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

Optimal. Leaf size=170 \[ \frac{3 b c \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{2 d^2}-\frac{3 b c \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{2 d^2}+\frac{3 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac{a+b \cosh ^{-1}(c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac{3 c \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2}-\frac{b c}{2 d^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )}{d^2} \]

[Out]

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

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Rubi [A]  time = 0.184093, antiderivative size = 170, 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 = {5746, 104, 21, 92, 205, 5689, 74, 5694, 4182, 2279, 2391} \[ \frac{3 b c \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{2 d^2}-\frac{3 b c \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{2 d^2}+\frac{3 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac{a+b \cosh ^{-1}(c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac{3 c \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2}-\frac{b c}{2 d^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )}{d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 104

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] &&
IntegersQ[2*m, 2*n, 2*p]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + 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 5689

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

Rule 74

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

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

Mathematica [A]  time = 0.744171, size = 283, normalized size = 1.66 \[ \frac{6 b c \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )-6 b c \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )-\frac{2 a c^2 x}{c^2 x^2-1}-3 a c \log (1-c x)+3 a c \log (c x+1)-\frac{4 a}{x}+\frac{4 b c \sqrt{c^2 x^2-1} \tan ^{-1}\left (\sqrt{c^2 x^2-1}\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^2 x \sqrt{\frac{c x-1}{c x+1}}}{1-c x}+b c \sqrt{\frac{c x-1}{c x+1}}+\frac{b c \sqrt{\frac{c x-1}{c x+1}}}{1-c x}+\frac{b c \cosh ^{-1}(c x)}{1-c x}-\frac{b c \cosh ^{-1}(c x)}{c x+1}-\frac{4 b \cosh ^{-1}(c x)}{x}-6 b c \cosh ^{-1}(c x) \log \left (1-e^{\cosh ^{-1}(c x)}\right )+6 b c \cosh ^{-1}(c x) \log \left (e^{\cosh ^{-1}(c x)}+1\right )}{4 d^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-4*a)/x + b*c*Sqrt[(-1 + c*x)/(1 + c*x)] + (b*c*Sqrt[(-1 + c*x)/(1 + c*x)])/(1 - c*x) + (b*c^2*x*Sqrt[(-1 +
c*x)/(1 + c*x)])/(1 - c*x) - (2*a*c^2*x)/(-1 + c^2*x^2) - (4*b*ArcCosh[c*x])/x + (b*c*ArcCosh[c*x])/(1 - c*x)
- (b*c*ArcCosh[c*x])/(1 + c*x) + (4*b*c*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1
+ c*x]) - 6*b*c*ArcCosh[c*x]*Log[1 - E^ArcCosh[c*x]] + 6*b*c*ArcCosh[c*x]*Log[1 + E^ArcCosh[c*x]] - 3*a*c*Log[
1 - c*x] + 3*a*c*Log[1 + c*x] + 6*b*c*PolyLog[2, -E^ArcCosh[c*x]] - 6*b*c*PolyLog[2, E^ArcCosh[c*x]])/(4*d^2)

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Maple [A]  time = 0.139, size = 259, normalized size = 1.5 \begin{align*} -{\frac{ca}{4\,{d}^{2} \left ( cx-1 \right ) }}-{\frac{3\,ca\ln \left ( cx-1 \right ) }{4\,{d}^{2}}}-{\frac{a}{{d}^{2}x}}-{\frac{ca}{4\,{d}^{2} \left ( cx+1 \right ) }}+{\frac{3\,ca\ln \left ( cx+1 \right ) }{4\,{d}^{2}}}-{\frac{3\,b{\rm arccosh} \left (cx\right ){c}^{2}x}{2\,{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{bc}{2\,{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}}+{\frac{b{\rm arccosh} \left (cx\right )}{{d}^{2}x \left ({c}^{2}{x}^{2}-1 \right ) }}+2\,{\frac{bc\arctan \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }{{d}^{2}}}+{\frac{3\,bc}{2\,{d}^{2}}{\it dilog} \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{3\,bc}{2\,{d}^{2}}{\it dilog} \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{3\,bc{\rm arccosh} \left (cx\right )}{2\,{d}^{2}}\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^2/(-c^2*d*x^2+d)^2,x)

[Out]

-1/4*c*a/d^2/(c*x-1)-3/4*c*a/d^2*ln(c*x-1)-a/d^2/x-1/4*c*a/d^2/(c*x+1)+3/4*c*a/d^2*ln(c*x+1)-3/2*b/d^2/(c^2*x^
2-1)*arccosh(c*x)*c^2*x-1/2*c*b/d^2/(c^2*x^2-1)*(c*x+1)^(1/2)*(c*x-1)^(1/2)+b/d^2/x/(c^2*x^2-1)*arccosh(c*x)+2
*c*b/d^2*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+3/2*c*b/d^2*dilog(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+3/2*c*b/d^
2*dilog(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+3/2*c*b/d^2*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*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(576*c^5*integrate(1/8*x^3*log(c*x - 1)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x) - 24*c^4*(2*x/(c^4*d^2*x^
2 - c^2*d^2) + log(c*x + 1)/(c^3*d^2) - log(c*x - 1)/(c^3*d^2)) - 192*c^4*integrate(1/8*x^2*log(c*x - 1)/(c^4*
d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x) + 9*(c*(2/(c^4*d^2*x - c^3*d^2) - log(c*x + 1)/(c^3*d^2) + log(c*x - 1)/(c^
3*d^2)) + 4*log(c*x - 1)/(c^4*d^2*x^2 - c^2*d^2))*c^3 + 16*c^2*(2*x/(c^2*d^2*x^2 - d^2) - log(c*x + 1)/(c*d^2)
+ log(c*x - 1)/(c*d^2)) + 192*c^2*integrate(1/8*log(c*x - 1)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x) - 4*(3*(
c^3*x^3 - c*x)*log(c*x + 1)^2 + 6*(c^3*x^3 - c*x)*log(c*x + 1)*log(c*x - 1) + 4*(6*c^2*x^2 - 3*(c^3*x^3 - c*x)
*log(c*x + 1) + 3*(c^3*x^3 - c*x)*log(c*x - 1) - 4)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^2*d^2*x^3 - d^2
*x) + 64*integrate(-1/4*(6*c^3*x^2 - 3*(c^4*x^3 - c^2*x)*log(c*x + 1) + 3*(c^4*x^3 - c^2*x)*log(c*x - 1) - 4*c
)/(c^5*d^2*x^6 - 2*c^3*d^2*x^4 + c*d^2*x^2 + (c^4*d^2*x^5 - 2*c^2*d^2*x^3 + d^2*x)*sqrt(c*x + 1)*sqrt(c*x - 1)
), x))*b - 1/4*a*(2*(3*c^2*x^2 - 2)/(c^2*d^2*x^3 - d^2*x) - 3*c*log(c*x + 1)/d^2 + 3*c*log(c*x - 1)/d^2)

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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^{4} d^{2} x^{6} - 2 \, c^{2} d^{2} x^{4} + d^{2} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )}^{2} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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