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

Optimal. Leaf size=200 \[ b c^2 d^2 \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\frac{c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{b}-2 c^2 d^2 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} b c^3 d^2 x \sqrt{c x-1} \sqrt{c x+1}-\frac{1}{4} b c^2 d^2 \cosh ^{-1}(c x)-\frac{b c d^2 (c x-1)^{3/2} (c x+1)^{3/2}}{2 x} \]

[Out]

(b*c^3*d^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/4 - (b*c*d^2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(2*x) - (b*c^2*d^2*A
rcCosh[c*x])/4 - c^2*d^2*(1 - c^2*x^2)*(a + b*ArcCosh[c*x]) - (d^2*(1 - c^2*x^2)^2*(a + b*ArcCosh[c*x]))/(2*x^
2) - (c^2*d^2*(a + b*ArcCosh[c*x])^2)/b - 2*c^2*d^2*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])] + b*c^2*
d^2*PolyLog[2, -E^(-2*ArcCosh[c*x])]

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Rubi [A]  time = 0.217591, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {5729, 97, 12, 38, 52, 5727, 5660, 3718, 2190, 2279, 2391} \[ -b c^2 d^2 \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{b}-2 c^2 d^2 \log \left (e^{2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} b c^3 d^2 x \sqrt{c x-1} \sqrt{c x+1}-\frac{1}{4} b c^2 d^2 \cosh ^{-1}(c x)-\frac{b c d^2 (c x-1)^{3/2} (c x+1)^{3/2}}{2 x} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(b*c^3*d^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/4 - (b*c*d^2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(2*x) - (b*c^2*d^2*A
rcCosh[c*x])/4 - c^2*d^2*(1 - c^2*x^2)*(a + b*ArcCosh[c*x]) - (d^2*(1 - c^2*x^2)^2*(a + b*ArcCosh[c*x]))/(2*x^
2) + (c^2*d^2*(a + b*ArcCosh[c*x])^2)/b - 2*c^2*d^2*(a + b*ArcCosh[c*x])*Log[1 + E^(2*ArcCosh[c*x])] - b*c^2*d
^2*PolyLog[2, -E^(2*ArcCosh[c*x])]

Rule 5729

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

Rule 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 12

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

Rule 38

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

Rule 52

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

Rule 5727

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

Rule 5660

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n/Coth[x], x], x, ArcCosh
[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 3718

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c + d*x)^(m +
1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(1 + E^(2*(-(I*e) + f*fz*x))), x],
x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{x^3} \, dx &=-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\left (2 c^2 d\right ) \int \frac{\left (d-c^2 d x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx+\frac{1}{2} \left (b c d^2\right ) \int \frac{(-1+c x)^{3/2} (1+c x)^{3/2}}{x^2} \, dx\\ &=-\frac{b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} \left (b c d^2\right ) \int 3 c^2 \sqrt{-1+c x} \sqrt{1+c x} \, dx-\left (2 c^2 d^2\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x} \, dx-\left (b c^3 d^2\right ) \int \sqrt{-1+c x} \sqrt{1+c x} \, dx\\ &=-\frac{1}{2} b c^3 d^2 x \sqrt{-1+c x} \sqrt{1+c x}-\frac{b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\left (2 c^2 d^2\right ) \operatorname{Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )+\frac{1}{2} \left (b c^3 d^2\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx+\frac{1}{2} \left (3 b c^3 d^2\right ) \int \sqrt{-1+c x} \sqrt{1+c x} \, dx\\ &=\frac{1}{4} b c^3 d^2 x \sqrt{-1+c x} \sqrt{1+c x}-\frac{b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}+\frac{1}{2} b c^2 d^2 \cosh ^{-1}(c x)-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{b}-\left (4 c^2 d^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )-\frac{1}{4} \left (3 b c^3 d^2\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{1}{4} b c^3 d^2 x \sqrt{-1+c x} \sqrt{1+c x}-\frac{b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}-\frac{1}{4} b c^2 d^2 \cosh ^{-1}(c x)-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{b}-2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )+\left (2 b c^2 d^2\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )\\ &=\frac{1}{4} b c^3 d^2 x \sqrt{-1+c x} \sqrt{1+c x}-\frac{b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}-\frac{1}{4} b c^2 d^2 \cosh ^{-1}(c x)-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{b}-2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )+\left (b c^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )\\ &=\frac{1}{4} b c^3 d^2 x \sqrt{-1+c x} \sqrt{1+c x}-\frac{b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}-\frac{1}{4} b c^2 d^2 \cosh ^{-1}(c x)-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{b}-2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )-b c^2 d^2 \text{Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )\\ \end{align*}

Mathematica [A]  time = 0.250183, size = 182, normalized size = 0.91 \[ \frac{d^2 \left (4 b c^2 x^2 \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )+2 a c^4 x^4-8 a c^2 x^2 \log (x)-2 a-b c^3 x^3 \sqrt{c x-1} \sqrt{c x+1}-4 b c^2 x^2 \cosh ^{-1}(c x)^2-2 b c^2 x^2 \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )+2 b \cosh ^{-1}(c x) \left (c^4 x^4-4 c^2 x^2 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )-1\right )+2 b c x \sqrt{c x-1} \sqrt{c x+1}\right )}{4 x^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(d^2*(-2*a + 2*a*c^4*x^4 + 2*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 4*b
*c^2*x^2*ArcCosh[c*x]^2 - 2*b*c^2*x^2*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]] + 2*b*ArcCosh[c*x]*(-1 + c^4*x^4 - 4
*c^2*x^2*Log[1 + E^(-2*ArcCosh[c*x])]) - 8*a*c^2*x^2*Log[x] + 4*b*c^2*x^2*PolyLog[2, -E^(-2*ArcCosh[c*x])]))/(
4*x^2)

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Maple [A]  time = 0.261, size = 220, normalized size = 1.1 \begin{align*}{\frac{{c}^{4}{d}^{2}a{x}^{2}}{2}}-2\,{c}^{2}{d}^{2}a\ln \left ( cx \right ) -{\frac{{d}^{2}a}{2\,{x}^{2}}}+{c}^{2}{d}^{2}b \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}+{\frac{{c}^{4}{d}^{2}b{\rm arccosh} \left (cx\right ){x}^{2}}{2}}-{\frac{b{c}^{3}{d}^{2}x}{4}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b{c}^{2}{d}^{2}{\rm arccosh} \left (cx\right )}{4}}-{\frac{{d}^{2}b{c}^{2}}{2}}+{\frac{{d}^{2}bc}{2\,x}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b{d}^{2}{\rm arccosh} \left (cx\right )}{2\,{x}^{2}}}-2\,{c}^{2}{d}^{2}b{\rm arccosh} \left (cx\right )\ln \left ( \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2}+1 \right ) -{c}^{2}{d}^{2}b{\it polylog} \left ( 2,- \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2} \right ) \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^3,x)

[Out]

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

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a c^{4} d^{2} x^{4} - 2 \, a c^{2} d^{2} x^{2} + a d^{2} +{\left (b c^{4} d^{2} x^{4} - 2 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \operatorname{arcosh}\left (c x\right )}{x^{3}}, x\right ) \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^3,x, algorithm="fricas")

[Out]

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

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

[Out]

d**2*(Integral(a/x**3, x) + Integral(-2*a*c**2/x, x) + Integral(a*c**4*x, x) + Integral(b*acosh(c*x)/x**3, x)
+ Integral(-2*b*c**2*acosh(c*x)/x, x) + Integral(b*c**4*x*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^{3}}\,{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^3,x, algorithm="giac")

[Out]

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