3.1.28 \(\int x^2 (a+b \log (c x^n)) \log (d (\frac {1}{d}+f x^2)) \, dx\) [28]

Optimal. Leaf size=241 \[ -\frac {8 b n x}{9 d f}+\frac {4}{27} b n x^3+\frac {2 b n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{9 d^{3/2} f^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}-\frac {1}{9} b n x^3 \log \left (1+d f x^2\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {i b n \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}-\frac {i b n \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}} \]

[Out]

-8/9*b*n*x/d/f+4/27*b*n*x^3+2/9*b*n*arctan(x*d^(1/2)*f^(1/2))/d^(3/2)/f^(3/2)+2/3*x*(a+b*ln(c*x^n))/d/f-2/9*x^
3*(a+b*ln(c*x^n))-2/3*arctan(x*d^(1/2)*f^(1/2))*(a+b*ln(c*x^n))/d^(3/2)/f^(3/2)-1/9*b*n*x^3*ln(d*f*x^2+1)+1/3*
x^3*(a+b*ln(c*x^n))*ln(d*f*x^2+1)+1/3*I*b*n*polylog(2,-I*x*d^(1/2)*f^(1/2))/d^(3/2)/f^(3/2)-1/3*I*b*n*polylog(
2,I*x*d^(1/2)*f^(1/2))/d^(3/2)/f^(3/2)

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Rubi [A]
time = 0.12, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {2505, 308, 211, 2423, 4940, 2438, 209} \begin {gather*} \frac {i b n \text {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}-\frac {i b n \text {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}-\frac {2 \text {ArcTan}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}+\frac {1}{3} x^3 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {2 b n \text {ArcTan}\left (\sqrt {d} \sqrt {f} x\right )}{9 d^{3/2} f^{3/2}}-\frac {1}{9} b n x^3 \log \left (d f x^2+1\right )-\frac {8 b n x}{9 d f}+\frac {4}{27} b n x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*Log[c*x^n])*Log[d*(d^(-1) + f*x^2)],x]

[Out]

(-8*b*n*x)/(9*d*f) + (4*b*n*x^3)/27 + (2*b*n*ArcTan[Sqrt[d]*Sqrt[f]*x])/(9*d^(3/2)*f^(3/2)) + (2*x*(a + b*Log[
c*x^n]))/(3*d*f) - (2*x^3*(a + b*Log[c*x^n]))/9 - (2*ArcTan[Sqrt[d]*Sqrt[f]*x]*(a + b*Log[c*x^n]))/(3*d^(3/2)*
f^(3/2)) - (b*n*x^3*Log[1 + d*f*x^2])/9 + (x^3*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/3 + ((I/3)*b*n*PolyLog[2,
(-I)*Sqrt[d]*Sqrt[f]*x])/(d^(3/2)*f^(3/2)) - ((I/3)*b*n*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x])/(d^(3/2)*f^(3/2))

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 2423

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)^r], x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist
[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] &
& RationalQ[q])) && NeQ[q, -1]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2505

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[(f*x)^(m +
1)*((a + b*Log[c*(d + e*x^n)^p])/(f*(m + 1))), x] - Dist[b*e*n*(p/(f*(m + 1))), Int[x^(n - 1)*((f*x)^(m + 1)/
(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rule 4940

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[I*(b/2), Int[Log[1 - I*c*x
]/x, x], x] - Dist[I*(b/2), Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

Rubi steps

\begin {align*} \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx &=\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-(b n) \int \left (\frac {2}{3 d f}-\frac {2 x^2}{9}-\frac {2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2} x}+\frac {1}{3} x^2 \log \left (1+d f x^2\right )\right ) \, dx\\ &=-\frac {2 b n x}{3 d f}+\frac {2}{27} b n x^3+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac {1}{3} (b n) \int x^2 \log \left (1+d f x^2\right ) \, dx+\frac {(2 b n) \int \frac {\tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{x} \, dx}{3 d^{3/2} f^{3/2}}\\ &=-\frac {2 b n x}{3 d f}+\frac {2}{27} b n x^3+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}-\frac {1}{9} b n x^3 \log \left (1+d f x^2\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {(i b n) \int \frac {\log \left (1-i \sqrt {d} \sqrt {f} x\right )}{x} \, dx}{3 d^{3/2} f^{3/2}}-\frac {(i b n) \int \frac {\log \left (1+i \sqrt {d} \sqrt {f} x\right )}{x} \, dx}{3 d^{3/2} f^{3/2}}+\frac {1}{9} (2 b d f n) \int \frac {x^4}{1+d f x^2} \, dx\\ &=-\frac {2 b n x}{3 d f}+\frac {2}{27} b n x^3+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}-\frac {1}{9} b n x^3 \log \left (1+d f x^2\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {i b n \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}-\frac {i b n \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}+\frac {1}{9} (2 b d f n) \int \left (-\frac {1}{d^2 f^2}+\frac {x^2}{d f}+\frac {1}{d^2 f^2 \left (1+d f x^2\right )}\right ) \, dx\\ &=-\frac {8 b n x}{9 d f}+\frac {4}{27} b n x^3+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}-\frac {1}{9} b n x^3 \log \left (1+d f x^2\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {i b n \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}-\frac {i b n \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}+\frac {(2 b n) \int \frac {1}{1+d f x^2} \, dx}{9 d f}\\ &=-\frac {8 b n x}{9 d f}+\frac {4}{27} b n x^3+\frac {2 b n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{9 d^{3/2} f^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}-\frac {1}{9} b n x^3 \log \left (1+d f x^2\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {i b n \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}-\frac {i b n \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 364, normalized size = 1.51 \begin {gather*} \frac {2 a x}{3 d f}-\frac {2 a x^3}{9}-\frac {2 a \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}+\frac {2 b x \left (-n+3 \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{9 d f}-\frac {2}{27} b x^3 \left (-n+3 \left (-n \log (x)+\log \left (c x^n\right )\right )\right )-\frac {2 b \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (-n+3 \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{9 d^{3/2} f^{3/2}}+\frac {1}{3} a x^3 \log \left (1+d f x^2\right )+\frac {1}{9} b x^3 \left (-n+3 n \log (x)+3 \left (-n \log (x)+\log \left (c x^n\right )\right )\right ) \log \left (1+d f x^2\right )-\frac {2}{3} b d f n \left (-\frac {x (-1+\log (x))}{d^2 f^2}+\frac {-\frac {x^3}{9}+\frac {1}{3} x^3 \log (x)}{d f}-\frac {i \left (\log (x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+\text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )\right )}{2 d^{5/2} f^{5/2}}+\frac {i \left (\log (x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )\right )}{2 d^{5/2} f^{5/2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*Log[c*x^n])*Log[d*(d^(-1) + f*x^2)],x]

[Out]

(2*a*x)/(3*d*f) - (2*a*x^3)/9 - (2*a*ArcTan[Sqrt[d]*Sqrt[f]*x])/(3*d^(3/2)*f^(3/2)) + (2*b*x*(-n + 3*(-(n*Log[
x]) + Log[c*x^n])))/(9*d*f) - (2*b*x^3*(-n + 3*(-(n*Log[x]) + Log[c*x^n])))/27 - (2*b*ArcTan[Sqrt[d]*Sqrt[f]*x
]*(-n + 3*(-(n*Log[x]) + Log[c*x^n])))/(9*d^(3/2)*f^(3/2)) + (a*x^3*Log[1 + d*f*x^2])/3 + (b*x^3*(-n + 3*n*Log
[x] + 3*(-(n*Log[x]) + Log[c*x^n]))*Log[1 + d*f*x^2])/9 - (2*b*d*f*n*(-((x*(-1 + Log[x]))/(d^2*f^2)) + (-1/9*x
^3 + (x^3*Log[x])/3)/(d*f) - ((I/2)*(Log[x]*Log[1 + I*Sqrt[d]*Sqrt[f]*x] + PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x])
)/(d^(5/2)*f^(5/2)) + ((I/2)*(Log[x]*Log[1 - I*Sqrt[d]*Sqrt[f]*x] + PolyLog[2, I*Sqrt[d]*Sqrt[f]*x]))/(d^(5/2)
*f^(5/2))))/3

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.15, size = 891, normalized size = 3.70

method	result	size
risch	\(\frac {x^{3} \ln \left (d f \,x^{2}+1\right ) a}{3}-\frac {2 \ln \left (c \right ) b \,x^{3}}{9}+\frac {2 b x \ln \left (x^{n}\right )}{3 d f}-\frac {2 x^{3} a}{9}+\frac {2 b \arctan \left (\frac {x d f}{\sqrt {d f}}\right ) n \ln \left (x \right )}{3 d f \sqrt {d f}}+\frac {b n \sqrt {-d f}\, \ln \left (x \right ) \ln \left (1+x \sqrt {-d f}\right )}{3 d^{2} f^{2}}-\frac {b n \sqrt {-d f}\, \ln \left (x \right ) \ln \left (1-x \sqrt {-d f}\right )}{3 d^{2} f^{2}}+\frac {4 b n \,x^{3}}{27}+\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x}{3 d f}+\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{3 d f \sqrt {d f}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x}{3 d f}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x^{3} \ln \left (d f \,x^{2}+1\right )}{6}-\frac {2 a \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{3 d f \sqrt {d f}}+\frac {2 a x}{3 d f}+\frac {b \ln \left (d f \,x^{2}+1\right ) \ln \left (x^{n}\right ) x^{3}}{3}+\frac {b \ln \left (c \right ) x^{3} \ln \left (d f \,x^{2}+1\right )}{3}-\frac {2 x^{3} b \ln \left (x^{n}\right )}{9}+\frac {b n \sqrt {-d f}\, \dilog \left (1+x \sqrt {-d f}\right )}{3 d^{2} f^{2}}-\frac {b n \sqrt {-d f}\, \dilog \left (1-x \sqrt {-d f}\right )}{3 d^{2} f^{2}}+\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{3}}{9}+\frac {2 b \ln \left (c \right ) x}{3 d f}-\frac {8 b n x}{9 d f}-\frac {2 b \ln \left (c \right ) \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{3 d f \sqrt {d f}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{3 d f \sqrt {d f}}-\frac {b n \,x^{3} \ln \left (d f \,x^{2}+1\right )}{9}+\frac {2 b n \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{9 d f \sqrt {d f}}-\frac {2 b \arctan \left (\frac {x d f}{\sqrt {d f}}\right ) \ln \left (x^{n}\right )}{3 d f \sqrt {d f}}-\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{3}}{9}-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{3} \ln \left (d f \,x^{2}+1\right )}{6}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{3}}{9}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{3} \ln \left (d f \,x^{2}+1\right )}{6}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x^{3}}{9}+\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{3} \ln \left (d f \,x^{2}+1\right )}{6}-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x}{3 d f}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{3 d f \sqrt {d f}}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x}{3 d f}-\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{3 d f \sqrt {d f}}\)	\(891\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*ln(c*x^n))*ln(d*(1/d+f*x^2)),x,method=_RETURNVERBOSE)

[Out]

1/3*x^3*ln(d*f*x^2+1)*a+1/3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d/f*x+1/3*I*b*Pi*csgn(I*c*x^n)^3/d/f/(d*f)^(1/2
)*arctan(x*d*f/(d*f)^(1/2))-2/9*ln(c)*b*x^3+1/6*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2*x^3*ln(d*f*x^2+1)+2/3*b/d/f*x
*ln(x^n)-2/9*x^3*a+2/3*b/d/f/(d*f)^(1/2)*arctan(x*d*f/(d*f)^(1/2))*n*ln(x)+1/3*b*n/d^2/f^2*(-d*f)^(1/2)*ln(x)*
ln(1+x*(-d*f)^(1/2))-1/3*b*n/d^2/f^2*(-d*f)^(1/2)*ln(x)*ln(1-x*(-d*f)^(1/2))+4/27*b*n*x^3+1/3*I*b*Pi*csgn(I*c)
*csgn(I*c*x^n)^2/d/f*x-1/6*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^3*ln(d*f*x^2+1)-2/3*a/d/f/(d*f)^(1/2)*
arctan(x*d*f/(d*f)^(1/2))+2/3*a/d/f*x+1/3*b*ln(d*f*x^2+1)*ln(x^n)*x^3+1/9*I*b*Pi*csgn(I*c*x^n)^3*x^3+1/3*b*ln(
c)*x^3*ln(d*f*x^2+1)+1/9*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^3+1/6*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2
*x^3*ln(d*f*x^2+1)-1/3*I*b*Pi*csgn(I*c*x^n)^3/d/f*x-2/9*x^3*b*ln(x^n)-1/3*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/d/f
/(d*f)^(1/2)*arctan(x*d*f/(d*f)^(1/2))+2/3*b*ln(c)/d/f*x-1/3*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)/d/f*x-
8/9*b*n*x/d/f-1/3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d/f/(d*f)^(1/2)*arctan(x*d*f/(d*f)^(1/2))-2/3*b*ln(c)/d/f
/(d*f)^(1/2)*arctan(x*d*f/(d*f)^(1/2))-1/9*b*n*x^3*ln(d*f*x^2+1)-1/9*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*x^3+2/
9*b*n/d/f/(d*f)^(1/2)*arctan(x*d*f/(d*f)^(1/2))+1/3*b*n/d^2/f^2*(-d*f)^(1/2)*dilog(1+x*(-d*f)^(1/2))-1/3*b*n/d
^2/f^2*(-d*f)^(1/2)*dilog(1-x*(-d*f)^(1/2))-1/6*I*b*Pi*csgn(I*c*x^n)^3*x^3*ln(d*f*x^2+1)-2/3*b/d/f/(d*f)^(1/2)
*arctan(x*d*f/(d*f)^(1/2))*ln(x^n)-1/9*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2*x^3+1/3*I*b*Pi*csgn(I*c)*csgn(I*x^n)*c
sgn(I*c*x^n)/d/f/(d*f)^(1/2)*arctan(x*d*f/(d*f)^(1/2))

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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(x^2*(a+b*log(c*x^n))*log(d*(1/d+f*x^2)),x, algorithm="maxima")

[Out]

1/9*(3*b*x^3*log(x^n) - (b*(n - 3*log(c)) - 3*a)*x^3)*log(d*f*x^2 + 1) - integrate(2/9*(3*b*d*f*x^4*log(x^n) +
(3*a*d*f - (d*f*n - 3*d*f*log(c))*b)*x^4)/(d*f*x^2 + 1), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))*log(d*(1/d+f*x^2)),x, algorithm="fricas")

[Out]

integral(b*x^2*log(d*f*x^2 + 1)*log(c*x^n) + a*x^2*log(d*f*x^2 + 1), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*ln(c*x**n))*ln(d*(1/d+f*x**2)),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))*log(d*(1/d+f*x^2)),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*x^2*log((f*x^2 + 1/d)*d), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*log(d*(f*x^2 + 1/d))*(a + b*log(c*x^n)),x)

[Out]

int(x^2*log(d*(f*x^2 + 1/d))*(a + b*log(c*x^n)), x)

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