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

Optimal. Leaf size=37 \[ -\frac{\sqrt{c x-1}}{b c \sqrt{1-c x} \left (a+b \cosh ^{-1}(c x)\right )} \]

[Out]

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

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Rubi [A]  time = 0.215333, antiderivative size = 50, normalized size of antiderivative = 1.35, 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, 5676} \[ -\frac{\sqrt{c x-1} \sqrt{c x+1}}{b c \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 5676

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

Rubi steps

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

Mathematica [A]  time = 0.0313497, size = 50, normalized size = 1.35 \[ -\frac{\sqrt{c x-1} \sqrt{c x+1}}{b c \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.04, size = 57, normalized size = 1.5 \begin{align*}{\frac{1}{c \left ({c}^{2}{x}^{2}-1 \right ) \left ( a+b{\rm arccosh} \left (cx\right ) \right ) b}\sqrt{- \left ( cx-1 \right ) \left ( cx+1 \right ) }\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))^2/(-c^2*x^2+1)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{c^{3} x^{3} +{\left (c^{2} x^{2} - 1\right )} \sqrt{c x + 1} \sqrt{c x - 1} - c x}{{\left ({\left (c x + 1\right )} \sqrt{c x - 1} b^{2} c^{2} x +{\left (b^{2} c^{3} x^{2} - b^{2} c\right )} \sqrt{c x + 1}\right )} \sqrt{-c x + 1} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) +{\left ({\left (c x + 1\right )} \sqrt{c x - 1} a b c^{2} x +{\left (a b c^{3} x^{2} - a b c\right )} \sqrt{c x + 1}\right )} \sqrt{-c x + 1}} + \int -\frac{c^{2} x^{2} -{\left (c x + 1\right )}{\left (c x - 1\right )} - 1}{{\left ({\left (c x + 1\right )}^{\frac{3}{2}}{\left (c x - 1\right )} b^{2} c^{2} x^{2} + 2 \,{\left (b^{2} c^{3} x^{3} - b^{2} c x\right )}{\left (c x + 1\right )} \sqrt{c x - 1} +{\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \sqrt{c x + 1}\right )} \sqrt{-c x + 1} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) +{\left ({\left (c x + 1\right )}^{\frac{3}{2}}{\left (c x - 1\right )} a b c^{2} x^{2} + 2 \,{\left (a b c^{3} x^{3} - a b c x\right )}{\left (c x + 1\right )} \sqrt{c x - 1} +{\left (a b c^{4} x^{4} - 2 \, a b c^{2} x^{2} + a b\right )} \sqrt{c x + 1}\right )} \sqrt{-c x + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c^3*x^3 + (c^2*x^2 - 1)*sqrt(c*x + 1)*sqrt(c*x - 1) - c*x)/(((c*x + 1)*sqrt(c*x - 1)*b^2*c^2*x + (b^2*c^3*x^
2 - b^2*c)*sqrt(c*x + 1))*sqrt(-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + ((c*x + 1)*sqrt(c*x - 1)*a*b
*c^2*x + (a*b*c^3*x^2 - a*b*c)*sqrt(c*x + 1))*sqrt(-c*x + 1)) + integrate(-(c^2*x^2 - (c*x + 1)*(c*x - 1) - 1)
/(((c*x + 1)^(3/2)*(c*x - 1)*b^2*c^2*x^2 + 2*(b^2*c^3*x^3 - b^2*c*x)*(c*x + 1)*sqrt(c*x - 1) + (b^2*c^4*x^4 -
2*b^2*c^2*x^2 + b^2)*sqrt(c*x + 1))*sqrt(-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + ((c*x + 1)^(3/2)*(
c*x - 1)*a*b*c^2*x^2 + 2*(a*b*c^3*x^3 - a*b*c*x)*(c*x + 1)*sqrt(c*x - 1) + (a*b*c^4*x^4 - 2*a*b*c^2*x^2 + a*b)
*sqrt(c*x + 1))*sqrt(-c*x + 1)), x)

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Fricas [B]  time = 2.04339, size = 153, normalized size = 4.14 \begin{align*} \frac{\sqrt{c^{2} x^{2} - 1} \sqrt{-c^{2} x^{2} + 1}}{a b c^{3} x^{2} - a b c +{\left (b^{2} c^{3} x^{2} - b^{2} c\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(-(c*x - 1)*(c*x + 1))*(a + b*acosh(c*x))**2), 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 )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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