3.2.91 \(\int (d+e x^2) \cosh ^{-1}(a x) \log (c x^n) \, dx\) [191]

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

[Out]

1/27*e*n*(a*x-1)^(3/2)*(a*x+1)^(3/2)/a^3-d*n*x*arccosh(a*x)-1/9*e*n*x^3*arccosh(a*x)-1/9*(9*a^2*d+2*e)*n*arcta
n((a*x-1)^(1/2)*(a*x+1)^(1/2))/a^3+d*x*arccosh(a*x)*ln(c*x^n)+1/3*e*x^3*arccosh(a*x)*ln(c*x^n)+d*n*(a*x-1)^(1/
2)*(a*x+1)^(1/2)/a+2/27*e*n*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3+1/9*(9*a^2*d+2*e)*n*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^
3+1/27*e*n*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a-1/9*(9*a^2*d+2*e)*ln(c*x^n)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3-1/9*e
*x^2*ln(c*x^n)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a

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Rubi [A]
time = 0.14, antiderivative size = 312, normalized size of antiderivative = 1.00, 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 = {5908, 471, 75, 2434, 103, 94, 211, 5879, 5883, 102, 12} \begin {gather*} \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}-\frac {n \text {ArcTan}\left (\sqrt {a x-1} \sqrt {a x+1}\right ) \left (9 a^2 d+2 e\right )}{9 a^3}-\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}+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 x^2 \sqrt {a x-1} \sqrt {a x+1} \log \left (c x^n\right )}{9 a}+\frac {d n \sqrt {a x-1} \sqrt {a x+1}}{a}-d n x \cosh ^{-1}(a x)-\frac {1}{9} e n x^3 \cosh ^{-1}(a x)+\frac {e n x^2 \sqrt {a x-1} \sqrt {a x+1}}{27 a} \end {gather*}

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[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 94

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 102

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 103

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 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 471

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 2434

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 5879

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 5883

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)/(Sqrt
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5908

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])

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 ) \text {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*}

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Mathematica [A]
time = 0.17, size = 145, normalized size = 0.46 \begin {gather*} \frac {3 \left (9 a^2 d+2 e\right ) n \tan ^{-1}\left (\frac {1}{\sqrt {-1+a x} \sqrt {1+a x}}\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 )+\sqrt {-1+a x} \sqrt {1+a x} \left (n \left (7 e+2 a^2 \left (27 d+e x^2\right )\right )-3 \left (2 e+a^2 \left (9 d+e x^2\right )\right ) \log \left (c x^n\right )\right )}{27 a^3} \end {gather*}

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] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 3.38, size = 4732, normalized size = 15.17

method	result	size
default	\(\text {Expression too large to display}\)	\(4732\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)*csgn(I/a)*x^2*
e*n+1/6*I*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*csgn(I*(1+(a*x+(
a*x-1)^(1/2)*(a*x+1)^(1/2))^2))*arccosh(a*x)*Pi*x^3*e*n+1/2*I*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*arccosh(a*x)*Pi*csgn(I/a)*x*d*n-1/9*e*n*x^3*arccosh(a*x)-2/9/a^3*(a*x+1)
^(1/2)*(a*x-1)^(1/2)*e*(ln(c*x^n)-n*ln(x))-1/a*(a*x+1)^(1/2)*(a*x-1)^(1/2)*d*(ln(c*x^n)-n*ln(x))+I/a*n*ln(a*x+
(a*x-1)^(1/2)*(a*x+1)^(1/2)-I)*d-1/3*ln(a)*arccosh(a*x)*x^3*e*n-1/3*arccosh(a*x)*ln(2)*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-ln(a)*arccosh(a*x)*x*d*n-arccosh(a*x)*ln(2)*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+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-1/18*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))^2*csgn(I*(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-1/18*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*Pi*(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+1/6*I*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*arccosh(a*x)*Pi*x^3*e*n+1/6*I*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*arccosh(a*x)*Pi*csgn(I/a)*x^3*e*n+1/2*I*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*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*arccosh(a*x
)*Pi*x*d*n+1/2*I*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*csgn(I*(1
+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))*arccosh(a*x)*Pi*x*d*n+1/2*I*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*arccosh(a*x)*Pi*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*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*csgn(I/(a*x+(a*x-
1)^(1/2)*(a*x+1)^(1/2)))*arccosh(a*x)*Pi*x^3*e*n-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))-d*n*x*arccosh(a*x)+1/a*ln(a)*(a*x+1)^(1/2)*(a*x-1)^(1/2)*d*n+1/a*ln(2)*(a*x+1)^
(1/2)*(a*x-1)^(1/2)*d*n-1/a*ln(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)*(a*x+1)^(1/2)*(a*x-1)^(1/2)*d*n+2/9/a^3*
ln(a)*(a*x+1)^(1/2)*(a*x-1)^(1/2)*e*n+2/9/a^3*ln(2)*(a*x+1)^(1/2)*(a*x-1)^(1/2)*e*n-2/9/a^3*ln(1+(a*x+(a*x-1)^
(1/2)*(a*x+1)^(1/2))^2)*(a*x+1)^(1/2)*(a*x-1)^(1/2)*e*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+1/2*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*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))*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))*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)*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*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))*Pi*(a*x+1)^(1/2)*(a*x-1)^(1/2)*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))^2*csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*Pi*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*e*
n+2/27*e*n*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a+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*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*csgn(I*(1...

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

[Out]

1/6*(3*a^2*d*n + n*e)*(log(a*x + 1)*log(x) + dilog(-a*x))/a^3 - 1/6*(3*a^2*d*n + n*e)*(log(-a*x + 1)*log(x) +
dilog(a*x))/a^3 - 1/18*(9*(d*n - d*log(c))*a^2 + (n - 3*log(c))*e)*log(a*x + 1)/a^3 + 1/18*(9*(d*n - d*log(c))
*a^2 + (n - 3*log(c))*e)*log(a*x - 1)/a^3 + 1/54*(2*a^3*(2*n - 3*log(c))*x^3*e - 9*(3*a^2*d*n + n*e)*log(a*x +
1)*log(x) + 9*(3*a^2*d*n + n*e)*log(a*x - 1)*log(x) + 6*(9*(2*d*n - d*log(c))*a^3 + a*(4*n - 3*log(c))*e)*x -
6*(a^3*(n - 3*log(c))*x^3*e + 9*(d*n - d*log(c))*a^3*x - 3*(a^3*x^3*e + 3*a^3*d*x)*log(x^n))*log(a*x + sqrt(a
*x + 1)*sqrt(a*x - 1)) - 3*(2*a^3*x^3*e + 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*(a*(n - 3*log(c))*x^3*e + 9*(d*n - d*log(c))*a*x - 3*(a*x^3*e +
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 = 0.49, size = 390, normalized size = 1.25 \begin {gather*} -\frac {6 \, {\left (9 \, a^{2} d n + 2 \, n \cosh \left (1\right ) + 2 \, n \sinh \left (1\right )\right )} \arctan \left (-a x + \sqrt {a^{2} x^{2} - 1}\right ) + 3 \, {\left (9 \, a^{3} d n x - 9 \, a^{3} d n + {\left (a^{3} n x^{3} - a^{3} n\right )} \cosh \left (1\right ) - 3 \, {\left (3 \, a^{3} d x - 3 \, a^{3} d + {\left (a^{3} x^{3} - a^{3}\right )} \cosh \left (1\right ) + {\left (a^{3} x^{3} - a^{3}\right )} \sinh \left (1\right )\right )} \log \left (c\right ) - 3 \, {\left (a^{3} n x^{3} \cosh \left (1\right ) + a^{3} n x^{3} \sinh \left (1\right ) + 3 \, a^{3} d n x\right )} \log \left (x\right ) + {\left (a^{3} n x^{3} - a^{3} n\right )} \sinh \left (1\right )\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - 3 \, {\left (9 \, a^{3} d n + a^{3} n \cosh \left (1\right ) + a^{3} n \sinh \left (1\right ) - 3 \, {\left (3 \, a^{3} d + a^{3} \cosh \left (1\right ) + a^{3} \sinh \left (1\right )\right )} \log \left (c\right )\right )} \log \left (-a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (54 \, a^{2} d n + {\left (2 \, a^{2} n x^{2} + 7 \, n\right )} \cosh \left (1\right ) - 3 \, {\left (9 \, a^{2} d + {\left (a^{2} x^{2} + 2\right )} \cosh \left (1\right ) + {\left (a^{2} x^{2} + 2\right )} \sinh \left (1\right )\right )} \log \left (c\right ) - 3 \, {\left (9 \, a^{2} d n + {\left (a^{2} n x^{2} + 2 \, n\right )} \cosh \left (1\right ) + {\left (a^{2} n x^{2} + 2 \, n\right )} \sinh \left (1\right )\right )} \log \left (x\right ) + {\left (2 \, a^{2} n x^{2} + 7 \, n\right )} \sinh \left (1\right )\right )} \sqrt {a^{2} x^{2} - 1}}{27 \, a^{3}} \end {gather*}

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*n + 2*n*cosh(1) + 2*n*sinh(1))*arctan(-a*x + sqrt(a^2*x^2 - 1)) + 3*(9*a^3*d*n*x - 9*a^3*d*n
+ (a^3*n*x^3 - a^3*n)*cosh(1) - 3*(3*a^3*d*x - 3*a^3*d + (a^3*x^3 - a^3)*cosh(1) + (a^3*x^3 - a^3)*sinh(1))*l
og(c) - 3*(a^3*n*x^3*cosh(1) + a^3*n*x^3*sinh(1) + 3*a^3*d*n*x)*log(x) + (a^3*n*x^3 - a^3*n)*sinh(1))*log(a*x
+ sqrt(a^2*x^2 - 1)) - 3*(9*a^3*d*n + a^3*n*cosh(1) + a^3*n*sinh(1) - 3*(3*a^3*d + a^3*cosh(1) + a^3*sinh(1))*
log(c))*log(-a*x + sqrt(a^2*x^2 - 1)) - (54*a^2*d*n + (2*a^2*n*x^2 + 7*n)*cosh(1) - 3*(9*a^2*d + (a^2*x^2 + 2)
*cosh(1) + (a^2*x^2 + 2)*sinh(1))*log(c) - 3*(9*a^2*d*n + (a^2*n*x^2 + 2*n)*cosh(1) + (a^2*n*x^2 + 2*n)*sinh(1
))*log(x) + (2*a^2*n*x^2 + 7*n)*sinh(1))*sqrt(a^2*x^2 - 1))/a^3

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d + e x^{2}\right ) \log {\left (c x^{n} \right )} \operatorname {acosh}{\left (a x \right )}\, dx \end {gather*}

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]

Integral((d + e*x**2)*log(c*x**n)*acosh(a*x), x)

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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((e*x^2+d)*arccosh(a*x)*log(c*x^n),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 = -0.00 \begin {gather*} \int \ln \left (c\,x^n\right )\,\mathrm {acosh}\left (a\,x\right )\,\left (e\,x^2+d\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*x^n)*acosh(a*x)*(d + e*x^2),x)

[Out]

int(log(c*x^n)*acosh(a*x)*(d + e*x^2), x)

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