3.36 \(\int \frac{a+b \cosh ^{-1}(c x)}{x^4 (d-c^2 d x^2)} \, dx\)

Optimal. Leaf size=157 \[ \frac{b c^3 \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{d}-\frac{b c^3 \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{d}-\frac{c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d x}+\frac{2 c^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d}-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3}+\frac{7 b c^3 \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )}{6 d}+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{6 d x^2} \]

[Out]

(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*d*x^2) - (a + b*ArcCosh[c*x])/(3*d*x^3) - (c^2*(a + b*ArcCosh[c*x]))/(d*
x) + (7*b*c^3*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(6*d) + (2*c^3*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]
])/d + (b*c^3*PolyLog[2, -E^ArcCosh[c*x]])/d - (b*c^3*PolyLog[2, E^ArcCosh[c*x]])/d

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Rubi [A]  time = 0.234406, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {5746, 103, 12, 92, 205, 5694, 4182, 2279, 2391} \[ \frac{b c^3 \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{d}-\frac{b c^3 \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{d}-\frac{c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d x}+\frac{2 c^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d}-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3}+\frac{7 b c^3 \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )}{6 d}+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{6 d x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*d*x^2) - (a + b*ArcCosh[c*x])/(3*d*x^3) - (c^2*(a + b*ArcCosh[c*x]))/(d*
x) + (7*b*c^3*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(6*d) + (2*c^3*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]
])/d + (b*c^3*PolyLog[2, -E^ArcCosh[c*x]])/d - (b*c^3*PolyLog[2, E^ArcCosh[c*x]])/d

Rule 5746

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
((f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(d*f*(m + 1)), x] + (Dist[(b*c*n*(-d)^p)/(f*(m + 1)
), Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] + Dist[(c^2
*(m + 2*p + 3))/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x]) /; FreeQ[{a, b,
c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegerQ[m] && IntegerQ[p]

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 12

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

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

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]

Rule 5694

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

Rule 4182

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*Ar
cTanh[E^(-(I*e) + f*fz*x)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 - E^(-(I*e) + f*
fz*x)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e) + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, 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{a+b \cosh ^{-1}(c x)}{x^4 \left (d-c^2 d x^2\right )} \, dx &=-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3}+c^2 \int \frac{a+b \cosh ^{-1}(c x)}{x^2 \left (d-c^2 d x^2\right )} \, dx+\frac{(b c) \int \frac{1}{x^3 \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 d}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{6 d x^2}-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3}-\frac{c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d x}+c^4 \int \frac{a+b \cosh ^{-1}(c x)}{d-c^2 d x^2} \, dx+\frac{(b c) \int \frac{c^2}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{6 d}+\frac{\left (b c^3\right ) \int \frac{1}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{d}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{6 d x^2}-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3}-\frac{c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d x}-\frac{c^3 \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{d}+\frac{\left (b c^3\right ) \int \frac{1}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{6 d}+\frac{\left (b c^4\right ) \operatorname{Subst}\left (\int \frac{1}{c+c x^2} \, dx,x,\sqrt{-1+c x} \sqrt{1+c x}\right )}{d}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{6 d x^2}-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3}-\frac{c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d x}+\frac{b c^3 \tan ^{-1}\left (\sqrt{-1+c x} \sqrt{1+c x}\right )}{d}+\frac{2 c^3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d}+\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d}-\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d}+\frac{\left (b c^4\right ) \operatorname{Subst}\left (\int \frac{1}{c+c x^2} \, dx,x,\sqrt{-1+c x} \sqrt{1+c x}\right )}{6 d}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{6 d x^2}-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3}-\frac{c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d x}+\frac{7 b c^3 \tan ^{-1}\left (\sqrt{-1+c x} \sqrt{1+c x}\right )}{6 d}+\frac{2 c^3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d}+\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d}-\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{6 d x^2}-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3}-\frac{c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d x}+\frac{7 b c^3 \tan ^{-1}\left (\sqrt{-1+c x} \sqrt{1+c x}\right )}{6 d}+\frac{2 c^3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d}+\frac{b c^3 \text{Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{d}-\frac{b c^3 \text{Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{d}\\ \end{align*}

Mathematica [A]  time = 0.342144, size = 223, normalized size = 1.42 \[ \frac{6 b c^3 \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )-6 b c^3 \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )-\frac{6 a c^2}{x}-6 a c^3 \log \left (1-e^{\cosh ^{-1}(c x)}\right )+6 a c^3 \log \left (e^{\cosh ^{-1}(c x)}+1\right )-\frac{2 a}{x^3}+\frac{7 b c^3 \sqrt{c^2 x^2-1} \tan ^{-1}\left (\sqrt{c^2 x^2-1}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{6 b c^2 \cosh ^{-1}(c x)}{x}-6 b c^3 \cosh ^{-1}(c x) \log \left (1-e^{\cosh ^{-1}(c x)}\right )+6 b c^3 \cosh ^{-1}(c x) \log \left (e^{\cosh ^{-1}(c x)}+1\right )+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{x^2}-\frac{2 b \cosh ^{-1}(c x)}{x^3}}{6 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-2*a)/x^3 - (6*a*c^2)/x + (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/x^2 - (2*b*ArcCosh[c*x])/x^3 - (6*b*c^2*ArcCosh
[c*x])/x + (7*b*c^3*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - 6*a*c^3*Lo
g[1 - E^ArcCosh[c*x]] - 6*b*c^3*ArcCosh[c*x]*Log[1 - E^ArcCosh[c*x]] + 6*a*c^3*Log[1 + E^ArcCosh[c*x]] + 6*b*c
^3*ArcCosh[c*x]*Log[1 + E^ArcCosh[c*x]] + 6*b*c^3*PolyLog[2, -E^ArcCosh[c*x]] - 6*b*c^3*PolyLog[2, E^ArcCosh[c
*x]])/(6*d)

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Maple [A]  time = 0.126, size = 225, normalized size = 1.4 \begin{align*} -{\frac{{c}^{3}a\ln \left ( cx-1 \right ) }{2\,d}}-{\frac{a}{3\,d{x}^{3}}}-{\frac{{c}^{2}a}{dx}}+{\frac{{c}^{3}a\ln \left ( cx+1 \right ) }{2\,d}}-{\frac{{c}^{2}b{\rm arccosh} \left (cx\right )}{dx}}+{\frac{bc}{6\,d{x}^{2}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b{\rm arccosh} \left (cx\right )}{3\,d{x}^{3}}}+{\frac{7\,b{c}^{3}}{3\,d}\arctan \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{b{c}^{3}}{d}{\it dilog} \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{b{c}^{3}}{d}{\it dilog} \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{b{c}^{3}{\rm arccosh} \left (cx\right )}{d}\ln \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*c^3*a/d*ln(c*x-1)-1/3*a/d/x^3-c^2*a/d/x+1/2*c^3*a/d*ln(c*x+1)-c^2*b/d*arccosh(c*x)/x+1/6*b*c*(c*x-1)^(1/2
)*(c*x+1)^(1/2)/d/x^2-1/3*b/d*arccosh(c*x)/x^3+7/3*c^3*b/d*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+c^3*b/d*dil
og(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+c^3*b/d*dilog(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+c^3*b/d*arccosh(c*x)*ln(1
+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{6} \,{\left (\frac{3 \, c^{3} \log \left (c x + 1\right )}{d} - \frac{3 \, c^{3} \log \left (c x - 1\right )}{d} - \frac{2 \,{\left (3 \, c^{2} x^{2} + 1\right )}}{d x^{3}}\right )} a + \frac{1}{24} \,{\left (216 \, c^{5} \int \frac{x^{3} \log \left (c x - 1\right )}{12 \,{\left (c^{2} d x^{4} - d x^{2}\right )}}\,{d x} - 12 \, c^{4}{\left (\frac{\log \left (c x + 1\right )}{c d} - \frac{\log \left (c x - 1\right )}{c d}\right )} - 72 \, c^{4} \int \frac{x^{2} \log \left (c x - 1\right )}{12 \,{\left (c^{2} d x^{4} - d x^{2}\right )}}\,{d x} - 4 \, c^{2}{\left (\frac{c \log \left (c x + 1\right )}{d} - \frac{c \log \left (c x - 1\right )}{d} - \frac{2}{d x}\right )} - \frac{3 \, c^{3} x^{3} \log \left (c x + 1\right )^{2} + 6 \, c^{3} x^{3} \log \left (c x + 1\right ) \log \left (c x - 1\right ) - 4 \,{\left (3 \, c^{3} x^{3} \log \left (c x + 1\right ) - 3 \, c^{3} x^{3} \log \left (c x - 1\right ) - 6 \, c^{2} x^{2} - 2\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{d x^{3}} + 24 \, \int \frac{3 \, c^{4} x^{3} \log \left (c x + 1\right ) - 3 \, c^{4} x^{3} \log \left (c x - 1\right ) - 6 \, c^{3} x^{2} - 2 \, c}{6 \,{\left (c^{3} d x^{6} - c d x^{4} +{\left (c^{2} d x^{5} - d x^{3}\right )} \sqrt{c x + 1} \sqrt{c x - 1}\right )}}\,{d x}\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d),x, algorithm="maxima")

[Out]

1/6*(3*c^3*log(c*x + 1)/d - 3*c^3*log(c*x - 1)/d - 2*(3*c^2*x^2 + 1)/(d*x^3))*a + 1/24*(216*c^5*integrate(1/12
*x^3*log(c*x - 1)/(c^2*d*x^4 - d*x^2), x) - 12*c^4*(log(c*x + 1)/(c*d) - log(c*x - 1)/(c*d)) - 72*c^4*integrat
e(1/12*x^2*log(c*x - 1)/(c^2*d*x^4 - d*x^2), x) - 4*c^2*(c*log(c*x + 1)/d - c*log(c*x - 1)/d - 2/(d*x)) - (3*c
^3*x^3*log(c*x + 1)^2 + 6*c^3*x^3*log(c*x + 1)*log(c*x - 1) - 4*(3*c^3*x^3*log(c*x + 1) - 3*c^3*x^3*log(c*x -
1) - 6*c^2*x^2 - 2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)))/(d*x^3) + 24*integrate(1/6*(3*c^4*x^3*log(c*x + 1)
- 3*c^4*x^3*log(c*x - 1) - 6*c^3*x^2 - 2*c)/(c^3*d*x^6 - c*d*x^4 + (c^2*d*x^5 - d*x^3)*sqrt(c*x + 1)*sqrt(c*x
- 1)), x))*b

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{b \operatorname{arcosh}\left (c x\right ) + a}{c^{2} d x^{6} - d x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d),x, algorithm="fricas")

[Out]

integral(-(b*arccosh(c*x) + a)/(c^2*d*x^6 - d*x^4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d),x, algorithm="giac")

[Out]

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