3.1.37 \(\int \frac {x^4 (a+b \cosh ^{-1}(c x))}{(d-c^2 d x^2)^2} \, dx\) [37]

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

[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 = 0.16, 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 = {5934, 5938, 5903, 4267, 2317, 2438, 75, 100, 21} \begin {gather*} -\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(b*x^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])
*ArcTanh[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^ArcCos
h[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 75

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 100

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 2317

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 2438

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 4267

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 5903

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 5934

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

Rule 5938

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[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] - Dist[b*f*(n/(c*(m + 2*p + 1))
)*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], 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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && I
GtQ[m, 1] && NeQ[m + 2*p + 1, 0]

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 \text {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) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^5 d^2}+\frac {(3 b) \text {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) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 c^5 d^2}+\frac {(3 b) \text {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 = 0.66, size = 244, normalized size = 1.38 \begin {gather*} \frac {4 a c x-3 b \sqrt {\frac {-1+c x}{1+c x}}-4 b c x \sqrt {\frac {-1+c x}{1+c x}}+\frac {b \sqrt {\frac {-1+c x}{1+c x}}}{1-c x}+\frac {b c x \sqrt {\frac {-1+c x}{1+c x}}}{1-c x}-\frac {2 a c x}{-1+c^2 x^2}+4 b c x \cosh ^{-1}(c x)+\frac {b \cosh ^{-1}(c x)}{1-c x}-\frac {b \cosh ^{-1}(c x)}{1+c x}+6 b \cosh ^{-1}(c x) \log \left (1-e^{\cosh ^{-1}(c x)}\right )-6 b \cosh ^{-1}(c x) \log \left (1+e^{\cosh ^{-1}(c x)}\right )+3 a \log (1-c x)-3 a \log (1+c x)-6 b \text {PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )+6 b \text {PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{4 c^5 d^2} \end {gather*}

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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Maple [A]
time = 7.88, size = 268, normalized size = 1.51

method	result	size
derivativedivides	\(\frac {\frac {a c x}{d^{2}}-\frac {a}{4 d^{2} \left (c x -1\right )}+\frac {3 a \ln \left (c x -1\right )}{4 d^{2}}-\frac {a}{4 d^{2} \left (c x +1\right )}-\frac {3 a \ln \left (c x +1\right )}{4 d^{2}}+\frac {b \,\mathrm {arccosh}\left (c x \right ) c x}{d^{2}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{d^{2}}-\frac {b \,\mathrm {arccosh}\left (c x \right ) c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {3 b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}-\frac {3 b \polylog \left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}+\frac {3 b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}+\frac {3 b \polylog \left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}}{c^{5}}\)	\(268\)
default	\(\frac {\frac {a c x}{d^{2}}-\frac {a}{4 d^{2} \left (c x -1\right )}+\frac {3 a \ln \left (c x -1\right )}{4 d^{2}}-\frac {a}{4 d^{2} \left (c x +1\right )}-\frac {3 a \ln \left (c x +1\right )}{4 d^{2}}+\frac {b \,\mathrm {arccosh}\left (c x \right ) c x}{d^{2}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{d^{2}}-\frac {b \,\mathrm {arccosh}\left (c x \right ) c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {3 b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}-\frac {3 b \polylog \left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}+\frac {3 b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}+\frac {3 b \polylog \left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}}{c^{5}}\)	\(268\)

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,method=_RETURNVERBOSE)

[Out]

1/c^5*(a/d^2*c*x-1/4*a/d^2/(c*x-1)+3/4*a/d^2*ln(c*x-1)-1/4*a/d^2/(c*x+1)-3/4*a/d^2*ln(c*x+1)+b/d^2*arccosh(c*x
)*c*x-b/d^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)-1/2*b/d^2/(c^2*x^2-1)*arccosh(c*x)*c*x-1/2*b/d^2/(c^2*x^2-1)*(c*x-1)^(
1/2)*(c*x+1)^(1/2)-3/2*b/d^2*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-3/2*b/d^2*polylog(2,-c*x-(c*x-
1)^(1/2)*(c*x+1)^(1/2))+3/2*b/d^2*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))+3/2*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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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}} \end {gather*}

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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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

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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