3.2.86 \(\int (d+e x^2) \sin ^{-1}(a x) \log (c x^n) \, dx\) [186]

Optimal. Leaf size=246 \[ -\frac {d n \sqrt {1-a^2 x^2}}{a}-\frac {\left (3 a^2 d+e\right ) n \sqrt {1-a^2 x^2}}{3 a^3}+\frac {2 e n \left (1-a^2 x^2\right )^{3/2}}{27 a^3}-d n x \sin ^{-1}(a x)-\frac {1}{9} e n x^3 \sin ^{-1}(a x)-\frac {e n \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{9 a^3}+\frac {\left (3 a^2 d+e\right ) n \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{3 a^3}+\frac {\left (3 a^2 d+e\right ) \sqrt {1-a^2 x^2} \log \left (c x^n\right )}{3 a^3}-\frac {e \left (1-a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \sin ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-1}(a x) \log \left (c x^n\right ) \]

[Out]

2/27*e*n*(-a^2*x^2+1)^(3/2)/a^3-d*n*x*arcsin(a*x)-1/9*e*n*x^3*arcsin(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*arcsin(a*x)*l
n(c*x^n)+1/3*e*x^3*arcsin(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 = 246, 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 = {4755, 455, 45, 2434, 272, 52, 65, 214, 4715, 267, 4723} \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 {ArcSin}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \text {ArcSin}(a x) \log \left (c x^n\right )-d n x \text {ArcSin}(a x)-\frac {1}{9} e n x^3 \text {ArcSin}(a x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)*ArcSin[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*ArcSin[a*x] - (e*n*x^3*ArcSin[a*x])/9 - (e*n*ArcTanh[Sqrt[1 - a^2*x^2]])/(9*a^3) + ((3*a^2*d + e)*
n*ArcTanh[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*ArcSin[a*x]*Log[c*x^n] + (e*x^3*ArcSin[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 4715

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

Rule 4723

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSi
n[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSin[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 4755

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2)
^p, x]}, Dist[a + b*ArcSin[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 ) \sin ^{-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 \sin ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-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 \sin ^{-1}(a x)+\frac {1}{3} e x^2 \sin ^{-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 \sin ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-1}(a x) \log \left (c x^n\right )-(d n) \int \sin ^{-1}(a x) \, dx-\frac {1}{3} (e n) \int x^2 \sin ^{-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 \sin ^{-1}(a x)-\frac {1}{9} e n x^3 \sin ^{-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 \sin ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-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 \sin ^{-1}(a x)-\frac {1}{9} e n x^3 \sin ^{-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 \sin ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-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 \sin ^{-1}(a x)-\frac {1}{9} e n x^3 \sin ^{-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 \sin ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-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 \sin ^{-1}(a x)-\frac {1}{9} e n x^3 \sin ^{-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 \sin ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-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 \sin ^{-1}(a x)-\frac {1}{9} e n x^3 \sin ^{-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 \sin ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-1}(a x) \log \left (c x^n\right )\\ \end {align*}

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Mathematica [A]
time = 0.12, 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 \sin ^{-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)*ArcSin[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*ArcSin[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)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.98, size = 6894, normalized size = 28.02

method	result	size
default	\(\text {Expression too large to display}\)	\(6894\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)*arcsin(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)*arcsin(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(a*x, sqrt(a*x + 1)*sqrt(-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*(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(a
*x, sqrt(a*x + 1)*sqrt(-a*x + 1)) + 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 = 0.57, size = 231, normalized size = 0.94 \begin {gather*} \frac {18 \, {\left (a^{3} x^{3} e + 3 \, a^{3} d x\right )} \arcsin \left (a x\right ) \log \left (c\right ) + 18 \, {\left (a^{3} n x^{3} e + 3 \, a^{3} d n x\right )} \arcsin \left (a x\right ) \log \left (x\right ) - 6 \, {\left (a^{3} n x^{3} e + 9 \, a^{3} d n x\right )} \arcsin \left (a x\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)*arcsin(a*x)*log(c*x^n),x, algorithm="fricas")

[Out]

1/54*(18*(a^3*x^3*e + 3*a^3*d*x)*arcsin(a*x)*log(c) + 18*(a^3*n*x^3*e + 3*a^3*d*n*x)*arcsin(a*x)*log(x) - 6*(a
^3*n*x^3*e + 9*a^3*d*n*x)*arcsin(a*x) + 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 {asin}{\left (a x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 5311 vs. \(2 (224) = 448\).
time = 6.32, size = 5311, normalized size = 21.59 \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)*arcsin(a*x)*log(c*x^n),x, algorithm="giac")

[Out]

1/54*(54*a^3*d*n*x*arcsin(a*x)*log(a*x) - 54*a^3*d*n*x*arcsin(a*x)*log(a) + 54*a^3*d*x*arcsin(a*x)*log(c) - 10
8*a^3*d*n*x*arcsin(a*x)/(sqrt(-a^2*x^2 + 1)*a^2*x^2/(sqrt(-a^2*x^2 + 1) + 1)^2 + a^2*x^2/(sqrt(-a^2*x^2 + 1) +
1)^2 + sqrt(-a^2*x^2 + 1) + 1) - 54*a^4*d*n*x^2*log(abs(a)*abs(x))/((a^2*x^2/(sqrt(-a^2*x^2 + 1) + 1)^2 + 1)*
(sqrt(-a^2*x^2 + 1) + 1)^2) + 18*(a^2*x^2 - 1)*a*x*arcsin(a*x)*e*log(c) + 54*a^4*d*n*x^2*log(sqrt(-a^2*x^2 + 1
) + 1)/((a^2*x^2/(sqrt(-a^2*x^2 + 1) + 1)^2 + 1)*(sqrt(-a^2*x^2 + 1) + 1)^2) + 54*sqrt(-a^2*x^2 + 1)*a^2*d*n*l
og(a*x) - 54*sqrt(-a^2*x^2 + 1)*a^2*d*n*log(a) + 108*a^4*d*n*x^2/((a^2*x^2/(sqrt(-a^2*x^2 + 1) + 1)^2 + 1)*(sq
rt(-a^2*x^2 + 1) + 1)^2) + 18*a*x*arcsin(a*x)*e*log(c) + 54*sqrt(-a^2*x^2 + 1)*a^2*d*log(c) - 54*a^2*d*n*log(a
bs(a)*abs(x))/(a^2*x^2/(sqrt(-a^2*x^2 + 1) + 1)^2 + 1) + 54*a^2*d*n*log(sqrt(-a^2*x^2 + 1) + 1)/(a^2*x^2/(sqrt
(-a^2*x^2 + 1) + 1)^2 + 1) - 6*(-a^2*x^2 + 1)^(3/2)*e*log(c) - 108*a^2*d*n/(a^2*x^2/(sqrt(-a^2*x^2 + 1) + 1)^2
+ 1) + (18*(a^2*x^2 - 1)*a*x*arcsin(a*x)*log(a*x) - 18*(a^2*x^2 - 1)*a*x*arcsin(a*x)*log(a) + 18*a*x*arcsin(a
*x)*log(a*x) - 18*a*x*arcsin(a*x)*log(a) - 6*(-a^2*x^2 + 1)^(3/2)*log(a*x) + 6*(-a^2*x^2 + 1)^(3/2)*log(a) + 1
8*sqrt(-a^2*x^2 + 1)*log(a*x) - 18*sqrt(-a^2*x^2 + 1)*log(a) - (192*(a^2*x^2 - 1)^2*a^8*x^8*log(abs(a)*abs(x))
/((4*(-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1) + 1)^2*(sqrt(-a^2*x^2 + 1) + 1)^6) - 192*(a^2*x^2 - 1)^2*a^8*
x^8*log(sqrt(-a^2*x^2 + 1) + 1)/((4*(-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1) + 1)^2*(sqrt(-a^2*x^2 + 1) + 1
)^6) - 224*(a^2*x^2 - 1)^2*a^8*x^8/((4*(-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1) + 1)^2*(sqrt(-a^2*x^2 + 1)
+ 1)^6) + 144*(a^2*x^2 - 1)^2*a^7*x^7*arcsin(a*x)/((4*(-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1) + 1)^2*(sqrt
(-a^2*x^2 + 1) + 1)^5) + 96*(a^2*x^2 - 1)*a^8*x^8*log(abs(a)*abs(x))/((4*(-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*x^
2 + 1) + 1)^2*(sqrt(-a^2*x^2 + 1) + 1)^6) - 96*(a^2*x^2 - 1)*a^8*x^8*log(sqrt(-a^2*x^2 + 1) + 1)/((4*(-a^2*x^2
+ 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1) + 1)^2*(sqrt(-a^2*x^2 + 1) + 1)^6) - 112*(a^2*x^2 - 1)*a^8*x^8/((4*(-a^2*x^
2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1) + 1)^2*(sqrt(-a^2*x^2 + 1) + 1)^6) + 12*(a^2*x^2 - 1)*a^7*x^7*arcsin(a*x)/
((4*(-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1) + 1)*(sqrt(-a^2*x^2 + 1) + 1)^6) + 72*(a^2*x^2 - 1)*a^7*x^7*ar
csin(a*x)/((4*(-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1) + 1)^2*(sqrt(-a^2*x^2 + 1) + 1)^5) + 12*a^8*x^8*log(
abs(a)*abs(x))/((4*(-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1) + 1)^2*(sqrt(-a^2*x^2 + 1) + 1)^6) + 576*(a^2*x
^2 - 1)^2*a^6*x^6*log(abs(a)*abs(x))/((4*(-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1) + 1)^2*(sqrt(-a^2*x^2 + 1
) + 1)^4) - 12*a^8*x^8*log(sqrt(-a^2*x^2 + 1) + 1)/((4*(-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1) + 1)^2*(sqr
t(-a^2*x^2 + 1) + 1)^6) - 576*(a^2*x^2 - 1)^2*a^6*x^6*log(sqrt(-a^2*x^2 + 1) + 1)/((4*(-a^2*x^2 + 1)^(3/2) - 3
*sqrt(-a^2*x^2 + 1) + 1)^2*(sqrt(-a^2*x^2 + 1) + 1)^4) - 14*a^8*x^8/((4*(-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2
+ 1) + 1)^2*(sqrt(-a^2*x^2 + 1) + 1)^6) - 336*(a^2*x^2 - 1)^2*a^6*x^6/((4*(-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*
x^2 + 1) + 1)^2*(sqrt(-a^2*x^2 + 1) + 1)^4) + 3*a^7*x^7*arcsin(a*x)/((4*(-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2
+ 1) + 1)*(sqrt(-a^2*x^2 + 1) + 1)^6) + 9*a^7*x^7*arcsin(a*x)/((4*(-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1)
+ 1)^2*(sqrt(-a^2*x^2 + 1) + 1)^5) + 288*(a^2*x^2 - 1)^2*a^5*x^5*arcsin(a*x)/((4*(-a^2*x^2 + 1)^(3/2) - 3*sqr
t(-a^2*x^2 + 1) + 1)^2*(sqrt(-a^2*x^2 + 1) + 1)^3) + 288*(a^2*x^2 - 1)*a^6*x^6*log(abs(a)*abs(x))/((4*(-a^2*x^
2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1) + 1)^2*(sqrt(-a^2*x^2 + 1) + 1)^4) - 288*(a^2*x^2 - 1)*a^6*x^6*log(sqrt(-a
^2*x^2 + 1) + 1)/((4*(-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1) + 1)^2*(sqrt(-a^2*x^2 + 1) + 1)^4) - 168*(a^2
*x^2 - 1)*a^6*x^6/((4*(-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1) + 1)^2*(sqrt(-a^2*x^2 + 1) + 1)^4) + 36*(a^2
*x^2 - 1)*a^5*x^5*arcsin(a*x)/((4*(-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1) + 1)*(sqrt(-a^2*x^2 + 1) + 1)^4)
+ 144*(a^2*x^2 - 1)*a^5*x^5*arcsin(a*x)/((4*(-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1) + 1)^2*(sqrt(-a^2*x^2
+ 1) + 1)^3) + 12*a^6*x^6*log(abs(a)*abs(x))/(sqrt(-a^2*x^2 + 1) + 1)^6 + 36*a^6*x^6*log(abs(a)*abs(x))/((4*(
-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1) + 1)^2*(sqrt(-a^2*x^2 + 1) + 1)^4) + 576*(a^2*x^2 - 1)^2*a^4*x^4*lo
g(abs(a)*abs(x))/((4*(-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1) + 1)^2*(sqrt(-a^2*x^2 + 1) + 1)^2) - 12*a^6*x
^6*log(sqrt(-a^2*x^2 + 1) + 1)/(sqrt(-a^2*x^2 + 1) + 1)^6 - 36*a^6*x^6*log(sqrt(-a^2*x^2 + 1) + 1)/((4*(-a^2*x
^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1) + 1)^2*(sqrt(-a^2*x^2 + 1) + 1)^4) - 576*(a^2*x^2 - 1)^2*a^4*x^4*log(sqrt
(-a^2*x^2 + 1) + 1)/((4*(-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1) + 1)^2*(sqrt(-a^2*x^2 + 1) + 1)^2) - 15*a^
6*x^6/(sqrt(-a^2*x^2 + 1) + 1)^6 - 21*a^6*x^6/((4*(-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1) + 1)^2*(sqrt(-a^
2*x^2 + 1) + 1)^4) + 384*(a^2*x^2 - 1)^2*a^4*x^4/((4*(-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1) + 1)^2*(sqrt(
-a^2*x^2 + 1) + 1)^2) + 9*a^5*x^5*arcsin(a*x)/(...

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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 {asin}\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)*asin(a*x)*(d + e*x^2),x)

[Out]

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

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