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

Optimal. Leaf size=118 \[ \frac{b c^2 \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{2 d}-\frac{b c^2 \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{2 d}+\frac{2 c^2 \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d}-\frac{a+b \cosh ^{-1}(c x)}{2 d x^2}+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{2 d x} \]

[Out]

(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*d*x) - (a + b*ArcCosh[c*x])/(2*d*x^2) + (2*c^2*(a + b*ArcCosh[c*x])*ArcT
anh[E^(2*ArcCosh[c*x])])/d + (b*c^2*PolyLog[2, -E^(2*ArcCosh[c*x])])/(2*d) - (b*c^2*PolyLog[2, E^(2*ArcCosh[c*
x])])/(2*d)

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Rubi [A]  time = 0.19727, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {5746, 95, 5721, 5461, 4182, 2279, 2391} \[ \frac{b c^2 \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{2 d}-\frac{b c^2 \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{2 d}+\frac{2 c^2 \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d}-\frac{a+b \cosh ^{-1}(c x)}{2 d x^2}+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{2 d x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*d*x) - (a + b*ArcCosh[c*x])/(2*d*x^2) + (2*c^2*(a + b*ArcCosh[c*x])*ArcT
anh[E^(2*ArcCosh[c*x])])/d + (b*c^2*PolyLog[2, -E^(2*ArcCosh[c*x])])/(2*d) - (b*c^2*PolyLog[2, E^(2*ArcCosh[c*
x])])/(2*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 95

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] /; FreeQ[{a, b, c, d,
e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0
] && NeQ[m, -1]

Rule 5721

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

Rule 5461

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]

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

Mathematica [A]  time = 0.527178, size = 144, normalized size = 1.22 \[ -\frac{b c^2 \left (\text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )-\text{PolyLog}\left (2,e^{-2 \cosh ^{-1}(c x)}\right )+\frac{\cosh ^{-1}(c x)}{c^2 x^2}-\frac{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}{c x}+2 \cosh ^{-1}(c x) \log \left (1-e^{-2 \cosh ^{-1}(c x)}\right )-2 \cosh ^{-1}(c x) \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )\right )+a c^2 \log \left (1-c^2 x^2\right )-2 a c^2 \log (x)+\frac{a}{x^2}}{2 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.092, size = 301, normalized size = 2.6 \begin{align*} -{\frac{{c}^{2}a\ln \left ( cx-1 \right ) }{2\,d}}-{\frac{a}{2\,d{x}^{2}}}+{\frac{{c}^{2}a\ln \left ( cx \right ) }{d}}-{\frac{{c}^{2}a\ln \left ( cx+1 \right ) }{2\,d}}+{\frac{bc}{2\,dx}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{{c}^{2}b}{2\,d}}-{\frac{b{\rm arccosh} \left (cx\right )}{2\,d{x}^{2}}}+{\frac{{c}^{2}b{\rm arccosh} \left (cx\right )}{d}\ln \left ( \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2}+1 \right ) }+{\frac{{c}^{2}b}{2\,d}{\it polylog} \left ( 2,- \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2} \right ) }-{\frac{{c}^{2}b{\rm arccosh} \left (cx\right )}{d}\ln \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }-{\frac{{c}^{2}b}{d}{\it polylog} \left ( 2,-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) }-{\frac{{c}^{2}b{\rm arccosh} \left (cx\right )}{d}\ln \left ( 1-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) }-{\frac{{c}^{2}b}{d}{\it polylog} \left ( 2,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^3/(-c^2*d*x^2+d),x)

[Out]

-1/2*c^2*a/d*ln(c*x-1)-1/2*a/d/x^2+c^2*a/d*ln(c*x)-1/2*c^2*a/d*ln(c*x+1)+1/2*b*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d
/x-1/2*c^2*b/d-1/2*b/d*arccosh(c*x)/x^2+c^2*b/d*arccosh(c*x)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2+1)+1/2*b*c
^2*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/d-c^2*b/d*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)
)-c^2*b/d*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))-c^2*b/d*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2
))-c^2*b/d*polylog(2,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}{2} \,{\left (\frac{c^{2} \log \left (c x + 1\right )}{d} + \frac{c^{2} \log \left (c x - 1\right )}{d} - \frac{2 \, c^{2} \log \left (x\right )}{d} + \frac{1}{d x^{2}}\right )} a - b \int \frac{\log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{c^{2} d x^{5} - d x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(c^2*log(c*x + 1)/d + c^2*log(c*x - 1)/d - 2*c^2*log(x)/d + 1/(d*x^2))*a - b*integrate(log(c*x + sqrt(c*x
+ 1)*sqrt(c*x - 1))/(c^2*d*x^5 - d*x^3), x)

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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^{5} - d x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(b*arccosh(c*x) + a)/(c^2*d*x^5 - d*x^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{a}{c^{2} x^{5} - x^{3}}\, dx + \int \frac{b \operatorname{acosh}{\left (c x \right )}}{c^{2} x^{5} - x^{3}}\, dx}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(Integral(a/(c**2*x**5 - x**3), x) + Integral(b*acosh(c*x)/(c**2*x**5 - x**3), 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^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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