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

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

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Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {2 \sinh ^{-1}\left (\frac {\sqrt {a+b x-1}}{\sqrt {2}}\right )}{b} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 1.00 \begin {gather*} \frac {2 \sinh ^{-1}\left (\frac {\sqrt {a+b x-1}}{\sqrt {2}}\right )}{b} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.06, size = 27, normalized size = 1.23 \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x-1}}{\sqrt {a+b x+1}}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.36, size = 31, normalized size = 1.41 \begin {gather*} -\frac {\log \left (-b x + \sqrt {b x + a + 1} \sqrt {b x + a - 1} - a\right )}{b} \end {gather*}

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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giac [A]  time = 1.17, size = 25, normalized size = 1.14 \begin {gather*} -\frac {2 \, \log \left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}{b} \end {gather*}

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

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maple [B]  time = 0.00, size = 94, normalized size = 4.27 \begin {gather*} \frac {\sqrt {\left (b x +a -1\right ) \left (b x +a +1\right )}\, \ln \left (\frac {b^{2} x +\frac {\left (a -1\right ) b}{2}+\frac {\left (a +1\right ) b}{2}}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+\left (\left (a -1\right ) b +\left (a +1\right ) b \right ) x +\left (a -1\right ) \left (a +1\right )}\right )}{\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \sqrt {b^{2}}} \end {gather*}

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

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maxima [A]  time = 0.53, size = 38, normalized size = 1.73 \begin {gather*} \frac {\log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{b} \end {gather*}

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]

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

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mupad [B]  time = 1.08, size = 53, normalized size = 2.41 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )\,\sqrt {-b^2}}\right )}{\sqrt {-b^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a + b x - 1} \sqrt {a + b x + 1}}\, dx \end {gather*}

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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