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

Optimal. Leaf size=135 \[ \frac{1}{3} c^4 d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-2 c^2 d^2 x \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac{1}{9} b c d^2 (c x-1)^{3/2} (c x+1)^{3/2}+\frac{5}{3} b c d^2 \sqrt{c x-1} \sqrt{c x+1}+b c d^2 \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right ) \]

[Out]

(5*b*c*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/3 - (b*c*d^2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/9 - (d^2*(a + b*ArcCos
h[c*x]))/x - 2*c^2*d^2*x*(a + b*ArcCosh[c*x]) + (c^4*d^2*x^3*(a + b*ArcCosh[c*x]))/3 + b*c*d^2*ArcTan[Sqrt[-1
+ c*x]*Sqrt[1 + c*x]]

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Rubi [A]  time = 0.229905, antiderivative size = 182, normalized size of antiderivative = 1.35, number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {270, 5731, 12, 520, 1251, 897, 1153, 205} \[ \frac{1}{3} c^4 d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-2 c^2 d^2 x \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac{b c d^2 \left (1-c^2 x^2\right )^2}{9 \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 b c d^2 \left (1-c^2 x^2\right )}{3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c d^2 \sqrt{c^2 x^2-1} \tan ^{-1}\left (\sqrt{c^2 x^2-1}\right )}{\sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*b*c*d^2*(1 - c^2*x^2))/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^2*(1 - c^2*x^2)^2)/(9*Sqrt[-1 + c*x]*Sqrt
[1 + c*x]) - (d^2*(a + b*ArcCosh[c*x]))/x - 2*c^2*d^2*x*(a + b*ArcCosh[c*x]) + (c^4*d^2*x^3*(a + b*ArcCosh[c*x
]))/3 + (b*c*d^2*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 5731

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[1
+ c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 520

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.) + (e_.)*(x_)^(n2_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b
2_.)*(x_)^(non2_.))^(p_.), x_Symbol] :> Dist[((a1 + b1*x^(n/2))^FracPart[p]*(a2 + b2*x^(n/2))^FracPart[p])/(a1
*a2 + b1*b2*x^n)^FracPart[p], Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n + e*x^(2*n))^q, x], x] /; FreeQ[{a1, b1,
a2, b2, c, d, e, n, p, q}, x] && EqQ[non2, n/2] && EqQ[n2, 2*n] && EqQ[a2*b1 + a1*b2, 0]

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 897

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - ((2*c*d - b*e)*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{x^2} \, dx &=-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac{d^2 \left (-3-6 c^2 x^2+c^4 x^4\right )}{3 x \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} \left (b c d^2\right ) \int \frac{-3-6 c^2 x^2+c^4 x^4}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c d^2 \sqrt{-1+c^2 x^2}\right ) \int \frac{-3-6 c^2 x^2+c^4 x^4}{x \sqrt{-1+c^2 x^2}} \, dx}{3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c d^2 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{-3-6 c^2 x+c^4 x^2}{x \sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{6 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b d^2 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{-8-4 x^2+x^4}{\frac{1}{c^2}+\frac{x^2}{c^2}} \, dx,x,\sqrt{-1+c^2 x^2}\right )}{3 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b d^2 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (-5 c^2+c^2 x^2-\frac{3}{\frac{1}{c^2}+\frac{x^2}{c^2}}\right ) \, dx,x,\sqrt{-1+c^2 x^2}\right )}{3 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{5 b c d^2 \left (1-c^2 x^2\right )}{3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c d^2 \left (1-c^2 x^2\right )^2}{9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{\left (b d^2 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c^2}+\frac{x^2}{c^2}} \, dx,x,\sqrt{-1+c^2 x^2}\right )}{c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{5 b c d^2 \left (1-c^2 x^2\right )}{3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c d^2 \left (1-c^2 x^2\right )^2}{9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{b c d^2 \sqrt{-1+c^2 x^2} \tan ^{-1}\left (\sqrt{-1+c^2 x^2}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 0.158013, size = 131, normalized size = 0.97 \[ \frac{d^2 \left (3 a c^4 x^4-18 a c^2 x^2-9 a-b c^3 x^3 \sqrt{c x-1} \sqrt{c x+1}+3 b \left (c^4 x^4-6 c^2 x^2-3\right ) \cosh ^{-1}(c x)+16 b c x \sqrt{c x-1} \sqrt{c x+1}-9 b c x \tan ^{-1}\left (\frac{1}{\sqrt{c x-1} \sqrt{c x+1}}\right )\right )}{9 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(-9*a - 18*a*c^2*x^2 + 3*a*c^4*x^4 + 16*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - b*c^3*x^3*Sqrt[-1 + c*x]*Sqr
t[1 + c*x] + 3*b*(-3 - 6*c^2*x^2 + c^4*x^4)*ArcCosh[c*x] - 9*b*c*x*ArcTan[1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])]))/
(9*x)

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Maple [A]  time = 0.017, size = 167, normalized size = 1.2 \begin{align*}{\frac{{d}^{2}a{c}^{4}{x}^{3}}{3}}-2\,{d}^{2}a{c}^{2}x-{\frac{{d}^{2}a}{x}}+{\frac{{d}^{2}b{\rm arccosh} \left (cx\right ){c}^{4}{x}^{3}}{3}}-2\,{d}^{2}b{\rm arccosh} \left (cx\right ){c}^{2}x-{\frac{b{d}^{2}{\rm arccosh} \left (cx\right )}{x}}-{\frac{{d}^{2}b{c}^{3}{x}^{2}}{9}\sqrt{cx-1}\sqrt{cx+1}}+{\frac{16\,{d}^{2}bc}{9}\sqrt{cx-1}\sqrt{cx+1}}-{{d}^{2}bc\sqrt{cx-1}\sqrt{cx+1}\arctan \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*d^2*a*c^4*x^3-2*d^2*a*c^2*x-d^2*a/x+1/3*d^2*b*arccosh(c*x)*c^4*x^3-2*d^2*b*arccosh(c*x)*c^2*x-d^2*b*arccos
h(c*x)/x-1/9*d^2*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^2+16/9*b*c*d^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)-c*d^2*b*(c*x-1
)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)^(1/2)*arctan(1/(c^2*x^2-1)^(1/2))

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Maxima [A]  time = 1.86869, size = 196, normalized size = 1.45 \begin{align*} \frac{1}{3} \, a c^{4} d^{2} x^{3} + \frac{1}{9} \,{\left (3 \, x^{3} \operatorname{arcosh}\left (c x\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b c^{4} d^{2} - 2 \, a c^{2} d^{2} x - 2 \,{\left (c x \operatorname{arcosh}\left (c x\right ) - \sqrt{c^{2} x^{2} - 1}\right )} b c d^{2} -{\left (c \arcsin \left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right ) + \frac{\operatorname{arcosh}\left (c x\right )}{x}\right )} b d^{2} - \frac{a d^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*c^4*d^2*x^3 + 1/9*(3*x^3*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/c^4))*b*c^4*d
^2 - 2*a*c^2*d^2*x - 2*(c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*b*c*d^2 - (c*arcsin(1/(sqrt(c^2)*abs(x))) + arcc
osh(c*x)/x)*b*d^2 - a*d^2/x

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Fricas [A]  time = 2.06974, size = 440, normalized size = 3.26 \begin{align*} \frac{3 \, a c^{4} d^{2} x^{4} - 18 \, a c^{2} d^{2} x^{2} + 18 \, b c d^{2} x \arctan \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) - 3 \,{\left (b c^{4} - 6 \, b c^{2} - 3 \, b\right )} d^{2} x \log \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) - 9 \, a d^{2} + 3 \,{\left (b c^{4} d^{2} x^{4} - 6 \, b c^{2} d^{2} x^{2} -{\left (b c^{4} - 6 \, b c^{2} - 3 \, b\right )} d^{2} x - 3 \, b d^{2}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (b c^{3} d^{2} x^{3} - 16 \, b c d^{2} x\right )} \sqrt{c^{2} x^{2} - 1}}{9 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(3*a*c^4*d^2*x^4 - 18*a*c^2*d^2*x^2 + 18*b*c*d^2*x*arctan(-c*x + sqrt(c^2*x^2 - 1)) - 3*(b*c^4 - 6*b*c^2 -
3*b)*d^2*x*log(-c*x + sqrt(c^2*x^2 - 1)) - 9*a*d^2 + 3*(b*c^4*d^2*x^4 - 6*b*c^2*d^2*x^2 - (b*c^4 - 6*b*c^2 -
3*b)*d^2*x - 3*b*d^2)*log(c*x + sqrt(c^2*x^2 - 1)) - (b*c^3*d^2*x^3 - 16*b*c*d^2*x)*sqrt(c^2*x^2 - 1))/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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