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

Optimal. Leaf size=61 \[ \frac{a+b \cosh ^{-1}(c x)}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{b x}{2 c d^2 \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

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

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Rubi [A]  time = 0.0521277, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {5716, 39} \[ \frac{a+b \cosh ^{-1}(c x)}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{b x}{2 c d^2 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5716

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

Rule 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]

Rubi steps

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

Mathematica [A]  time = 0.134365, size = 53, normalized size = 0.87 \[ \frac{a+b c x \sqrt{c x-1} \sqrt{c x+1}+b \cosh ^{-1}(c x)}{2 c^2 d^2-2 c^4 d^2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + b*ArcCosh[c*x])/(2*c^2*d^2 - 2*c^4*d^2*x^2)

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Maple [A]  time = 0.013, size = 64, normalized size = 1.1 \begin{align*}{\frac{1}{{c}^{2}} \left ( -{\frac{a}{2\,{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{b}{{d}^{2}} \left ( -{\frac{{\rm arccosh} \left (cx\right )}{2\,{c}^{2}{x}^{2}-2}}-{\frac{cx}{2}{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.27197, size = 223, normalized size = 3.66 \begin{align*} -\frac{1}{4} \,{\left (\frac{{\left (\frac{\sqrt{c^{2} x^{2} - 1} c^{2} d^{2}}{c^{6} d^{4} + \sqrt{c^{6} d^{4}} c^{4} d^{2} x} - \frac{\sqrt{c^{2} x^{2} - 1} c^{2} d^{2}}{c^{6} d^{4} - \sqrt{c^{6} d^{4}} c^{4} d^{2} x}\right )} c^{5} d^{2}}{\sqrt{c^{6} d^{4}}} + \frac{2 \, \operatorname{arcosh}\left (c x\right )}{c^{4} d^{2} x^{2} - c^{2} d^{2}}\right )} b - \frac{a}{2 \,{\left (c^{4} d^{2} x^{2} - c^{2} d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((sqrt(c^2*x^2 - 1)*c^2*d^2/(c^6*d^4 + sqrt(c^6*d^4)*c^4*d^2*x) - sqrt(c^2*x^2 - 1)*c^2*d^2/(c^6*d^4 - sq
rt(c^6*d^4)*c^4*d^2*x))*c^5*d^2/sqrt(c^6*d^4) + 2*arccosh(c*x)/(c^4*d^2*x^2 - c^2*d^2))*b - 1/2*a/(c^4*d^2*x^2
- c^2*d^2)

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Fricas [A]  time = 1.76537, size = 136, normalized size = 2.23 \begin{align*} -\frac{a c^{2} x^{2} + \sqrt{c^{2} x^{2} - 1} b c x + b \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )}{2 \,{\left (c^{4} d^{2} x^{2} - c^{2} d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(a*x/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(b*x*acosh(c*x)/(c**4*x**4 - 2*c**2*x**2 + 1), x))/d
**2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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