf size=22 \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{a+b x-1}}{\sqrt{2}}\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.0074095, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {63, 215} \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{a+b x-1}}{\sqrt{2}}\right )}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 215

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

Rubi steps

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

Mathematica [A]  time = 0.0111303, size = 22, normalized size = 1. \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{a+b x-1}}{\sqrt{2}}\right )}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.005, size = 94, normalized size = 4.3 \begin{align*}{\sqrt{ \left ( bx+a-1 \right ) \left ( bx+a+1 \right ) }\ln \left ({ \left ({\frac{b \left ( a-1 \right ) }{2}}+{\frac{b \left ( 1+a \right ) }{2}}+{b}^{2}x \right ){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+ \left ( b \left ( a-1 \right ) +b \left ( 1+a \right ) \right ) x+ \left ( 1+a \right ) \left ( a-1 \right ) } \right ){\frac{1}{\sqrt{bx+a-1}}}{\frac{1}{\sqrt{bx+a+1}}}{\frac{1}{\sqrt{{b}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.53664, size = 76, normalized size = 3.45 \begin{align*} -\frac{\log \left (-b x + \sqrt{b x + a + 1} \sqrt{b x + a - 1} - a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + b x - 1} \sqrt{a + b x + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 2.43867, size = 35, normalized size = 1.59 \begin{align*} -\frac{2 \, \log \left ({\left | -\sqrt{b x + a + 1} + \sqrt{b x + a - 1} \right |}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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