3.119 \(\int \frac{a+b \cosh ^{-1}(c x)}{(d-c^2 d x^2)^{3/2}} \, dx\)

Optimal. Leaf size=84 \[ \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{c x-1} \sqrt{c x+1} \log \left (1-c^2 x^2\right )}{2 c d \sqrt{d-c^2 d x^2}} \]

[Out]

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

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Rubi [A]  time = 0.144078, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {5713, 5688, 260} \[ \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{c x-1} \sqrt{c x+1} \log \left (1-c^2 x^2\right )}{2 c d \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^(3/2),x]

[Out]

(x*(a + b*ArcCosh[c*x]))/(d*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 - c^2*x^2])/(2*c*d*Sq
rt[d - c^2*d*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 5688

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(((d1_) + (e1_.)*(x_))^(3/2)*((d2_) + (e2_.)*(x_))^(3/2)), x_Sym
bol] :> Simp[(x*(a + b*ArcCosh[c*x])^n)/(d1*d2*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), x] + Dist[(b*c*n*Sqrt[1 + c*x
]*Sqrt[-1 + c*x])/(d1*d2*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), Int[(x*(a + b*ArcCosh[c*x])^(n - 1))/(1 - c^2*x^2),
x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[n, 0]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{a+b \cosh ^{-1}(c x)}{\left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}+\frac{\left (b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x}{1-c^2 x^2} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{-1+c x} \sqrt{1+c x} \log \left (1-c^2 x^2\right )}{2 c d \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0290793, size = 72, normalized size = 0.86 \[ \frac{2 a c x-b \sqrt{c x-1} \sqrt{c x+1} \log \left (1-c^2 x^2\right )+2 b c x \cosh ^{-1}(c x)}{2 c d \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^(3/2),x]

[Out]

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

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Maple [B]  time = 0.102, size = 180, normalized size = 2.1 \begin{align*}{\frac{ax}{d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}}-{\frac{b{\rm arccosh} \left (cx\right )}{c{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b{\rm arccosh} \left (cx\right )x}{{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{b}{c{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}\ln \left ( \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2}-1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.20032, size = 95, normalized size = 1.13 \begin{align*} -\frac{b c \sqrt{-\frac{1}{c^{4} d}} \log \left (x^{2} - \frac{1}{c^{2}}\right )}{2 \, d} + \frac{b x \operatorname{arcosh}\left (c x\right )}{\sqrt{-c^{2} d x^{2} + d} d} + \frac{a x}{\sqrt{-c^{2} d x^{2} + d} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="maxima")

[Out]

-1/2*b*c*sqrt(-1/(c^4*d))*log(x^2 - 1/c^2)/d + b*x*arccosh(c*x)/(sqrt(-c^2*d*x^2 + d)*d) + a*x/(sqrt(-c^2*d*x^
2 + d)*d)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-c^{2} d x^{2} + d}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(c*x))/(-c**2*d*x**2+d)**(3/2),x)

[Out]

Integral((a + b*acosh(c*x))/(-d*(c*x - 1)*(c*x + 1))**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)/(-c^2*d*x^2 + d)^(3/2), x)