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

Optimal. Leaf size=91 \[ \frac{a+b \cosh ^{-1}(c x)}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{b x}{6 c d^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x}{12 c d^3 (c x-1)^{3/2} (c x+1)^{3/2}} \]

[Out]

(b*x)/(12*c*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) - (b*x)/(6*c*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (a + b*ArcC
osh[c*x])/(4*c^2*d^3*(1 - c^2*x^2)^2)

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Rubi [A]  time = 0.0582914, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {5716, 40, 39} \[ \frac{a+b \cosh ^{-1}(c x)}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{b x}{6 c d^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x}{12 c d^3 (c x-1)^{3/2} (c x+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x)/(12*c*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) - (b*x)/(6*c*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (a + b*ArcC
osh[c*x])/(4*c^2*d^3*(1 - c^2*x^2)^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 40

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(x*(a + b*x)^(m + 1)*(c + d*x)^(m +
1))/(2*a*c*(m + 1)), x] + Dist[(2*m + 3)/(2*a*c*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(m + 1), x], x] /; F
reeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && ILtQ[m + 3/2, 0]

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 )^3} \, dx &=\frac{a+b \cosh ^{-1}(c x)}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{b \int \frac{1}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{4 c d^3}\\ &=\frac{b x}{12 c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac{a+b \cosh ^{-1}(c x)}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac{b \int \frac{1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{6 c d^3}\\ &=\frac{b x}{12 c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{b x}{6 c d^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{a+b \cosh ^{-1}(c x)}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.20381, size = 64, normalized size = 0.7 \[ \frac{3 a+b c x \sqrt{c x-1} \sqrt{c x+1} \left (3-2 c^2 x^2\right )+3 b \cosh ^{-1}(c x)}{12 c^2 d^3 \left (c^2 x^2-1\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(3 - 2*c^2*x^2) + 3*b*ArcCosh[c*x])/(12*c^2*d^3*(-1 + c^2*x^2)^2)

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Maple [A]  time = 0.015, size = 86, normalized size = 1. \begin{align*}{\frac{1}{{c}^{2}} \left ({\frac{a}{4\,{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) ^{2}}}-{\frac{b}{{d}^{3}} \left ( -{\frac{{\rm arccosh} \left (cx\right )}{4\, \left ({c}^{2}{x}^{2}-1 \right ) ^{2}}}+{\frac{cx \left ( 2\,{c}^{2}{x}^{2}-3 \right ) }{12\,{c}^{2}{x}^{2}-12}{\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)^3,x)

[Out]

1/c^2*(1/4*a/d^3/(c^2*x^2-1)^2-b/d^3*(-1/4/(c^2*x^2-1)^2*arccosh(c*x)+1/12/(c*x-1)^(1/2)/(c*x+1)^(1/2)*c*x*(2*
c^2*x^2-3)/(c^2*x^2-1)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{16} \, b{\left (\frac{4 \, \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) + 1}{c^{6} d^{3} x^{4} - 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}} + 16 \, \int \frac{1}{4 \,{\left (c^{8} d^{3} x^{7} - 3 \, c^{6} d^{3} x^{5} + 3 \, c^{4} d^{3} x^{3} - c^{2} d^{3} x +{\left (c^{7} d^{3} x^{6} - 3 \, c^{5} d^{3} x^{4} + 3 \, c^{3} d^{3} x^{2} - c d^{3}\right )} e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (c x - 1\right )\right )}\right )}}\,{d x}\right )} + \frac{a}{4 \,{\left (c^{6} d^{3} x^{4} - 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\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)^3,x, algorithm="maxima")

[Out]

1/16*b*((4*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + 1)/(c^6*d^3*x^4 - 2*c^4*d^3*x^2 + c^2*d^3) + 16*integrate(
1/4/(c^8*d^3*x^7 - 3*c^6*d^3*x^5 + 3*c^4*d^3*x^3 - c^2*d^3*x + (c^7*d^3*x^6 - 3*c^5*d^3*x^4 + 3*c^3*d^3*x^2 -
c*d^3)*e^(1/2*log(c*x + 1) + 1/2*log(c*x - 1))), x)) + 1/4*a/(c^6*d^3*x^4 - 2*c^4*d^3*x^2 + c^2*d^3)

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Fricas [A]  time = 1.86284, size = 208, normalized size = 2.29 \begin{align*} -\frac{3 \, a c^{4} x^{4} - 6 \, a c^{2} x^{2} - 3 \, b \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) +{\left (2 \, b c^{3} x^{3} - 3 \, b c x\right )} \sqrt{c^{2} x^{2} - 1}}{12 \,{\left (c^{6} d^{3} x^{4} - 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\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)^3,x, algorithm="fricas")

[Out]

-1/12*(3*a*c^4*x^4 - 6*a*c^2*x^2 - 3*b*log(c*x + sqrt(c^2*x^2 - 1)) + (2*b*c^3*x^3 - 3*b*c*x)*sqrt(c^2*x^2 - 1
))/(c^6*d^3*x^4 - 2*c^4*d^3*x^2 + c^2*d^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{a x}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac{b x \operatorname{acosh}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx}{d^{3}} \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)**3,x)

[Out]

-(Integral(a*x/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x) + Integral(b*x*acosh(c*x)/(c**6*x**6 - 3*c**4*x
**4 + 3*c**2*x**2 - 1), x))/d**3

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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 )}^{3}}\,{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)^3,x, algorithm="giac")

[Out]

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