3.191 \(\int (d+e x^2) \cosh ^{-1}(a x) \log (c x^n) \, dx\)

Optimal. Leaf size=312 \[ -\frac{\sqrt{a x-1} \sqrt{a x+1} \left (9 a^2 d+2 e\right ) \log \left (c x^n\right )}{9 a^3}+\frac{n \sqrt{a x-1} \sqrt{a x+1} \left (9 a^2 d+2 e\right )}{9 a^3}-\frac{n \left (9 a^2 d+2 e\right ) \tan ^{-1}\left (\sqrt{a x-1} \sqrt{a x+1}\right )}{9 a^3}+\frac{e n (a x-1)^{3/2} (a x+1)^{3/2}}{27 a^3}+\frac{2 e n \sqrt{a x-1} \sqrt{a x+1}}{27 a^3}+d x \cosh ^{-1}(a x) \log \left (c x^n\right )-\frac{e x^2 \sqrt{a x-1} \sqrt{a x+1} \log \left (c x^n\right )}{9 a}+\frac{1}{3} e x^3 \cosh ^{-1}(a x) \log \left (c x^n\right )+\frac{d n \sqrt{a x-1} \sqrt{a x+1}}{a}-d n x \cosh ^{-1}(a x)+\frac{e n x^2 \sqrt{a x-1} \sqrt{a x+1}}{27 a}-\frac{1}{9} e n x^3 \cosh ^{-1}(a x) \]

[Out]

(d*n*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/a + (2*e*n*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(27*a^3) + ((9*a^2*d + 2*e)*n*Sqrt
[-1 + a*x]*Sqrt[1 + a*x])/(9*a^3) + (e*n*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(27*a) + (e*n*(-1 + a*x)^(3/2)*(1 +
a*x)^(3/2))/(27*a^3) - d*n*x*ArcCosh[a*x] - (e*n*x^3*ArcCosh[a*x])/9 - ((9*a^2*d + 2*e)*n*ArcTan[Sqrt[-1 + a*
x]*Sqrt[1 + a*x]])/(9*a^3) - ((9*a^2*d + 2*e)*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Log[c*x^n])/(9*a^3) - (e*x^2*Sqrt[-
1 + a*x]*Sqrt[1 + a*x]*Log[c*x^n])/(9*a) + d*x*ArcCosh[a*x]*Log[c*x^n] + (e*x^3*ArcCosh[a*x]*Log[c*x^n])/3

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Rubi [A]  time = 0.210302, antiderivative size = 312, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 11, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.611, Rules used = {5705, 460, 74, 2387, 101, 92, 205, 5654, 5662, 100, 12} \[ -\frac{\sqrt{a x-1} \sqrt{a x+1} \left (9 a^2 d+2 e\right ) \log \left (c x^n\right )}{9 a^3}+\frac{n \sqrt{a x-1} \sqrt{a x+1} \left (9 a^2 d+2 e\right )}{9 a^3}-\frac{n \left (9 a^2 d+2 e\right ) \tan ^{-1}\left (\sqrt{a x-1} \sqrt{a x+1}\right )}{9 a^3}+\frac{e n (a x-1)^{3/2} (a x+1)^{3/2}}{27 a^3}+\frac{2 e n \sqrt{a x-1} \sqrt{a x+1}}{27 a^3}+d x \cosh ^{-1}(a x) \log \left (c x^n\right )-\frac{e x^2 \sqrt{a x-1} \sqrt{a x+1} \log \left (c x^n\right )}{9 a}+\frac{1}{3} e x^3 \cosh ^{-1}(a x) \log \left (c x^n\right )+\frac{d n \sqrt{a x-1} \sqrt{a x+1}}{a}-d n x \cosh ^{-1}(a x)+\frac{e n x^2 \sqrt{a x-1} \sqrt{a x+1}}{27 a}-\frac{1}{9} e n x^3 \cosh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)*ArcCosh[a*x]*Log[c*x^n],x]

[Out]

(d*n*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/a + (2*e*n*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(27*a^3) + ((9*a^2*d + 2*e)*n*Sqrt
[-1 + a*x]*Sqrt[1 + a*x])/(9*a^3) + (e*n*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(27*a) + (e*n*(-1 + a*x)^(3/2)*(1 +
a*x)^(3/2))/(27*a^3) - d*n*x*ArcCosh[a*x] - (e*n*x^3*ArcCosh[a*x])/9 - ((9*a^2*d + 2*e)*n*ArcTan[Sqrt[-1 + a*
x]*Sqrt[1 + a*x]])/(9*a^3) - ((9*a^2*d + 2*e)*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Log[c*x^n])/(9*a^3) - (e*x^2*Sqrt[-
1 + a*x]*Sqrt[1 + a*x]*Log[c*x^n])/(9*a) + d*x*ArcCosh[a*x]*Log[c*x^n] + (e*x^3*ArcCosh[a*x]*Log[c*x^n])/3

Rule 5705

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x
], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])

Rule 460

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.)*((c_) + (d_.)
*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1))/(b1*b2*e*
(m + n*(p + 1) + 1)), x] - Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(b1*b2*(m + n*(p + 1) + 1)), I
nt[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] &&
EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]

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 2387

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(Px_.)*(F_)[(d_.)*((e_.) + (f_.)*(x_))]^(m_.), x_Symbol] :> With[{u
= IntHide[Px*F[d*(e + f*x)]^m, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist[1/x, u, x], x], x]] /; F
reeQ[{a, b, c, d, e, f, n}, x] && PolynomialQ[Px, x] && IGtQ[m, 0] && MemberQ[{ArcSin, ArcCos, ArcSinh, ArcCos
h}, F]

Rule 101

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

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 5654

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 100

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

Rule 12

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

Rubi steps

\begin{align*} \int \left (d+e x^2\right ) \cosh ^{-1}(a x) \log \left (c x^n\right ) \, dx &=-\frac{\left (9 a^2 d+2 e\right ) \sqrt{-1+a x} \sqrt{1+a x} \log \left (c x^n\right )}{9 a^3}-\frac{e x^2 \sqrt{-1+a x} \sqrt{1+a x} \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \cosh ^{-1}(a x) \log \left (c x^n\right )-n \int \left (-\frac{\left (9 a^2 d+2 e\right ) \sqrt{-1+a x} \sqrt{1+a x}}{9 a^3 x}-\frac{e x \sqrt{-1+a x} \sqrt{1+a x}}{9 a}+d \cosh ^{-1}(a x)+\frac{1}{3} e x^2 \cosh ^{-1}(a x)\right ) \, dx\\ &=-\frac{\left (9 a^2 d+2 e\right ) \sqrt{-1+a x} \sqrt{1+a x} \log \left (c x^n\right )}{9 a^3}-\frac{e x^2 \sqrt{-1+a x} \sqrt{1+a x} \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \cosh ^{-1}(a x) \log \left (c x^n\right )-(d n) \int \cosh ^{-1}(a x) \, dx-\frac{1}{3} (e n) \int x^2 \cosh ^{-1}(a x) \, dx+\frac{(e n) \int x \sqrt{-1+a x} \sqrt{1+a x} \, dx}{9 a}+\frac{\left (\left (9 a^2 d+2 e\right ) n\right ) \int \frac{\sqrt{-1+a x} \sqrt{1+a x}}{x} \, dx}{9 a^3}\\ &=\frac{\left (9 a^2 d+2 e\right ) n \sqrt{-1+a x} \sqrt{1+a x}}{9 a^3}+\frac{e n (-1+a x)^{3/2} (1+a x)^{3/2}}{27 a^3}-d n x \cosh ^{-1}(a x)-\frac{1}{9} e n x^3 \cosh ^{-1}(a x)-\frac{\left (9 a^2 d+2 e\right ) \sqrt{-1+a x} \sqrt{1+a x} \log \left (c x^n\right )}{9 a^3}-\frac{e x^2 \sqrt{-1+a x} \sqrt{1+a x} \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \cosh ^{-1}(a x) \log \left (c x^n\right )+(a d n) \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx+\frac{1}{9} (a e n) \int \frac{x^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx-\frac{\left (\left (9 a^2 d+2 e\right ) n\right ) \int \frac{1}{x \sqrt{-1+a x} \sqrt{1+a x}} \, dx}{9 a^3}\\ &=\frac{d n \sqrt{-1+a x} \sqrt{1+a x}}{a}+\frac{\left (9 a^2 d+2 e\right ) n \sqrt{-1+a x} \sqrt{1+a x}}{9 a^3}+\frac{e n x^2 \sqrt{-1+a x} \sqrt{1+a x}}{27 a}+\frac{e n (-1+a x)^{3/2} (1+a x)^{3/2}}{27 a^3}-d n x \cosh ^{-1}(a x)-\frac{1}{9} e n x^3 \cosh ^{-1}(a x)-\frac{\left (9 a^2 d+2 e\right ) \sqrt{-1+a x} \sqrt{1+a x} \log \left (c x^n\right )}{9 a^3}-\frac{e x^2 \sqrt{-1+a x} \sqrt{1+a x} \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \cosh ^{-1}(a x) \log \left (c x^n\right )+\frac{(e n) \int \frac{2 x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{27 a}-\frac{\left (\left (9 a^2 d+2 e\right ) n\right ) \operatorname{Subst}\left (\int \frac{1}{a+a x^2} \, dx,x,\sqrt{-1+a x} \sqrt{1+a x}\right )}{9 a^2}\\ &=\frac{d n \sqrt{-1+a x} \sqrt{1+a x}}{a}+\frac{\left (9 a^2 d+2 e\right ) n \sqrt{-1+a x} \sqrt{1+a x}}{9 a^3}+\frac{e n x^2 \sqrt{-1+a x} \sqrt{1+a x}}{27 a}+\frac{e n (-1+a x)^{3/2} (1+a x)^{3/2}}{27 a^3}-d n x \cosh ^{-1}(a x)-\frac{1}{9} e n x^3 \cosh ^{-1}(a x)-\frac{\left (9 a^2 d+2 e\right ) n \tan ^{-1}\left (\sqrt{-1+a x} \sqrt{1+a x}\right )}{9 a^3}-\frac{\left (9 a^2 d+2 e\right ) \sqrt{-1+a x} \sqrt{1+a x} \log \left (c x^n\right )}{9 a^3}-\frac{e x^2 \sqrt{-1+a x} \sqrt{1+a x} \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \cosh ^{-1}(a x) \log \left (c x^n\right )+\frac{(2 e n) \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{27 a}\\ &=\frac{d n \sqrt{-1+a x} \sqrt{1+a x}}{a}+\frac{2 e n \sqrt{-1+a x} \sqrt{1+a x}}{27 a^3}+\frac{\left (9 a^2 d+2 e\right ) n \sqrt{-1+a x} \sqrt{1+a x}}{9 a^3}+\frac{e n x^2 \sqrt{-1+a x} \sqrt{1+a x}}{27 a}+\frac{e n (-1+a x)^{3/2} (1+a x)^{3/2}}{27 a^3}-d n x \cosh ^{-1}(a x)-\frac{1}{9} e n x^3 \cosh ^{-1}(a x)-\frac{\left (9 a^2 d+2 e\right ) n \tan ^{-1}\left (\sqrt{-1+a x} \sqrt{1+a x}\right )}{9 a^3}-\frac{\left (9 a^2 d+2 e\right ) \sqrt{-1+a x} \sqrt{1+a x} \log \left (c x^n\right )}{9 a^3}-\frac{e x^2 \sqrt{-1+a x} \sqrt{1+a x} \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \cosh ^{-1}(a x) \log \left (c x^n\right )\\ \end{align*}

Mathematica [A]  time = 0.224192, size = 145, normalized size = 0.46 \[ \frac{\sqrt{a x-1} \sqrt{a x+1} \left (n \left (2 a^2 \left (27 d+e x^2\right )+7 e\right )-3 \left (a^2 \left (9 d+e x^2\right )+2 e\right ) \log \left (c x^n\right )\right )-3 a^3 x \cosh ^{-1}(a x) \left (n \left (9 d+e x^2\right )-3 \left (3 d+e x^2\right ) \log \left (c x^n\right )\right )+3 n \left (9 a^2 d+2 e\right ) \tan ^{-1}\left (\frac{1}{\sqrt{a x-1} \sqrt{a x+1}}\right )}{27 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2)*ArcCosh[a*x]*Log[c*x^n],x]

[Out]

(3*(9*a^2*d + 2*e)*n*ArcTan[1/(Sqrt[-1 + a*x]*Sqrt[1 + a*x])] - 3*a^3*x*ArcCosh[a*x]*(n*(9*d + e*x^2) - 3*(3*d
+ e*x^2)*Log[c*x^n]) + Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(n*(7*e + 2*a^2*(27*d + e*x^2)) - 3*(2*e + a^2*(9*d + e*x
^2))*Log[c*x^n]))/(27*a^3)

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Maple [C]  time = 1.968, size = 4732, normalized size = 15.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)*arccosh(a*x)*ln(c*x^n),x)

[Out]

-1/9/a^3*n*(3*arccosh(a*x)*x^3*a^3*e-(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*a^2*e+9*arccosh(a*x)*x*a^3*d-9*(a*x+1)^(1
/2)*(a*x-1)^(1/2)*a^2*d-2*(a*x+1)^(1/2)*(a*x-1)^(1/2)*e)*ln(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))+2*d*n*(a*x-1)^(1/
2)*(a*x+1)^(1/2)/a+7/27*e*n*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3+2/27*e*n*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a-1/9*e*n
*x^3*arccosh(a*x)-d*n*x*arccosh(a*x)-1/2*I*arccosh(a*x)*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+(a*x+(a*x-
1)^(1/2)*(a*x+1)^(1/2))^2))*csgn(I/a*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))
)*Pi*csgn(I/a)*x*d*n+1/18*I/a*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)
)^3*Pi*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*e*n+1/3*arccosh(a*x)*x^3*e*(ln(c*x^n)-n*ln(x))+arccosh(a*x)*x*d*(ln(c*x
^n)-n*ln(x))+1/18*I/a*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))*csgn(I
/a*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*Pi*(a*x+1)^(1/2)*(a*x-1)^(1/2)*c
sgn(I/a)*x^2*e*n+1/9*I/a^3*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))*c
sgn(I/a*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*Pi*(a*x+1)^(1/2)*(a*x-1)^(1
/2)*csgn(I/a)*e*n-1/18*I/a*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*
(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))^2*Pi*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*e*n-1/18*I/a*csgn(I*(1+(a*x+(a*x
-1)^(1/2)*(a*x+1)^(1/2))^2))*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))
^2*Pi*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*e*n-1/18*I/a*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+(a*x+(a*x-1)^(1
/2)*(a*x+1)^(1/2))^2))*csgn(I/a*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))^2*P
i*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*e*n-1/18*I/a*csgn(I/a*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)/(a*x+(a*x-1)^(
1/2)*(a*x+1)^(1/2)))^2*Pi*(a*x+1)^(1/2)*(a*x-1)^(1/2)*csgn(I/a)*x^2*e*n+1/2*I/a*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x
+1)^(1/2)))*csgn(I*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+(a*x+(
a*x-1)^(1/2)*(a*x+1)^(1/2))^2))*Pi*(a*x+1)^(1/2)*(a*x-1)^(1/2)*d*n+1/2*I/a*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(
1/2))*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))*csgn(I/a*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)/(a*x+(a*x-1)^(
1/2)*(a*x+1)^(1/2)))*Pi*(a*x+1)^(1/2)*(a*x-1)^(1/2)*csgn(I/a)*d*n+1/9*I/a^3*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^
(1/2)))*csgn(I*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+(a*x+(a*x-
1)^(1/2)*(a*x+1)^(1/2))^2))*Pi*(a*x+1)^(1/2)*(a*x-1)^(1/2)*e*n+1/2*I*arccosh(a*x)*csgn(I/(a*x+(a*x-1)^(1/2)*(a
*x+1)^(1/2)))*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))^2*Pi*x*d*n+1/2
*I*arccosh(a*x)*csgn(I*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+(a
*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))^2*Pi*x*d*n+1/2*I*arccosh(a*x)*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+
(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))*csgn(I/a*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)/(a*x+(a*x-1)^(1/2)*(a*x
+1)^(1/2)))^2*Pi*x*d*n+1/2*I*arccosh(a*x)*csgn(I/a*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)/(a*x+(a*x-1)^(1/2)*
(a*x+1)^(1/2)))^2*Pi*csgn(I/a)*x*d*n+1/2*I/a*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+(a*x+(a*x-1)^(1/2)*(a
*x+1)^(1/2))^2))^3*Pi*(a*x+1)^(1/2)*(a*x-1)^(1/2)*d*n+1/2*I/a*csgn(I/a*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)
/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))^3*Pi*(a*x+1)^(1/2)*(a*x-1)^(1/2)*d*n+1/9*I/a^3*csgn(I/(a*x+(a*x-1)^(1/2)*(
a*x+1)^(1/2))*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))^3*Pi*(a*x+1)^(1/2)*(a*x-1)^(1/2)*e*n+1/9*I/a^3*csgn(I/a
*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))^3*Pi*(a*x+1)^(1/2)*(a*x-1)^(1/2)*e
*n+1/6*I*arccosh(a*x)*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+(a
*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))^2*Pi*x^3*e*n+1/6*I*arccosh(a*x)*csgn(I*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)
)^2))*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))^2*Pi*x^3*e*n+1/6*I*arc
cosh(a*x)*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))*csgn(I/a*(1+(a*x+(
a*x-1)^(1/2)*(a*x+1)^(1/2))^2)/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))^2*Pi*x^3*e*n+1/6*I*arccosh(a*x)*csgn(I/a*(1+
(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))^2*Pi*csgn(I/a)*x^3*e*n+1/18*I/a*csgn(I
/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*csgn(I*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))*csgn(I/(a*x+(a*x-1)^(1/2)*
(a*x+1)^(1/2))*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))*Pi*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*e*n+1/a*ln(2)*(a*x+
1)^(1/2)*(a*x-1)^(1/2)*d*n-1/a*(a*x+1)^(1/2)*(a*x-1)^(1/2)*d*(ln(c*x^n)-n*ln(x))-1/2*I/a*csgn(I/a*(1+(a*x+(a*x
-1)^(1/2)*(a*x+1)^(1/2))^2)/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))^2*Pi*(a*x+1)^(1/2)*(a*x-1)^(1/2)*csgn(I/a)*d*n-
1/9*I/a^3*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+(a*x+(a*x-1)^(
1/2)*(a*x+1)^(1/2))^2))^2*Pi*(a*x+1)^(1/2)*(a*x-1)^(1/2)*e*n-1/9*I/a^3*csgn(I*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1
/2))^2))*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))^2*Pi*(a*x+1)^(1/2)*
(a*x-1)^(1/2)*e*n-1/9*I/a^3*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))*
csgn(I/a*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))^2*Pi*(a*x+1)^(1/2)*(a*x-1)
^(1/2)*e*n-1/9*I/a^3*csgn(I/a*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))^2*Pi*
(a*x+1)^(1/2)*(a*x-1)^(1/2)*csgn(I/a)*e*n+1/18*I/a*csgn(I/a*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)/(a*x+(a*x-
1)^(1/2)*(a*x+1)^(1/2)))^3*Pi*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*e*n-1/2*I/a*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1
/2)))*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))^2*Pi*(a*x+1)^(1/2)*(a*
x-1)^(1/2)*d*n-1/2*I/a*csgn(I*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)
)*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))^2*Pi*(a*x+1)^(1/2)*(a*x-1)^(1/2)*d*n-1/2*I/a*csgn(I/(a*x+(a*x-1)^(1
/2)*(a*x+1)^(1/2))*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))*csgn(I/a*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)/(
a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))^2*Pi*(a*x+1)^(1/2)*(a*x-1)^(1/2)*d*n-1/6*I*arccosh(a*x)*csgn(I/(a*x+(a*x-1)^
(1/2)*(a*x+1)^(1/2)))*csgn(I*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))
*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))*Pi*x^3*e*n-1/6*I*arccosh(a*x)*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2
))*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))*csgn(I/a*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)/(a*x+(a*x-1)^(1/2
)*(a*x+1)^(1/2)))*Pi*csgn(I/a)*x^3*e*n-1/2*I*arccosh(a*x)*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*csgn(I*(1+
(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(
1/2))^2))*Pi*x*d*n-2/9/a^3*(a*x+1)^(1/2)*(a*x-1)^(1/2)*e*(ln(c*x^n)-n*ln(x))-1/3*arccosh(a*x)*ln(2)*x^3*e*n-1/
3*arccosh(a*x)*ln(a)*x^3*e*n+1/3*arccosh(a*x)*ln(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)*x^3*e*n-arccosh(a*x)*l
n(2)*x*d*n-arccosh(a*x)*ln(a)*x*d*n+arccosh(a*x)*ln(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)*x*d*n+I/a*n*ln(a*x+
(a*x-1)^(1/2)*(a*x+1)^(1/2)-I)*d-I/a*n*ln(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)+I)*d-2/9*I/a^3*n*ln(a*x+(a*x-1)^(1/2
)*(a*x+1)^(1/2)+I)*e+2/9*I/a^3*n*ln(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)-I)*e+1/a*ln(a)*(a*x+1)^(1/2)*(a*x-1)^(1/2)
*d*n-1/a*(a*x+1)^(1/2)*(a*x-1)^(1/2)*ln(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)*d*n+2/9/a^3*ln(2)*(a*x+1)^(1/2)
*(a*x-1)^(1/2)*e*n+2/9/a^3*ln(a)*(a*x+1)^(1/2)*(a*x-1)^(1/2)*e*n-2/9/a^3*(a*x+1)^(1/2)*(a*x-1)^(1/2)*ln(1+(a*x
+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)*e*n-1/9/a*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*e*(ln(c*x^n)-n*ln(x))-1/6*I*arccosh
(a*x)*csgn(I/a*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))^3*Pi*x^3*e*n-1/2*I*a
rccosh(a*x)*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))^3*Pi*x*d*n-1/2*I
*arccosh(a*x)*csgn(I/a*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))^3*Pi*x*d*n+1
/9/a*ln(2)*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*e*n+1/9/a*ln(a)*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*e*n-1/9/a*(a*x+1)^(
1/2)*(a*x-1)^(1/2)*ln(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)*x^2*e*n-1/6*I*arccosh(a*x)*csgn(I/(a*x+(a*x-1)^(1
/2)*(a*x+1)^(1/2))*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))^3*Pi*x^3*e*n

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (3 \, a^{2} d n + e n\right )}{\left (\log \left (a x + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-a x\right )\right )}}{6 \, a^{3}} - \frac{{\left (3 \, a^{2} d n + e n\right )}{\left (\log \left (-a x + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (a x\right )\right )}}{6 \, a^{3}} - \frac{{\left (9 \,{\left (d n - d \log \left (c\right )\right )} a^{2} + e n - 3 \, e \log \left (c\right )\right )} \log \left (a x + 1\right )}{18 \, a^{3}} + \frac{{\left (9 \,{\left (d n - d \log \left (c\right )\right )} a^{2} + e n - 3 \, e \log \left (c\right )\right )} \log \left (a x - 1\right )}{18 \, a^{3}} + \frac{2 \,{\left (2 \, e n - 3 \, e \log \left (c\right )\right )} a^{3} x^{3} - 9 \,{\left (3 \, a^{2} d n + e n\right )} \log \left (a x + 1\right ) \log \left (x\right ) + 9 \,{\left (3 \, a^{2} d n + e n\right )} \log \left (a x - 1\right ) \log \left (x\right ) + 6 \,{\left (9 \,{\left (2 \, d n - d \log \left (c\right )\right )} a^{3} +{\left (4 \, e n - 3 \, e \log \left (c\right )\right )} a\right )} x - 6 \,{\left ({\left (e n - 3 \, e \log \left (c\right )\right )} a^{3} x^{3} + 9 \,{\left (d n - d \log \left (c\right )\right )} a^{3} x - 3 \,{\left (a^{3} e x^{3} + 3 \, a^{3} d x\right )} \log \left (x^{n}\right )\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right ) - 3 \,{\left (2 \, a^{3} e x^{3} + 6 \,{\left (3 \, a^{3} d + a e\right )} x - 3 \,{\left (3 \, a^{2} d + e\right )} \log \left (a x + 1\right ) + 3 \,{\left (3 \, a^{2} d + e\right )} \log \left (a x - 1\right )\right )} \log \left (x^{n}\right )}{54 \, a^{3}} + \int -\frac{{\left (e n - 3 \, e \log \left (c\right )\right )} a x^{3} + 9 \,{\left (d n - d \log \left (c\right )\right )} a x - 3 \,{\left (a e x^{3} + 3 \, a d x\right )} \log \left (x^{n}\right )}{9 \,{\left (a^{3} x^{3} +{\left (a^{2} x^{2} - 1\right )} \sqrt{a x + 1} \sqrt{a x - 1} - a x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*arccosh(a*x)*log(c*x^n),x, algorithm="maxima")

[Out]

1/6*(3*a^2*d*n + e*n)*(log(a*x + 1)*log(x) + dilog(-a*x))/a^3 - 1/6*(3*a^2*d*n + e*n)*(log(-a*x + 1)*log(x) +
dilog(a*x))/a^3 - 1/18*(9*(d*n - d*log(c))*a^2 + e*n - 3*e*log(c))*log(a*x + 1)/a^3 + 1/18*(9*(d*n - d*log(c))
*a^2 + e*n - 3*e*log(c))*log(a*x - 1)/a^3 + 1/54*(2*(2*e*n - 3*e*log(c))*a^3*x^3 - 9*(3*a^2*d*n + e*n)*log(a*x
+ 1)*log(x) + 9*(3*a^2*d*n + e*n)*log(a*x - 1)*log(x) + 6*(9*(2*d*n - d*log(c))*a^3 + (4*e*n - 3*e*log(c))*a)
*x - 6*((e*n - 3*e*log(c))*a^3*x^3 + 9*(d*n - d*log(c))*a^3*x - 3*(a^3*e*x^3 + 3*a^3*d*x)*log(x^n))*log(a*x +
sqrt(a*x + 1)*sqrt(a*x - 1)) - 3*(2*a^3*e*x^3 + 6*(3*a^3*d + a*e)*x - 3*(3*a^2*d + e)*log(a*x + 1) + 3*(3*a^2*
d + e)*log(a*x - 1))*log(x^n))/a^3 + integrate(-1/9*((e*n - 3*e*log(c))*a*x^3 + 9*(d*n - d*log(c))*a*x - 3*(a*
e*x^3 + 3*a*d*x)*log(x^n))/(a^3*x^3 + (a^2*x^2 - 1)*sqrt(a*x + 1)*sqrt(a*x - 1) - a*x), x)

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Fricas [A]  time = 1.36909, size = 630, normalized size = 2.02 \begin{align*} -\frac{6 \,{\left (9 \, a^{2} d + 2 \, e\right )} n \arctan \left (-a x + \sqrt{a^{2} x^{2} - 1}\right ) + 3 \,{\left (a^{3} e n x^{3} + 9 \, a^{3} d n x -{\left (9 \, a^{3} d + a^{3} e\right )} n - 3 \,{\left (a^{3} e x^{3} + 3 \, a^{3} d x - 3 \, a^{3} d - a^{3} e\right )} \log \left (c\right ) - 3 \,{\left (a^{3} e n x^{3} + 3 \, a^{3} d n x\right )} \log \left (x\right )\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - 3 \,{\left ({\left (9 \, a^{3} d + a^{3} e\right )} n - 3 \,{\left (3 \, a^{3} d + a^{3} e\right )} \log \left (c\right )\right )} \log \left (-a x + \sqrt{a^{2} x^{2} - 1}\right ) -{\left (2 \, a^{2} e n x^{2} +{\left (54 \, a^{2} d + 7 \, e\right )} n - 3 \,{\left (a^{2} e x^{2} + 9 \, a^{2} d + 2 \, e\right )} \log \left (c\right ) - 3 \,{\left (a^{2} e n x^{2} +{\left (9 \, a^{2} d + 2 \, e\right )} n\right )} \log \left (x\right )\right )} \sqrt{a^{2} x^{2} - 1}}{27 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*arccosh(a*x)*log(c*x^n),x, algorithm="fricas")

[Out]

-1/27*(6*(9*a^2*d + 2*e)*n*arctan(-a*x + sqrt(a^2*x^2 - 1)) + 3*(a^3*e*n*x^3 + 9*a^3*d*n*x - (9*a^3*d + a^3*e)
*n - 3*(a^3*e*x^3 + 3*a^3*d*x - 3*a^3*d - a^3*e)*log(c) - 3*(a^3*e*n*x^3 + 3*a^3*d*n*x)*log(x))*log(a*x + sqrt
(a^2*x^2 - 1)) - 3*((9*a^3*d + a^3*e)*n - 3*(3*a^3*d + a^3*e)*log(c))*log(-a*x + sqrt(a^2*x^2 - 1)) - (2*a^2*e
*n*x^2 + (54*a^2*d + 7*e)*n - 3*(a^2*e*x^2 + 9*a^2*d + 2*e)*log(c) - 3*(a^2*e*n*x^2 + (9*a^2*d + 2*e)*n)*log(x
))*sqrt(a^2*x^2 - 1))/a^3

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)*acosh(a*x)*ln(c*x**n),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x^{2} + d\right )} \operatorname{arcosh}\left (a x\right ) \log \left (c x^{n}\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*arccosh(a*x)*log(c*x^n),x, algorithm="giac")

[Out]

integrate((e*x^2 + d)*arccosh(a*x)*log(c*x^n), x)