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

Optimal. Leaf size=244 \[ -\frac{\sqrt{a^2 x^2+1} \left (3 a^2 d-e\right ) \log \left (c x^n\right )}{3 a^3}-\frac{e \left (a^2 x^2+1\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+\frac{n \sqrt{a^2 x^2+1} \left (3 a^2 d-e\right )}{3 a^3}-\frac{n \left (3 a^2 d-e\right ) \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right )}{3 a^3}+\frac{d n \sqrt{a^2 x^2+1}}{a}+\frac{2 e n \left (a^2 x^2+1\right )^{3/2}}{27 a^3}-\frac{e n \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right )}{9 a^3}+d x \sinh ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \sinh ^{-1}(a x) \log \left (c x^n\right )-d n x \sinh ^{-1}(a x)-\frac{1}{9} e n x^3 \sinh ^{-1}(a x) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.21857, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 11, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.611, Rules used = {5704, 444, 43, 2387, 266, 50, 63, 208, 5653, 261, 5661} \[ -\frac{\sqrt{a^2 x^2+1} \left (3 a^2 d-e\right ) \log \left (c x^n\right )}{3 a^3}-\frac{e \left (a^2 x^2+1\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+\frac{n \sqrt{a^2 x^2+1} \left (3 a^2 d-e\right )}{3 a^3}-\frac{n \left (3 a^2 d-e\right ) \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right )}{3 a^3}+\frac{d n \sqrt{a^2 x^2+1}}{a}+\frac{2 e n \left (a^2 x^2+1\right )^{3/2}}{27 a^3}-\frac{e n \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right )}{9 a^3}+d x \sinh ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \sinh ^{-1}(a x) \log \left (c x^n\right )-d n x \sinh ^{-1}(a x)-\frac{1}{9} e n x^3 \sinh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5704

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

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rule 5653

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 5661

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcS
inh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt
[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \left (d+e x^2\right ) \sinh ^{-1}(a x) \log \left (c x^n\right ) \, dx &=-\frac{\left (3 a^2 d-e\right ) \sqrt{1+a^2 x^2} \log \left (c x^n\right )}{3 a^3}-\frac{e \left (1+a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \sinh ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \sinh ^{-1}(a x) \log \left (c x^n\right )-n \int \left (-\frac{\left (3 a^2 d-e\right ) \sqrt{1+a^2 x^2}}{3 a^3 x}-\frac{e \left (1+a^2 x^2\right )^{3/2}}{9 a^3 x}+d \sinh ^{-1}(a x)+\frac{1}{3} e x^2 \sinh ^{-1}(a x)\right ) \, dx\\ &=-\frac{\left (3 a^2 d-e\right ) \sqrt{1+a^2 x^2} \log \left (c x^n\right )}{3 a^3}-\frac{e \left (1+a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \sinh ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \sinh ^{-1}(a x) \log \left (c x^n\right )-(d n) \int \sinh ^{-1}(a x) \, dx+\frac{\left (\left (3 a^2 d-e\right ) n\right ) \int \frac{\sqrt{1+a^2 x^2}}{x} \, dx}{3 a^3}-\frac{1}{3} (e n) \int x^2 \sinh ^{-1}(a x) \, dx+\frac{(e n) \int \frac{\left (1+a^2 x^2\right )^{3/2}}{x} \, dx}{9 a^3}\\ &=-d n x \sinh ^{-1}(a x)-\frac{1}{9} e n x^3 \sinh ^{-1}(a x)-\frac{\left (3 a^2 d-e\right ) \sqrt{1+a^2 x^2} \log \left (c x^n\right )}{3 a^3}-\frac{e \left (1+a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \sinh ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \sinh ^{-1}(a x) \log \left (c x^n\right )+(a d n) \int \frac{x}{\sqrt{1+a^2 x^2}} \, dx+\frac{\left (\left (3 a^2 d-e\right ) n\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+a^2 x}}{x} \, dx,x,x^2\right )}{6 a^3}+\frac{(e n) \operatorname{Subst}\left (\int \frac{\left (1+a^2 x\right )^{3/2}}{x} \, dx,x,x^2\right )}{18 a^3}+\frac{1}{9} (a e n) \int \frac{x^3}{\sqrt{1+a^2 x^2}} \, dx\\ &=\frac{d n \sqrt{1+a^2 x^2}}{a}+\frac{\left (3 a^2 d-e\right ) n \sqrt{1+a^2 x^2}}{3 a^3}+\frac{e n \left (1+a^2 x^2\right )^{3/2}}{27 a^3}-d n x \sinh ^{-1}(a x)-\frac{1}{9} e n x^3 \sinh ^{-1}(a x)-\frac{\left (3 a^2 d-e\right ) \sqrt{1+a^2 x^2} \log \left (c x^n\right )}{3 a^3}-\frac{e \left (1+a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \sinh ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \sinh ^{-1}(a x) \log \left (c x^n\right )+\frac{\left (\left (3 a^2 d-e\right ) n\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+a^2 x}} \, dx,x,x^2\right )}{6 a^3}+\frac{(e n) \operatorname{Subst}\left (\int \frac{\sqrt{1+a^2 x}}{x} \, dx,x,x^2\right )}{18 a^3}+\frac{1}{18} (a e n) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+a^2 x}} \, dx,x,x^2\right )\\ &=\frac{d n \sqrt{1+a^2 x^2}}{a}+\frac{\left (3 a^2 d-e\right ) n \sqrt{1+a^2 x^2}}{3 a^3}+\frac{e n \sqrt{1+a^2 x^2}}{9 a^3}+\frac{e n \left (1+a^2 x^2\right )^{3/2}}{27 a^3}-d n x \sinh ^{-1}(a x)-\frac{1}{9} e n x^3 \sinh ^{-1}(a x)-\frac{\left (3 a^2 d-e\right ) \sqrt{1+a^2 x^2} \log \left (c x^n\right )}{3 a^3}-\frac{e \left (1+a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \sinh ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \sinh ^{-1}(a x) \log \left (c x^n\right )+\frac{\left (\left (3 a^2 d-e\right ) n\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{a^2}+\frac{x^2}{a^2}} \, dx,x,\sqrt{1+a^2 x^2}\right )}{3 a^5}+\frac{(e n) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+a^2 x}} \, dx,x,x^2\right )}{18 a^3}+\frac{1}{18} (a e n) \operatorname{Subst}\left (\int \left (-\frac{1}{a^2 \sqrt{1+a^2 x}}+\frac{\sqrt{1+a^2 x}}{a^2}\right ) \, dx,x,x^2\right )\\ &=\frac{d n \sqrt{1+a^2 x^2}}{a}+\frac{\left (3 a^2 d-e\right ) n \sqrt{1+a^2 x^2}}{3 a^3}+\frac{2 e n \left (1+a^2 x^2\right )^{3/2}}{27 a^3}-d n x \sinh ^{-1}(a x)-\frac{1}{9} e n x^3 \sinh ^{-1}(a x)-\frac{\left (3 a^2 d-e\right ) n \tanh ^{-1}\left (\sqrt{1+a^2 x^2}\right )}{3 a^3}-\frac{\left (3 a^2 d-e\right ) \sqrt{1+a^2 x^2} \log \left (c x^n\right )}{3 a^3}-\frac{e \left (1+a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \sinh ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \sinh ^{-1}(a x) \log \left (c x^n\right )+\frac{(e n) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{a^2}+\frac{x^2}{a^2}} \, dx,x,\sqrt{1+a^2 x^2}\right )}{9 a^5}\\ &=\frac{d n \sqrt{1+a^2 x^2}}{a}+\frac{\left (3 a^2 d-e\right ) n \sqrt{1+a^2 x^2}}{3 a^3}+\frac{2 e n \left (1+a^2 x^2\right )^{3/2}}{27 a^3}-d n x \sinh ^{-1}(a x)-\frac{1}{9} e n x^3 \sinh ^{-1}(a x)-\frac{\left (3 a^2 d-e\right ) n \tanh ^{-1}\left (\sqrt{1+a^2 x^2}\right )}{3 a^3}-\frac{e n \tanh ^{-1}\left (\sqrt{1+a^2 x^2}\right )}{9 a^3}-\frac{\left (3 a^2 d-e\right ) \sqrt{1+a^2 x^2} \log \left (c x^n\right )}{3 a^3}-\frac{e \left (1+a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \sinh ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \sinh ^{-1}(a x) \log \left (c x^n\right )\\ \end{align*}

Mathematica [A]  time = 0.149568, size = 240, normalized size = 0.98 \[ \frac{-3 a^3 x \sinh ^{-1}(a x) \left (n \left (9 d+e x^2\right )-3 \left (3 d+e x^2\right ) \log \left (c x^n\right )\right )-27 a^2 d \sqrt{a^2 x^2+1} \log \left (c x^n\right )-3 a^2 e x^2 \sqrt{a^2 x^2+1} \log \left (c x^n\right )+6 e \sqrt{a^2 x^2+1} \log \left (c x^n\right )+3 n \log (x) \left (9 a^2 d-2 e\right )+54 a^2 d n \sqrt{a^2 x^2+1}-27 a^2 d n \log \left (\sqrt{a^2 x^2+1}+1\right )+2 a^2 e n x^2 \sqrt{a^2 x^2+1}-7 e n \sqrt{a^2 x^2+1}+6 e n \log \left (\sqrt{a^2 x^2+1}+1\right )}{27 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(54*a^2*d*n*Sqrt[1 + a^2*x^2] - 7*e*n*Sqrt[1 + a^2*x^2] + 2*a^2*e*n*x^2*Sqrt[1 + a^2*x^2] + 3*(9*a^2*d - 2*e)*
n*Log[x] - 27*a^2*d*Sqrt[1 + a^2*x^2]*Log[c*x^n] + 6*e*Sqrt[1 + a^2*x^2]*Log[c*x^n] - 3*a^2*e*x^2*Sqrt[1 + a^2
*x^2]*Log[c*x^n] - 3*a^3*x*ArcSinh[a*x]*(n*(9*d + e*x^2) - 3*(3*d + e*x^2)*Log[c*x^n]) - 27*a^2*d*n*Log[1 + Sq
rt[1 + a^2*x^2]] + 6*e*n*Log[1 + Sqrt[1 + a^2*x^2]])/(27*a^3)

________________________________________________________________________________________

Maple [C]  time = 1.861, size = 4077, normalized size = 16.7 \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)*arcsinh(a*x)*ln(c*x^n),x)

[Out]

-1/18*I/a*csgn(I/a)*Pi*(a^2*x^2+1)^(1/2)*csgn(I/a*((a*x+(a^2*x^2+1)^(1/2))^2-1)/(a*x+(a^2*x^2+1)^(1/2)))^2*x^2
*e*n-1/18*I/a*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x+(a^2*x^2+1)^(1/2))^2-1))^2*csgn(I/(a*x+(a^2*x^2+1)^(1/2)))*
Pi*(a^2*x^2+1)^(1/2)*x^2*e*n-1/18*I/a*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x+(a^2*x^2+1)^(1/2))^2-1))^2*Pi*(a^2*
x^2+1)^(1/2)*csgn(I*((a*x+(a^2*x^2+1)^(1/2))^2-1))*x^2*e*n-1/18*I/a*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x+(a^2*
x^2+1)^(1/2))^2-1))*Pi*(a^2*x^2+1)^(1/2)*csgn(I/a*((a*x+(a^2*x^2+1)^(1/2))^2-1)/(a*x+(a^2*x^2+1)^(1/2)))^2*x^2
*e*n+1/2*I/a*csgn(I/a)*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x+(a^2*x^2+1)^(1/2))^2-1))*Pi*(a^2*x^2+1)^(1/2)*csgn
(I/a*((a*x+(a^2*x^2+1)^(1/2))^2-1)/(a*x+(a^2*x^2+1)^(1/2)))*d*n+1/2*I/a*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x+(
a^2*x^2+1)^(1/2))^2-1))*csgn(I/(a*x+(a^2*x^2+1)^(1/2)))*Pi*(a^2*x^2+1)^(1/2)*csgn(I*((a*x+(a^2*x^2+1)^(1/2))^2
-1))*d*n+2*d*n*(a^2*x^2+1)^(1/2)/a-1/9*e*n*x^3*arcsinh(a*x)-d*n*x*arcsinh(a*x)+1/18*I/a*csgn(I/a)*csgn(I/(a*x+
(a^2*x^2+1)^(1/2))*((a*x+(a^2*x^2+1)^(1/2))^2-1))*Pi*(a^2*x^2+1)^(1/2)*csgn(I/a*((a*x+(a^2*x^2+1)^(1/2))^2-1)/
(a*x+(a^2*x^2+1)^(1/2)))*x^2*e*n+1/18*I/a*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x+(a^2*x^2+1)^(1/2))^2-1))*csgn(I
/(a*x+(a^2*x^2+1)^(1/2)))*Pi*(a^2*x^2+1)^(1/2)*csgn(I*((a*x+(a^2*x^2+1)^(1/2))^2-1))*x^2*e*n-1/9*I/a^3*csgn(I/
a)*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x+(a^2*x^2+1)^(1/2))^2-1))*Pi*(a^2*x^2+1)^(1/2)*csgn(I/a*((a*x+(a^2*x^2+
1)^(1/2))^2-1)/(a*x+(a^2*x^2+1)^(1/2)))*e*n-1/9*I/a^3*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x+(a^2*x^2+1)^(1/2))^
2-1))*csgn(I/(a*x+(a^2*x^2+1)^(1/2)))*Pi*(a^2*x^2+1)^(1/2)*csgn(I*((a*x+(a^2*x^2+1)^(1/2))^2-1))*e*n-1/6*I*arc
sinh(a*x)*csgn(I/a)*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x+(a^2*x^2+1)^(1/2))^2-1))*Pi*csgn(I/a*((a*x+(a^2*x^2+1
)^(1/2))^2-1)/(a*x+(a^2*x^2+1)^(1/2)))*x^3*e*n-1/6*I*arcsinh(a*x)*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x+(a^2*x^
2+1)^(1/2))^2-1))*csgn(I/(a*x+(a^2*x^2+1)^(1/2)))*Pi*csgn(I*((a*x+(a^2*x^2+1)^(1/2))^2-1))*x^3*e*n-1/2*I*arcsi
nh(a*x)*csgn(I/a)*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x+(a^2*x^2+1)^(1/2))^2-1))*Pi*csgn(I/a*((a*x+(a^2*x^2+1)^
(1/2))^2-1)/(a*x+(a^2*x^2+1)^(1/2)))*x*d*n-1/2*I*arcsinh(a*x)*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x+(a^2*x^2+1)
^(1/2))^2-1))*csgn(I/(a*x+(a^2*x^2+1)^(1/2)))*Pi*csgn(I*((a*x+(a^2*x^2+1)^(1/2))^2-1))*x*d*n+1/18*I/a*Pi*(a^2*
x^2+1)^(1/2)*csgn(I/a*((a*x+(a^2*x^2+1)^(1/2))^2-1)/(a*x+(a^2*x^2+1)^(1/2)))^3*x^2*e*n-1/2*I/a*csgn(I/a)*Pi*(a
^2*x^2+1)^(1/2)*csgn(I/a*((a*x+(a^2*x^2+1)^(1/2))^2-1)/(a*x+(a^2*x^2+1)^(1/2)))^2*d*n-1/2*I/a*csgn(I/(a*x+(a^2
*x^2+1)^(1/2))*((a*x+(a^2*x^2+1)^(1/2))^2-1))^2*csgn(I/(a*x+(a^2*x^2+1)^(1/2)))*Pi*(a^2*x^2+1)^(1/2)*d*n-1/2*I
/a*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x+(a^2*x^2+1)^(1/2))^2-1))^2*Pi*(a^2*x^2+1)^(1/2)*csgn(I*((a*x+(a^2*x^2+
1)^(1/2))^2-1))*d*n+1/6*I*arcsinh(a*x)*csgn(I/a)*Pi*csgn(I/a*((a*x+(a^2*x^2+1)^(1/2))^2-1)/(a*x+(a^2*x^2+1)^(1
/2)))^2*x^3*e*n+1/6*I*arcsinh(a*x)*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x+(a^2*x^2+1)^(1/2))^2-1))^2*csgn(I/(a*x
+(a^2*x^2+1)^(1/2)))*Pi*x^3*e*n+1/6*I*arcsinh(a*x)*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x+(a^2*x^2+1)^(1/2))^2-1
))^2*Pi*csgn(I*((a*x+(a^2*x^2+1)^(1/2))^2-1))*x^3*e*n+1/6*I*arcsinh(a*x)*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x+
(a^2*x^2+1)^(1/2))^2-1))*Pi*csgn(I/a*((a*x+(a^2*x^2+1)^(1/2))^2-1)/(a*x+(a^2*x^2+1)^(1/2)))^2*x^3*e*n+1/2*I*ar
csinh(a*x)*csgn(I/a)*Pi*csgn(I/a*((a*x+(a^2*x^2+1)^(1/2))^2-1)/(a*x+(a^2*x^2+1)^(1/2)))^2*x*d*n+1/2*I*arcsinh(
a*x)*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x+(a^2*x^2+1)^(1/2))^2-1))^2*csgn(I/(a*x+(a^2*x^2+1)^(1/2)))*Pi*x*d*n+
1/2*I*arcsinh(a*x)*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x+(a^2*x^2+1)^(1/2))^2-1))^2*Pi*csgn(I*((a*x+(a^2*x^2+1)
^(1/2))^2-1))*x*d*n+1/2*I*arcsinh(a*x)*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x+(a^2*x^2+1)^(1/2))^2-1))*Pi*csgn(I
/a*((a*x+(a^2*x^2+1)^(1/2))^2-1)/(a*x+(a^2*x^2+1)^(1/2)))^2*x*d*n-1/2*I/a*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x
+(a^2*x^2+1)^(1/2))^2-1))*Pi*(a^2*x^2+1)^(1/2)*csgn(I/a*((a*x+(a^2*x^2+1)^(1/2))^2-1)/(a*x+(a^2*x^2+1)^(1/2)))
^2*d*n+1/9*I/a^3*csgn(I/a)*Pi*(a^2*x^2+1)^(1/2)*csgn(I/a*((a*x+(a^2*x^2+1)^(1/2))^2-1)/(a*x+(a^2*x^2+1)^(1/2))
)^2*e*n+1/9*I/a^3*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x+(a^2*x^2+1)^(1/2))^2-1))^2*csgn(I/(a*x+(a^2*x^2+1)^(1/2
)))*Pi*(a^2*x^2+1)^(1/2)*e*n+1/9*I/a^3*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x+(a^2*x^2+1)^(1/2))^2-1))^2*Pi*(a^2
*x^2+1)^(1/2)*csgn(I*((a*x+(a^2*x^2+1)^(1/2))^2-1))*e*n+1/9*I/a^3*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x+(a^2*x^
2+1)^(1/2))^2-1))*Pi*(a^2*x^2+1)^(1/2)*csgn(I/a*((a*x+(a^2*x^2+1)^(1/2))^2-1)/(a*x+(a^2*x^2+1)^(1/2)))^2*e*n+1
/18*I/a*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x+(a^2*x^2+1)^(1/2))^2-1))^3*Pi*(a^2*x^2+1)^(1/2)*x^2*e*n+2/27/a*(a
^2*x^2+1)^(1/2)*x^2*e*n+1/a*ln(a)*(a^2*x^2+1)^(1/2)*d*n+1/a*ln(2)*(a^2*x^2+1)^(1/2)*d*n-1/a*ln((a*x+(a^2*x^2+1
)^(1/2))^2-1)*(a^2*x^2+1)^(1/2)*d*n-2/9/a^3*ln(a)*(a^2*x^2+1)^(1/2)*e*n-2/9/a^3*ln(2)*(a^2*x^2+1)^(1/2)*e*n+2/
9/a^3*ln((a*x+(a^2*x^2+1)^(1/2))^2-1)*(a^2*x^2+1)^(1/2)*e*n-1/3*arcsinh(a*x)*ln(a)*x^3*e*n-1/3*arcsinh(a*x)*ln
(2)*x^3*e*n+1/3*arcsinh(a*x)*ln((a*x+(a^2*x^2+1)^(1/2))^2-1)*x^3*e*n-arcsinh(a*x)*ln(a)*x*d*n-arcsinh(a*x)*ln(
2)*x*d*n+arcsinh(a*x)*ln((a*x+(a^2*x^2+1)^(1/2))^2-1)*x*d*n-1/9/a*ln((a*x+(a^2*x^2+1)^(1/2))^2-1)*(a^2*x^2+1)^
(1/2)*x^2*e*n+1/9/a*ln(a)*(a^2*x^2+1)^(1/2)*x^2*e*n+1/9/a*ln(2)*(a^2*x^2+1)^(1/2)*x^2*e*n-1/9/a*(a^2*x^2+1)^(1
/2)*x^2*e*(ln(c*x^n)-n*ln(x))-1/6*I*arcsinh(a*x)*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x+(a^2*x^2+1)^(1/2))^2-1))
^3*Pi*x^3*e*n-1/6*I*arcsinh(a*x)*Pi*csgn(I/a*((a*x+(a^2*x^2+1)^(1/2))^2-1)/(a*x+(a^2*x^2+1)^(1/2)))^3*x^3*e*n-
1/2*I*arcsinh(a*x)*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x+(a^2*x^2+1)^(1/2))^2-1))^3*Pi*x*d*n-1/2*I*arcsinh(a*x)
*Pi*csgn(I/a*((a*x+(a^2*x^2+1)^(1/2))^2-1)/(a*x+(a^2*x^2+1)^(1/2)))^3*x*d*n+1/2*I/a*csgn(I/(a*x+(a^2*x^2+1)^(1
/2))*((a*x+(a^2*x^2+1)^(1/2))^2-1))^3*Pi*(a^2*x^2+1)^(1/2)*d*n+1/2*I/a*Pi*(a^2*x^2+1)^(1/2)*csgn(I/a*((a*x+(a^
2*x^2+1)^(1/2))^2-1)/(a*x+(a^2*x^2+1)^(1/2)))^3*d*n-1/9*I/a^3*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*((a*x+(a^2*x^2+1)
^(1/2))^2-1))^3*Pi*(a^2*x^2+1)^(1/2)*e*n-1/9*I/a^3*Pi*(a^2*x^2+1)^(1/2)*csgn(I/a*((a*x+(a^2*x^2+1)^(1/2))^2-1)
/(a*x+(a^2*x^2+1)^(1/2)))^3*e*n-1/a*ln(1+a*x+(a^2*x^2+1)^(1/2))*d*n+1/a*ln(a*x+(a^2*x^2+1)^(1/2)-1)*d*n-7/27/a
^3*(a^2*x^2+1)^(1/2)*e*n-1/9/a^3*n*(3*arcsinh(a*x)*x^3*a^3*e+9*arcsinh(a*x)*x*a^3*d-(a^2*x^2+1)^(1/2)*x^2*a^2*
e-9*(a^2*x^2+1)^(1/2)*a^2*d+2*(a^2*x^2+1)^(1/2)*e)*ln(a*x+(a^2*x^2+1)^(1/2))+2/9/a^3*ln(1+a*x+(a^2*x^2+1)^(1/2
))*e*n-2/9/a^3*ln(a*x+(a^2*x^2+1)^(1/2)-1)*e*n+2/9/a^3*(a^2*x^2+1)^(1/2)*e*(ln(c*x^n)-n*ln(x))+1/3*arcsinh(a*x
)*x^3*e*(ln(c*x^n)-n*ln(x))+arcsinh(a*x)*x*d*(ln(c*x^n)-n*ln(x))-1/a*(a^2*x^2+1)^(1/2)*d*(ln(c*x^n)-n*ln(x))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a^{2} d n{\left (\frac{2 \, x}{a^{2}} + \frac{i \,{\left (\log \left (i \, a x + 1\right ) - \log \left (-i \, a x + 1\right )\right )}}{a^{3}}\right )} + \frac{1}{54} \, a^{2} e n{\left (\frac{2 \,{\left (a^{2} x^{3} - 3 \, x\right )}}{a^{4}} - \frac{3 i \,{\left (\log \left (i \, a x + 1\right ) - \log \left (-i \, a x + 1\right )\right )}}{a^{5}}\right )} - 3 \, a^{2} e n \int \frac{x^{4} \log \left (x\right )}{9 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} - 9 \, a^{2} d n \int \frac{x^{2} \log \left (x\right )}{9 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} - \frac{1}{2} \, a^{2} d{\left (\frac{2 \, x}{a^{2}} + \frac{i \,{\left (\log \left (i \, a x + 1\right ) - \log \left (-i \, a x + 1\right )\right )}}{a^{3}}\right )} \log \left (c\right ) - \frac{1}{18} \, a^{2} e{\left (\frac{2 \,{\left (a^{2} x^{3} - 3 \, x\right )}}{a^{4}} - \frac{3 i \,{\left (\log \left (i \, a x + 1\right ) - \log \left (-i \, a x + 1\right )\right )}}{a^{5}}\right )} \log \left (c\right ) - \frac{1}{9} \,{\left ({\left (e n - 3 \, e \log \left (c\right )\right )} x^{3} + 9 \,{\left (d n - d \log \left (c\right )\right )} x - 3 \,{\left (e x^{3} + 3 \, d x\right )} \log \left (x^{n}\right )\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - \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} + a x +{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*d*n*(2*x/a^2 + I*(log(I*a*x + 1) - log(-I*a*x + 1))/a^3) + 1/54*a^2*e*n*(2*(a^2*x^3 - 3*x)/a^4 - 3*I*(
log(I*a*x + 1) - log(-I*a*x + 1))/a^5) - 3*a^2*e*n*integrate(1/9*x^4*log(x)/(a^2*x^2 + 1), x) - 9*a^2*d*n*inte
grate(1/9*x^2*log(x)/(a^2*x^2 + 1), x) - 1/2*a^2*d*(2*x/a^2 + I*(log(I*a*x + 1) - log(-I*a*x + 1))/a^3)*log(c)
- 1/18*a^2*e*(2*(a^2*x^3 - 3*x)/a^4 - 3*I*(log(I*a*x + 1) - log(-I*a*x + 1))/a^5)*log(c) - 1/9*((e*n - 3*e*lo
g(c))*x^3 + 9*(d*n - d*log(c))*x - 3*(e*x^3 + 3*d*x)*log(x^n))*log(a*x + sqrt(a^2*x^2 + 1)) - 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*x + (a^2*x^2
+ 1)^(3/2)), x)

________________________________________________________________________________________

Fricas [A]  time = 1.32468, size = 707, normalized size = 2.9 \begin{align*} -\frac{3 \,{\left (9 \, a^{2} d - 2 \, e\right )} n \log \left (-a x + \sqrt{a^{2} x^{2} + 1} + 1\right ) - 3 \,{\left (9 \, a^{2} d - 2 \, e\right )} n \log \left (-a x + \sqrt{a^{2} x^{2} + 1} - 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)*arcsinh(a*x)*log(c*x^n),x, algorithm="fricas")

[Out]

-1/27*(3*(9*a^2*d - 2*e)*n*log(-a*x + sqrt(a^2*x^2 + 1) + 1) - 3*(9*a^2*d - 2*e)*n*log(-a*x + sqrt(a^2*x^2 + 1
) - 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

________________________________________________________________________________________

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)*asinh(a*x)*ln(c*x**n),x)

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

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