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

Optimal. Leaf size=177 \[ -\frac {3 \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^5 d^2}+\frac {3 x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d^2}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {3 b \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{2 c^5 d^2}+\frac {3 b \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{2 c^5 d^2}-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{2 c^5 d^2}-\frac {b x^2}{2 c^3 d^2 \sqrt {c x-1} \sqrt {c x+1}} \]

[Out]

3/2*x*(a+b*arccosh(c*x))/c^4/d^2+1/2*x^3*(a+b*arccosh(c*x))/c^2/d^2/(-c^2*x^2+1)-3*(a+b*arccosh(c*x))*arctanh(
c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c^5/d^2-3/2*b*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/c^5/d^2+3/2*b*polyl
og(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c^5/d^2-1/2*b*x^2/c^3/d^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/2*b*(c*x-1)^(1/2
)*(c*x+1)^(1/2)/c^5/d^2

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Rubi [A]  time = 0.23, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {5750, 98, 21, 74, 5766, 5694, 4182, 2279, 2391} \[ -\frac {3 b \text {PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{2 c^5 d^2}+\frac {3 b \text {PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{2 c^5 d^2}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {3 x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d^2}-\frac {3 \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^5 d^2}-\frac {b x^2}{2 c^3 d^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{2 c^5 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*x^2)/(2*c^3*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*c^5*d^2) + (3*x*(a + b
*ArcCosh[c*x]))/(2*c^4*d^2) + (x^3*(a + b*ArcCosh[c*x]))/(2*c^2*d^2*(1 - c^2*x^2)) - (3*(a + b*ArcCosh[c*x])*A
rcTanh[E^ArcCosh[c*x]])/(c^5*d^2) - (3*b*PolyLog[2, -E^ArcCosh[c*x]])/(2*c^5*d^2) + (3*b*PolyLog[2, E^ArcCosh[
c*x]])/(2*c^5*d^2)

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 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 98

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

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 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 5750

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

Rule 5766

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

Rubi steps

\begin {align*} \int \frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx &=\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b \int \frac {x^3}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 c d^2}-\frac {3 \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{2 c^2 d}\\ &=-\frac {b x^2}{2 c^3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d^2}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {b \int \frac {x (-2-2 c x)}{\sqrt {-1+c x} (1+c x)^{3/2}} \, dx}{2 c^3 d^2}-\frac {(3 b) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 c^3 d^2}-\frac {3 \int \frac {a+b \cosh ^{-1}(c x)}{d-c^2 d x^2} \, dx}{2 c^4 d}\\ &=-\frac {b x^2}{2 c^3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b \sqrt {-1+c x} \sqrt {1+c x}}{2 c^5 d^2}+\frac {3 x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d^2}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {3 \operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^5 d^2}+\frac {b \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{c^3 d^2}\\ &=-\frac {b x^2}{2 c^3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{2 c^5 d^2}+\frac {3 x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d^2}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^5 d^2}-\frac {(3 b) \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^5 d^2}+\frac {(3 b) \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^5 d^2}\\ &=-\frac {b x^2}{2 c^3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{2 c^5 d^2}+\frac {3 x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d^2}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^5 d^2}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 c^5 d^2}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 c^5 d^2}\\ &=-\frac {b x^2}{2 c^3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{2 c^5 d^2}+\frac {3 x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d^2}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^5 d^2}-\frac {3 b \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{2 c^5 d^2}+\frac {3 b \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{2 c^5 d^2}\\ \end {align*}

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Mathematica [A]  time = 1.09, size = 244, normalized size = 1.38 \[ \frac {-\frac {2 a c x}{c^2 x^2-1}+4 a c x+3 a \log (1-c x)-3 a \log (c x+1)-6 b \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )+6 b \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )-4 b c x \sqrt {\frac {c x-1}{c x+1}}+\frac {b c x \sqrt {\frac {c x-1}{c x+1}}}{1-c x}+\frac {b \sqrt {\frac {c x-1}{c x+1}}}{1-c x}-3 b \sqrt {\frac {c x-1}{c x+1}}+4 b c x \cosh ^{-1}(c x)+\frac {b \cosh ^{-1}(c x)}{1-c x}-\frac {b \cosh ^{-1}(c x)}{c x+1}+6 b \cosh ^{-1}(c x) \log \left (1-e^{\cosh ^{-1}(c x)}\right )-6 b \cosh ^{-1}(c x) \log \left (e^{\cosh ^{-1}(c x)}+1\right )}{4 c^5 d^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(4*a*c*x - 3*b*Sqrt[(-1 + c*x)/(1 + c*x)] - 4*b*c*x*Sqrt[(-1 + c*x)/(1 + c*x)] + (b*Sqrt[(-1 + c*x)/(1 + c*x)]
)/(1 - c*x) + (b*c*x*Sqrt[(-1 + c*x)/(1 + c*x)])/(1 - c*x) - (2*a*c*x)/(-1 + c^2*x^2) + 4*b*c*x*ArcCosh[c*x] +
(b*ArcCosh[c*x])/(1 - c*x) - (b*ArcCosh[c*x])/(1 + c*x) + 6*b*ArcCosh[c*x]*Log[1 - E^ArcCosh[c*x]] - 6*b*ArcC
osh[c*x]*Log[1 + E^ArcCosh[c*x]] + 3*a*Log[1 - c*x] - 3*a*Log[1 + c*x] - 6*b*PolyLog[2, -E^ArcCosh[c*x]] + 6*b
*PolyLog[2, E^ArcCosh[c*x]])/(4*c^5*d^2)

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fricas [F]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{4} \operatorname {arcosh}\left (c x\right ) + a x^{4}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.61, size = 300, normalized size = 1.69 \[ \frac {a x}{c^{4} d^{2}}-\frac {a}{4 c^{5} d^{2} \left (c x -1\right )}+\frac {3 a \ln \left (c x -1\right )}{4 c^{5} d^{2}}-\frac {a}{4 c^{5} d^{2} \left (c x +1\right )}-\frac {3 a \ln \left (c x +1\right )}{4 c^{5} d^{2}}+\frac {b \,\mathrm {arccosh}\left (c x \right ) x}{c^{4} d^{2}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{c^{5} d^{2}}-\frac {b \,\mathrm {arccosh}\left (c x \right ) x}{2 c^{4} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {c x +1}\, \sqrt {c x -1}}{2 c^{5} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {3 b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 c^{5} d^{2}}-\frac {3 b \polylog \left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 c^{5} d^{2}}+\frac {3 b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 c^{5} d^{2}}+\frac {3 b \polylog \left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 c^{5} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^4*a/d^2*x-1/4/c^5*a/d^2/(c*x-1)+3/4/c^5*a/d^2*ln(c*x-1)-1/4/c^5*a/d^2/(c*x+1)-3/4/c^5*a/d^2*ln(c*x+1)+1/c^
4*b/d^2*arccosh(c*x)*x-b*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5/d^2-1/2/c^4*b/d^2/(c^2*x^2-1)*arccosh(c*x)*x-1/2/c^5*
b/d^2/(c^2*x^2-1)*(c*x+1)^(1/2)*(c*x-1)^(1/2)-3/2/c^5*b/d^2*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))
-3/2*b*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/c^5/d^2+3/2/c^5*b/d^2*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(
c*x+1)^(1/2))+3/2*b*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c^5/d^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{64} \, {\left (16 \, c^{4} {\left (\frac {2 \, x}{c^{10} d^{2} x^{2} - c^{8} d^{2}} - \frac {4 \, x}{c^{8} d^{2}} + \frac {3 \, \log \left (c x + 1\right )}{c^{9} d^{2}} - \frac {3 \, \log \left (c x - 1\right )}{c^{9} d^{2}}\right )} - 576 \, c^{3} \int \frac {x^{3} \log \left (c x - 1\right )}{8 \, {\left (c^{8} d^{2} x^{4} - 2 \, c^{6} d^{2} x^{2} + c^{4} d^{2}\right )}}\,{d x} - 24 \, c^{2} {\left (\frac {2 \, x}{c^{8} d^{2} x^{2} - c^{6} d^{2}} + \frac {\log \left (c x + 1\right )}{c^{7} d^{2}} - \frac {\log \left (c x - 1\right )}{c^{7} d^{2}}\right )} + 192 \, c^{2} \int \frac {x^{2} \log \left (c x - 1\right )}{8 \, {\left (c^{8} d^{2} x^{4} - 2 \, c^{6} d^{2} x^{2} + c^{4} d^{2}\right )}}\,{d x} - 9 \, {\left (c {\left (\frac {2}{c^{8} d^{2} x - c^{7} d^{2}} - \frac {\log \left (c x + 1\right )}{c^{7} d^{2}} + \frac {\log \left (c x - 1\right )}{c^{7} d^{2}}\right )} + \frac {4 \, \log \left (c x - 1\right )}{c^{8} d^{2} x^{2} - c^{6} d^{2}}\right )} c + \frac {4 \, {\left (3 \, {\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right )^{2} + 6 \, {\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) \log \left (c x - 1\right ) + 4 \, {\left (4 \, c^{3} x^{3} - 6 \, c x - 3 \, {\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) + 3 \, {\left (c^{2} x^{2} - 1\right )} \log \left (c x - 1\right )\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )\right )}}{c^{7} d^{2} x^{2} - c^{5} d^{2}} - 64 \, \int -\frac {4 \, c^{3} x^{3} - 6 \, c x - 3 \, {\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) + 3 \, {\left (c^{2} x^{2} - 1\right )} \log \left (c x - 1\right )}{4 \, {\left (c^{9} d^{2} x^{5} - 2 \, c^{7} d^{2} x^{3} + c^{5} d^{2} x + {\left (c^{8} d^{2} x^{4} - 2 \, c^{6} d^{2} x^{2} + c^{4} d^{2}\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )}}\,{d x} - 192 \, \int \frac {\log \left (c x - 1\right )}{8 \, {\left (c^{8} d^{2} x^{4} - 2 \, c^{6} d^{2} x^{2} + c^{4} d^{2}\right )}}\,{d x}\right )} b - \frac {1}{4} \, a {\left (\frac {2 \, x}{c^{6} d^{2} x^{2} - c^{4} d^{2}} - \frac {4 \, x}{c^{4} d^{2}} + \frac {3 \, \log \left (c x + 1\right )}{c^{5} d^{2}} - \frac {3 \, \log \left (c x - 1\right )}{c^{5} d^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(16*c^4*(2*x/(c^10*d^2*x^2 - c^8*d^2) - 4*x/(c^8*d^2) + 3*log(c*x + 1)/(c^9*d^2) - 3*log(c*x - 1)/(c^9*d^
2)) - 576*c^3*integrate(1/8*x^3*log(c*x - 1)/(c^8*d^2*x^4 - 2*c^6*d^2*x^2 + c^4*d^2), x) - 24*c^2*(2*x/(c^8*d^
2*x^2 - c^6*d^2) + log(c*x + 1)/(c^7*d^2) - log(c*x - 1)/(c^7*d^2)) + 192*c^2*integrate(1/8*x^2*log(c*x - 1)/(
c^8*d^2*x^4 - 2*c^6*d^2*x^2 + c^4*d^2), x) - 9*(c*(2/(c^8*d^2*x - c^7*d^2) - log(c*x + 1)/(c^7*d^2) + log(c*x
- 1)/(c^7*d^2)) + 4*log(c*x - 1)/(c^8*d^2*x^2 - c^6*d^2))*c + 4*(3*(c^2*x^2 - 1)*log(c*x + 1)^2 + 6*(c^2*x^2 -
1)*log(c*x + 1)*log(c*x - 1) + 4*(4*c^3*x^3 - 6*c*x - 3*(c^2*x^2 - 1)*log(c*x + 1) + 3*(c^2*x^2 - 1)*log(c*x
- 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^7*d^2*x^2 - c^5*d^2) - 64*integrate(-1/4*(4*c^3*x^3 - 6*c*x -
3*(c^2*x^2 - 1)*log(c*x + 1) + 3*(c^2*x^2 - 1)*log(c*x - 1))/(c^9*d^2*x^5 - 2*c^7*d^2*x^3 + c^5*d^2*x + (c^8*
d^2*x^4 - 2*c^6*d^2*x^2 + c^4*d^2)*sqrt(c*x + 1)*sqrt(c*x - 1)), x) - 192*integrate(1/8*log(c*x - 1)/(c^8*d^2*
x^4 - 2*c^6*d^2*x^2 + c^4*d^2), x))*b - 1/4*a*(2*x/(c^6*d^2*x^2 - c^4*d^2) - 4*x/(c^4*d^2) + 3*log(c*x + 1)/(c
^5*d^2) - 3*log(c*x - 1)/(c^5*d^2))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a x^{4}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b x^{4} \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(a*x**4/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(b*x**4*acosh(c*x)/(c**4*x**4 - 2*c**2*x**2 + 1),
x))/d**2

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