3.302 \(\int \frac{1}{\sqrt{1-c^2 x^2} (a+b \cosh ^{-1}(c x))} \, dx\)

Optimal. Leaf size=35 \[ \frac{\sqrt{c x-1} \log \left (a+b \cosh ^{-1}(c x)\right )}{b c \sqrt{1-c x}} \]

[Out]

(Sqrt[-1 + c*x]*Log[a + b*ArcCosh[c*x]])/(b*c*Sqrt[1 - c*x])

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Rubi [A]  time = 0.220579, antiderivative size = 48, normalized size of antiderivative = 1.37, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {5713, 5674} \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \log \left (a+b \cosh ^{-1}(c x)\right )}{b c \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[a + b*ArcCosh[c*x]])/(b*c*Sqrt[1 - c^2*x^2])

Rule 5713

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist[((-d)^IntPart[p]*(
d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(1 + c*x)^p*(-1 + c*x)^p*(a + b*Ar
cCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[c^2*d + e, 0] &&  !IntegerQ[p]

Rule 5674

Int[1/(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :>
Simp[Log[a + b*ArcCosh[c*x]]/(b*c*Sqrt[-(d1*d2)]), x] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1]
&& EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0]

Rubi steps

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

Mathematica [A]  time = 0.105507, size = 54, normalized size = 1.54 \[ \frac{\sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left (a+b \cosh ^{-1}(c x)\right )}{b c \sqrt{-(c x-1) (c x+1)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[a + b*ArcCosh[c*x]])/(b*c*Sqrt[-((-1 + c*x)*(1 + c*x))])

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Maple [A]  time = 0.079, size = 55, normalized size = 1.6 \begin{align*} -{\frac{\ln \left ( a+b{\rm arccosh} \left (cx\right ) \right ) }{c \left ({c}^{2}{x}^{2}-1 \right ) b}\sqrt{-{c}^{2}{x}^{2}+1}\sqrt{cx-1}\sqrt{cx+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-c^{2} x^{2} + 1}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(-c^2*x^2 + 1)*(b*arccosh(c*x) + a)), x)

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Fricas [B]  time = 1.90049, size = 136, normalized size = 3.89 \begin{align*} -\frac{\sqrt{c^{2} x^{2} - 1} \sqrt{-c^{2} x^{2} + 1} \log \left (\frac{b \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) + a}{b}\right )}{b c^{3} x^{2} - b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(c^2*x^2 - 1)*sqrt(-c^2*x^2 + 1)*log((b*log(c*x + sqrt(c^2*x^2 - 1)) + a)/b)/(b*c^3*x^2 - b*c)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-c^{2} x^{2} + 1}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(-c^2*x^2 + 1)*(b*arccosh(c*x) + a)), x)