3.2.87 \(\int (d+e x^2) \cos ^{-1}(a x) \log (c x^n) \, dx\) [187]

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

[Out]

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

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Rubi [A]
time = 0.16, antiderivative size = 245, normalized size of antiderivative = 1.00, 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 = {4756, 455, 45, 2434, 272, 52, 65, 214, 4716, 267, 4724} \begin {gather*} \frac {d n \sqrt {1-a^2 x^2}}{a}-\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 {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 \text {ArcCos}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \text {ArcCos}(a x) \log \left (c x^n\right )-d n x \text {ArcCos}(a x)-\frac {1}{9} e n x^3 \text {ArcCos}(a x) \end {gather*}

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 45

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 52

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 65

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 214

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

Rule 267

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 272

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 455

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 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 4716

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 4724

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]

Rule 4756

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])

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) \text {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 ) \text {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) \text {Subst}\left (\int \frac {\sqrt {1-a^2 x}}{x} \, dx,x,x^2\right )}{18 a^3}-\frac {1}{18} (a e n) \text {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 ) \text {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) \text {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) \text {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 ) \text {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) \text {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*}

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Mathematica [A]
time = 0.13, size = 248, normalized size = 1.01 \begin {gather*} -\frac {-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 \left (9 a^2 d+2 e\right ) n \log (x)+27 a^2 d \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 a^2 e x^2 \sqrt {1-a^2 x^2} \log \left (c x^n\right )+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 n \log \left (1+\sqrt {1-a^2 x^2}\right )+6 e n \log \left (1+\sqrt {1-a^2 x^2}\right )}{27 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.76, size = 5619, normalized size = 22.93

method	result	size
default	\(\text {Expression too large to display}\)	\(5619\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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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)*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*n*(2*(a^2*x^3 + 3*x)/a^4 - 3*log(a
*x + 1)/a^5 + 3*log(a*x - 1)/a^5)*e - 162*a^2*n*e*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*(2*(a^2*x^3 + 3*x)/a^4 - 3*log(a*x + 1)/a^5 + 3*log(a*x - 1)/a^5)*e*log(c))*a^3 - 2*(-2*I*a^3*n*e + 3*I*a^
3*e*log(c))*x^3 + 54*a^3*integrate(-1/9*((a*n*e - 3*a*e*log(c))*x^3 + 9*(a*d*n - a*d*log(c))*x - 3*(a*x^3*e +
3*a*d*x)*log(x^n))*sqrt(a*x + 1)*sqrt(-a*x + 1)/(a^2*x^2 - 1), x) - 9*(3*I*a^2*d + I*e)*n*dilog(a*x) - 9*(-3*I
*a^2*d - I*e)*n*dilog(-a*x) - 6*(9*I*a^3*d*log(c) + 3*I*a*e*log(c) + 2*(-9*I*a^3*d - 2*I*a*e)*n)*x + 6*((a^3*n
*e - 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) - 3*(-9*I*
a^2*d*log(c) + (9*I*a^2*d + I*e)*n - 3*I*e*log(c))*log(a*x + 1) - 3*(9*I*a^2*d*log(c) + (-9*I*a^2*d - I*e)*n +
3*I*e*log(c))*log(a*x - 1) - 3*(2*I*a^3*x^3*e + 6*(3*I*a^3*d + I*a*e)*x + 6*(a^3*x^3*e + 3*a^3*d*x)*arctan2(s
qrt(a*x + 1)*sqrt(-a*x + 1), a*x) + 3*(-3*I*a^2*d - I*e)*log(a*x + 1) + 3*(3*I*a^2*d + I*e)*log(-a*x + 1))*log
(x^n))/a^3

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Fricas [A]
time = 0.63, size = 320, normalized size = 1.31 \begin {gather*} \frac {18 \, {\left (3 \, a^{3} d x - 3 \, a^{3} d + {\left (a^{3} x^{3} - a^{3}\right )} e\right )} \arccos \left (a x\right ) \log \left (c\right ) + 18 \, {\left (a^{3} n x^{3} e + 3 \, a^{3} d n x\right )} \arccos \left (a x\right ) \log \left (x\right ) - 6 \, {\left (9 \, a^{3} d n x - 9 \, a^{3} d n + {\left (a^{3} n x^{3} - a^{3} n\right )} e\right )} \arccos \left (a x\right ) - 6 \, {\left (9 \, a^{3} d n + a^{3} n e - 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 ) - 3 \, {\left (9 \, a^{2} d n + 2 \, n e\right )} \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right ) + 3 \, {\left (9 \, a^{2} d n + 2 \, n e\right )} \log \left (\sqrt {-a^{2} x^{2} + 1} - 1\right ) + 2 \, {\left (54 \, a^{2} d n + {\left (2 \, a^{2} n x^{2} + 7 \, n\right )} e - 3 \, {\left (9 \, a^{2} d + {\left (a^{2} x^{2} + 2\right )} e\right )} \log \left (c\right ) - 3 \, {\left (9 \, a^{2} d n + {\left (a^{2} n x^{2} + 2 \, n\right )} e\right )} \log \left (x\right )\right )} \sqrt {-a^{2} x^{2} + 1}}{54 \, a^{3}} \end {gather*}

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*(3*a^3*d*x - 3*a^3*d + (a^3*x^3 - a^3)*e)*arccos(a*x)*log(c) + 18*(a^3*n*x^3*e + 3*a^3*d*n*x)*arccos(
a*x)*log(x) - 6*(9*a^3*d*n*x - 9*a^3*d*n + (a^3*n*x^3 - a^3*n)*e)*arccos(a*x) - 6*(9*a^3*d*n + a^3*n*e - 3*(3*
a^3*d + a^3*e)*log(c))*arctan(sqrt(-a^2*x^2 + 1)*a*x/(a^2*x^2 - 1)) - 3*(9*a^2*d*n + 2*n*e)*log(sqrt(-a^2*x^2
+ 1) + 1) + 3*(9*a^2*d*n + 2*n*e)*log(sqrt(-a^2*x^2 + 1) - 1) + 2*(54*a^2*d*n + (2*a^2*n*x^2 + 7*n)*e - 3*(9*a
^2*d + (a^2*x^2 + 2)*e)*log(c) - 3*(9*a^2*d*n + (a^2*n*x^2 + 2*n)*e)*log(x))*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 {acos}{\left (a x \right )}\, dx \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 2136 vs. \(2 (223) = 446\).
time = 7.26, size = 2136, normalized size = 8.72 \begin {gather*} \text {Too large to display} \end {gather*}

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]

1/3*n*x^3*arccos(a*x)*e*log(a*x) - 1/3*n*x^3*arccos(a*x)*e*log(a) + 1/3*x^3*arccos(a*x)*e*log(c) + d*n*x*arcco
s(a*x)*log(a*x) - d*n*x*arccos(a*x)*log(a) - 1/9*sqrt(-a^2*x^2 + 1)*n*x^2*e*log(a*x)/a + 1/9*sqrt(-a^2*x^2 + 1
)*n*x^2*e*log(a)/a + d*x*arccos(a*x)*log(c) - 1/9*sqrt(-a^2*x^2 + 1)*x^2*e*log(c)/a - sqrt(-a^2*x^2 + 1)*d*n*l
og(a*x)/a + sqrt(-a^2*x^2 + 1)*d*n*log(a)/a - sqrt(-a^2*x^2 + 1)*d*log(c)/a + d*n*arccos(a*x)/(a*((a^2*x^2 - 1
)/(a*x + 1)^2 - 1)) + d*n*log(abs(a*x + sqrt(-a^2*x^2 + 1) + 1))/(a*((a^2*x^2 - 1)/(a*x + 1)^2 - 1)) - d*n*log
(abs(-a*x + sqrt(-a^2*x^2 + 1) - 1))/(a*((a^2*x^2 - 1)/(a*x + 1)^2 - 1)) + 4*sqrt(-a^2*x^2 + 1)*d*n/((a*x - (a
^2*x^2 - 1)*a*x/(a*x + 1)^2 - (a^2*x^2 - 1)/(a*x + 1)^2 + 1)*a) - 2/9*sqrt(-a^2*x^2 + 1)*n*e*log(a*x)/a^3 + 2/
9*sqrt(-a^2*x^2 + 1)*n*e*log(a)/a^3 + (a^2*x^2 - 1)*d*n*arccos(a*x)/((a*x + 1)^2*a*((a^2*x^2 - 1)/(a*x + 1)^2
- 1)) - (a^2*x^2 - 1)*d*n*log(abs(a*x + sqrt(-a^2*x^2 + 1) + 1))/((a*x + 1)^2*a*((a^2*x^2 - 1)/(a*x + 1)^2 - 1
)) + (a^2*x^2 - 1)*d*n*log(abs(-a*x + sqrt(-a^2*x^2 + 1) - 1))/((a*x + 1)^2*a*((a^2*x^2 - 1)/(a*x + 1)^2 - 1))
- 2/9*sqrt(-a^2*x^2 + 1)*e*log(c)/a^3 + 1/9*n*arccos(a*x)*e/(a^3*(3*(a^2*x^2 - 1)/(a*x + 1)^2 - 3*(a^2*x^2 -
1)^2/(a*x + 1)^4 + (a^2*x^2 - 1)^3/(a*x + 1)^6 - 1)) + 2/9*n*e*log(abs(a*x + sqrt(-a^2*x^2 + 1) + 1))/(a^3*(3*
(a^2*x^2 - 1)/(a*x + 1)^2 - 3*(a^2*x^2 - 1)^2/(a*x + 1)^4 + (a^2*x^2 - 1)^3/(a*x + 1)^6 - 1)) - 2/9*n*e*log(ab
s(-a*x + sqrt(-a^2*x^2 + 1) - 1))/(a^3*(3*(a^2*x^2 - 1)/(a*x + 1)^2 - 3*(a^2*x^2 - 1)^2/(a*x + 1)^4 + (a^2*x^2
- 1)^3/(a*x + 1)^6 - 1)) + 2/3*sqrt(-a^2*x^2 + 1)*n*e/((a*x - 3*(a^2*x^2 - 1)*a*x/(a*x + 1)^2 + 3*(a^2*x^2 -
1)^2*a*x/(a*x + 1)^4 - 3*(a^2*x^2 - 1)/(a*x + 1)^2 - (a^2*x^2 - 1)^3*a*x/(a*x + 1)^6 + 3*(a^2*x^2 - 1)^2/(a*x
+ 1)^4 - (a^2*x^2 - 1)^3/(a*x + 1)^6 + 1)*a^3) + 1/3*(a^2*x^2 - 1)*n*arccos(a*x)*e/((a*x + 1)^2*a^3*(3*(a^2*x^
2 - 1)/(a*x + 1)^2 - 3*(a^2*x^2 - 1)^2/(a*x + 1)^4 + (a^2*x^2 - 1)^3/(a*x + 1)^6 - 1)) - 2/3*(a^2*x^2 - 1)*n*e
*log(abs(a*x + sqrt(-a^2*x^2 + 1) + 1))/((a*x + 1)^2*a^3*(3*(a^2*x^2 - 1)/(a*x + 1)^2 - 3*(a^2*x^2 - 1)^2/(a*x
+ 1)^4 + (a^2*x^2 - 1)^3/(a*x + 1)^6 - 1)) + 2/3*(a^2*x^2 - 1)*n*e*log(abs(-a*x + sqrt(-a^2*x^2 + 1) - 1))/((
a*x + 1)^2*a^3*(3*(a^2*x^2 - 1)/(a*x + 1)^2 - 3*(a^2*x^2 - 1)^2/(a*x + 1)^4 + (a^2*x^2 - 1)^3/(a*x + 1)^6 - 1)
) + 1/3*(a^2*x^2 - 1)^2*n*arccos(a*x)*e/((a*x + 1)^4*a^3*(3*(a^2*x^2 - 1)/(a*x + 1)^2 - 3*(a^2*x^2 - 1)^2/(a*x
+ 1)^4 + (a^2*x^2 - 1)^3/(a*x + 1)^6 - 1)) + 2/3*(a^2*x^2 - 1)^2*n*e*log(abs(a*x + sqrt(-a^2*x^2 + 1) + 1))/(
(a*x + 1)^4*a^3*(3*(a^2*x^2 - 1)/(a*x + 1)^2 - 3*(a^2*x^2 - 1)^2/(a*x + 1)^4 + (a^2*x^2 - 1)^3/(a*x + 1)^6 - 1
)) - 2/3*(a^2*x^2 - 1)^2*n*e*log(abs(-a*x + sqrt(-a^2*x^2 + 1) - 1))/((a*x + 1)^4*a^3*(3*(a^2*x^2 - 1)/(a*x +
1)^2 - 3*(a^2*x^2 - 1)^2/(a*x + 1)^4 + (a^2*x^2 - 1)^3/(a*x + 1)^6 - 1)) - 20/27*(-a^2*x^2 + 1)^(3/2)*n*e/((a*
x + 1)^3*a^3*(3*(a^2*x^2 - 1)/(a*x + 1)^2 - 3*(a^2*x^2 - 1)^2/(a*x + 1)^4 + (a^2*x^2 - 1)^3/(a*x + 1)^6 - 1))
+ 1/9*(a^2*x^2 - 1)^3*n*arccos(a*x)*e/((a*x + 1)^6*a^3*(3*(a^2*x^2 - 1)/(a*x + 1)^2 - 3*(a^2*x^2 - 1)^2/(a*x +
1)^4 + (a^2*x^2 - 1)^3/(a*x + 1)^6 - 1)) - 2/9*(a^2*x^2 - 1)^3*n*e*log(abs(a*x + sqrt(-a^2*x^2 + 1) + 1))/((a
*x + 1)^6*a^3*(3*(a^2*x^2 - 1)/(a*x + 1)^2 - 3*(a^2*x^2 - 1)^2/(a*x + 1)^4 + (a^2*x^2 - 1)^3/(a*x + 1)^6 - 1))
+ 2/9*(a^2*x^2 - 1)^3*n*e*log(abs(-a*x + sqrt(-a^2*x^2 + 1) - 1))/((a*x + 1)^6*a^3*(3*(a^2*x^2 - 1)/(a*x + 1)
^2 - 3*(a^2*x^2 - 1)^2/(a*x + 1)^4 + (a^2*x^2 - 1)^3/(a*x + 1)^6 - 1)) - 2/3*(a^2*x^2 - 1)^2*sqrt(-a^2*x^2 + 1
)*n*e/((a*x + 1)^5*a^3*(3*(a^2*x^2 - 1)/(a*x + 1)^2 - 3*(a^2*x^2 - 1)^2/(a*x + 1)^4 + (a^2*x^2 - 1)^3/(a*x + 1
)^6 - 1))

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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 {acos}\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)*acos(a*x)*(d + e*x^2),x)

[Out]

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

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