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

Optimal. Leaf size=152 \[ \frac {c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \left (1-c^2 x^2\right )}-\frac {a+b \cosh ^{-1}(c x)}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\frac {4 c^2 \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2}+\frac {b c^2 \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{d^2}-\frac {b c^2 \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{d^2}-\frac {b c}{2 d^2 x \sqrt {c x-1} \sqrt {c x+1}} \]

[Out]

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

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Rubi [A]  time = 0.26, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5746, 103, 12, 39, 5754, 5721, 5461, 4182, 2279, 2391} \[ \frac {b c^2 \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{d^2}-\frac {b c^2 \text {PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{d^2}+\frac {c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \left (1-c^2 x^2\right )}-\frac {a+b \cosh ^{-1}(c x)}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\frac {4 c^2 \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2}-\frac {b c}{2 d^2 x \sqrt {c x-1} \sqrt {c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

Rule 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]

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 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]

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 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 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 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 5754

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)/(2*d*f*(p + 1)), x] + (Dist[(m + 2*p + 3)/(2*d*(p
+ 1)), Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[(b*c*n*(-d)^p)/(2*f*(p + 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]) /; FreeQ[{a, b,
c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] &&  !GtQ[m, 1] && IntegerQ[p]

Rubi steps

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

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Mathematica [B]  time = 0.60, size = 319, normalized size = 2.10 \[ \frac {4 a c^4 x^4 \log (x)-2 a c^2 x^2-4 a c^2 x^2 \log (x)+2 a c^2 x^2 \log \left (1-c^2 x^2\right )-2 a c^4 x^4 \log \left (1-c^2 x^2\right )+a-4 b c^4 x^4 \cosh ^{-1}(c x) \log \left (1-e^{-2 \cosh ^{-1}(c x)}\right )+4 b c^4 x^4 \cosh ^{-1}(c x) \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )-2 b c^2 x^2 \left (c^2 x^2-1\right ) \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right )+2 b c^2 x^2 \left (c^2 x^2-1\right ) \text {Li}_2\left (e^{-2 \cosh ^{-1}(c x)}\right )-b c^2 x^2 \sqrt {\frac {c x-1}{c x+1}}-2 b c^2 x^2 \cosh ^{-1}(c x)+4 b c^2 x^2 \cosh ^{-1}(c x) \log \left (1-e^{-2 \cosh ^{-1}(c x)}\right )-4 b c^2 x^2 \cosh ^{-1}(c x) \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )-b c x \sqrt {\frac {c x-1}{c x+1}}+b \cosh ^{-1}(c x)}{2 d^2 x^2 \left (c^2 x^2-1\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [F]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arcosh}\left (c x\right ) + a}{c^{4} d^{2} x^{7} - 2 \, c^{2} d^{2} x^{5} + d^{2} x^{3}}, x\right ) \]

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)^2,x, algorithm="fricas")

[Out]

integral((b*arccosh(c*x) + a)/(c^4*d^2*x^7 - 2*c^2*d^2*x^5 + d^2*x^3), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2} x^{3}}\,{d x} \]

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)^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.34, size = 371, normalized size = 2.44 \[ -\frac {a}{2 d^{2} x^{2}}+\frac {2 c^{2} a \ln \left (c x \right )}{d^{2}}-\frac {c^{2} a}{4 d^{2} \left (c x -1\right )}-\frac {c^{2} a \ln \left (c x -1\right )}{d^{2}}+\frac {c^{2} a}{4 d^{2} \left (c x +1\right )}-\frac {c^{2} a \ln \left (c x +1\right )}{d^{2}}-\frac {c^{2} b \,\mathrm {arccosh}\left (c x \right )}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {c b \sqrt {c x +1}\, \sqrt {c x -1}}{2 d^{2} x \left (c^{2} x^{2}-1\right )}+\frac {b \,\mathrm {arccosh}\left (c x \right )}{2 d^{2} x^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 c^{2} b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{d^{2}}+\frac {b \,c^{2} \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{d^{2}}-\frac {2 c^{2} b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2}}-\frac {2 c^{2} b \polylog \left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2}}-\frac {2 c^{2} b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2}}-\frac {2 c^{2} b \polylog \left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2}} \]

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)^2,x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, a {\left (\frac {2 \, c^{2} \log \left (c x + 1\right )}{d^{2}} + \frac {2 \, c^{2} \log \left (c x - 1\right )}{d^{2}} - \frac {4 \, c^{2} \log \relax (x)}{d^{2}} + \frac {2 \, c^{2} x^{2} - 1}{c^{2} d^{2} x^{4} - d^{2} x^{2}}\right )} + b \int \frac {\log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{c^{4} d^{2} x^{7} - 2 \, c^{2} d^{2} x^{5} + d^{2} x^{3}}\,{d x} \]

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)^2,x, algorithm="maxima")

[Out]

-1/2*a*(2*c^2*log(c*x + 1)/d^2 + 2*c^2*log(c*x - 1)/d^2 - 4*c^2*log(x)/d^2 + (2*c^2*x^2 - 1)/(c^2*d^2*x^4 - d^
2*x^2)) + b*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(c^4*d^2*x^7 - 2*c^2*d^2*x^5 + d^2*x^3), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^3\,{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a}{c^{4} x^{7} - 2 c^{2} x^{5} + x^{3}}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{7} - 2 c^{2} x^{5} + x^{3}}\, dx}{d^{2}} \]

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)**2,x)

[Out]

(Integral(a/(c**4*x**7 - 2*c**2*x**5 + x**3), x) + Integral(b*acosh(c*x)/(c**4*x**7 - 2*c**2*x**5 + x**3), x))
/d**2

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