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

Optimal. Leaf size=245 \[ -\frac{\sqrt{1-a^2 x^2} \left (3 a^2 d+e\right ) \log \left (c x^n\right )}{3 a^3}+\frac{e \left (1-a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+\frac{n \sqrt{1-a^2 x^2} \left (3 a^2 d+e\right )}{3 a^3}-\frac{n \left (3 a^2 d+e\right ) \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{3 a^3}+\frac{d n \sqrt{1-a^2 x^2}}{a}-\frac{2 e n \left (1-a^2 x^2\right )^{3/2}}{27 a^3}+\frac{e n \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{9 a^3}+d x \cos ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \cos ^{-1}(a x) \log \left (c x^n\right )-d n x \cos ^{-1}(a x)-\frac{1}{9} e n x^3 \cos ^{-1}(a x) \]

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

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Rubi [A]  time = 0.227344, antiderivative size = 245, 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 = {4666, 444, 43, 2387, 266, 50, 63, 208, 4620, 261, 4628} \[ -\frac{\sqrt{1-a^2 x^2} \left (3 a^2 d+e\right ) \log \left (c x^n\right )}{3 a^3}+\frac{e \left (1-a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+\frac{n \sqrt{1-a^2 x^2} \left (3 a^2 d+e\right )}{3 a^3}-\frac{n \left (3 a^2 d+e\right ) \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{3 a^3}+\frac{d n \sqrt{1-a^2 x^2}}{a}-\frac{2 e n \left (1-a^2 x^2\right )^{3/2}}{27 a^3}+\frac{e n \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{9 a^3}+d x \cos ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \cos ^{-1}(a x) \log \left (c x^n\right )-d n x \cos ^{-1}(a x)-\frac{1}{9} e n x^3 \cos ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)*ArcCos[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*ArcCos[a*x] - (e*n*x^3*ArcCos[a*x])/9 + (e*n*ArcTanh[Sqrt[1 - a^2*x^2]])/(9*a^3) - ((3*a^2*d + e)*n*A
rcTanh[Sqrt[1 - a^2*x^2]])/(3*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*ArcCos[a*x]*Log[c*x^n] + (e*x^3*ArcCos[a*x]*Log[c*x^n])/3

Rule 4666

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2)
^p, x]}, Dist[a + b*ArcCos[c*x], u, x] + Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; F
reeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (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 4620

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCos[c*x])^n, x] + Dist[b*c*n, Int[
(x*(a + b*ArcCos[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 4628

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcCo
s[c*x])^n)/(d*(m + 1)), x] + Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCos[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 ) \cos ^{-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 \cos ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \cos ^{-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 \cos ^{-1}(a x)+\frac{1}{3} e x^2 \cos ^{-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 \cos ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \cos ^{-1}(a x) \log \left (c x^n\right )-(d n) \int \cos ^{-1}(a x) \, dx-\frac{1}{3} (e n) \int x^2 \cos ^{-1}(a x) \, dx-\frac{(e n) \int \frac{\left (1-a^2 x^2\right )^{3/2}}{x} \, dx}{9 a^3}+\frac{\left (\left (3 a^2 d+e\right ) n\right ) \int \frac{\sqrt{1-a^2 x^2}}{x} \, dx}{3 a^3}\\ &=-d n x \cos ^{-1}(a x)-\frac{1}{9} e n x^3 \cos ^{-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 \cos ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \cos ^{-1}(a x) \log \left (c x^n\right )-(a d n) \int \frac{x}{\sqrt{1-a^2 x^2}} \, dx-\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{\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{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 \cos ^{-1}(a x)-\frac{1}{9} e n x^3 \cos ^{-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 \cos ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \cos ^{-1}(a x) \log \left (c x^n\right )-\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{\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{d n \sqrt{1-a^2 x^2}}{a}-\frac{e n \sqrt{1-a^2 x^2}}{9 a^3}+\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 \cos ^{-1}(a x)-\frac{1}{9} e n x^3 \cos ^{-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 \cos ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \cos ^{-1}(a x) \log \left (c x^n\right )-\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{\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{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 \cos ^{-1}(a x)-\frac{1}{9} e n x^3 \cos ^{-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 \cos ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \cos ^{-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 \cos ^{-1}(a x)-\frac{1}{9} e n x^3 \cos ^{-1}(a x)+\frac{e n \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{9 a^3}-\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 \cos ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \cos ^{-1}(a x) \log \left (c x^n\right )\\ \end{align*}

Mathematica [A]  time = 0.175422, size = 248, normalized size = 1.01 \[ -\frac{3 a^3 x \cos ^{-1}(a x) \left (n \left (9 d+e x^2\right )-3 \left (3 d+e x^2\right ) \log \left (c x^n\right )\right )+27 a^2 d \sqrt{1-a^2 x^2} \log \left (c x^n\right )+3 a^2 e x^2 \sqrt{1-a^2 x^2} \log \left (c x^n\right )+6 e \sqrt{1-a^2 x^2} \log \left (c x^n\right )-3 n \log (x) \left (9 a^2 d+2 e\right )-54 a^2 d n \sqrt{1-a^2 x^2}+27 a^2 d n \log \left (\sqrt{1-a^2 x^2}+1\right )-2 a^2 e n x^2 \sqrt{1-a^2 x^2}-7 e n \sqrt{1-a^2 x^2}+6 e n \log \left (\sqrt{1-a^2 x^2}+1\right )}{27 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2)*ArcCos[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*ArcCos[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 + S
qrt[1 - a^2*x^2]] + 6*e*n*Log[1 + Sqrt[1 - a^2*x^2]])/(27*a^3)

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Maple [C]  time = 2.079, size = 8281, normalized size = 33.8 \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)*arccos(a*x)*ln(c*x^n),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/54*(-I*(27*a^2*d*n*(2*x/a^2 - log(a*x + 1)/a^3 + log(a*x - 1)/a^3) + a^2*e*n*(2*(a^2*x^3 + 3*x)/a^4 - 3*log
(a*x + 1)/a^5 + 3*log(a*x - 1)/a^5) - 162*a^2*e*n*integrate(1/9*x^4*log(x)/(a^2*x^2 - 1), x) - 486*a^2*d*n*int
egrate(1/9*x^2*log(x)/(a^2*x^2 - 1), x) - 27*a^2*d*(2*x/a^2 - log(a*x + 1)/a^3 + log(a*x - 1)/a^3)*log(c) - 3*
a^2*e*(2*(a^2*x^3 + 3*x)/a^4 - 3*log(a*x + 1)/a^5 + 3*log(a*x - 1)/a^5)*log(c))*a^3 + (4*I*a^3*e*n - 6*I*a^3*e
*log(c))*x^3 + 54*a^3*integrate(-1/9*((a*e*n - 3*a*e*log(c))*x^3 + 9*(a*d*n - a*d*log(c))*x - 3*(a*e*x^3 + 3*a
*d*x)*log(x^n))*sqrt(a*x + 1)*sqrt(-a*x + 1)/(a^2*x^2 - 1), x) + (-27*I*a^2*d - 9*I*e)*n*dilog(a*x) + (27*I*a^
2*d + 9*I*e)*n*dilog(-a*x) + (-54*I*a^3*d*log(c) - 18*I*a*e*log(c) + (108*I*a^3*d + 24*I*a*e)*n)*x + 6*((a^3*e
*n - 3*a^3*e*log(c))*x^3 + 9*(a^3*d*n - a^3*d*log(c))*x)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x) + (27*I*a^
2*d*log(c) + (-27*I*a^2*d - 3*I*e)*n + 9*I*e*log(c))*log(a*x + 1) + (-27*I*a^2*d*log(c) + (27*I*a^2*d + 3*I*e)
*n - 9*I*e*log(c))*log(a*x - 1) + (-6*I*a^3*e*x^3 + (-54*I*a^3*d - 18*I*a*e)*x - 18*(a^3*e*x^3 + 3*a^3*d*x)*ar
ctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x) + (27*I*a^2*d + 9*I*e)*log(a*x + 1) + (-27*I*a^2*d - 9*I*e)*log(-a*x
+ 1))*log(x^n))/a^3

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Fricas [A]  time = 1.76181, size = 726, normalized size = 2.96 \begin{align*} \frac{18 \,{\left (a^{3} e x^{3} + 3 \, a^{3} d x - 3 \, a^{3} d - a^{3} e\right )} \arccos \left (a x\right ) \log \left (c\right ) + 18 \,{\left (a^{3} e n x^{3} + 3 \, a^{3} d n x\right )} \arccos \left (a x\right ) \log \left (x\right ) - 3 \,{\left (9 \, a^{2} d + 2 \, e\right )} n \log \left (\sqrt{-a^{2} x^{2} + 1} + 1\right ) + 3 \,{\left (9 \, a^{2} d + 2 \, e\right )} n \log \left (\sqrt{-a^{2} x^{2} + 1} - 1\right ) - 6 \,{\left (a^{3} e n x^{3} + 9 \, a^{3} d n x -{\left (9 \, a^{3} d + a^{3} e\right )} n\right )} \arccos \left (a x\right ) - 6 \,{\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 )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} a x}{a^{2} x^{2} - 1}\right ) + 2 \,{\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}}{54 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/54*(18*(a^3*e*x^3 + 3*a^3*d*x - 3*a^3*d - a^3*e)*arccos(a*x)*log(c) + 18*(a^3*e*n*x^3 + 3*a^3*d*n*x)*arccos(
a*x)*log(x) - 3*(9*a^2*d + 2*e)*n*log(sqrt(-a^2*x^2 + 1) + 1) + 3*(9*a^2*d + 2*e)*n*log(sqrt(-a^2*x^2 + 1) - 1
) - 6*(a^3*e*n*x^3 + 9*a^3*d*n*x - (9*a^3*d + a^3*e)*n)*arccos(a*x) - 6*((9*a^3*d + a^3*e)*n - 3*(3*a^3*d + a^
3*e)*log(c))*arctan(sqrt(-a^2*x^2 + 1)*a*x/(a^2*x^2 - 1)) + 2*(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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d + e x^{2}\right ) \log{\left (c x^{n} \right )} \operatorname{acos}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [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)*arccos(a*x)*log(c*x^n),x, algorithm="giac")

[Out]

Timed out