3.1.96 \(\int (a+b \log (c x^n)) \log (d (e+f x^2)^m) \, dx\) [96]

Optimal. Leaf size=194 \[ 4 b m n x-\frac {2 b \sqrt {e} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}-2 m x \left (a+b \log \left (c x^n\right )\right )+\frac {2 \sqrt {e} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {f}}-b n x \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {i b \sqrt {e} m n \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+\frac {i b \sqrt {e} m n \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}} \]

[Out]

4*b*m*n*x-2*m*x*(a+b*ln(c*x^n))-b*n*x*ln(d*(f*x^2+e)^m)+x*(a+b*ln(c*x^n))*ln(d*(f*x^2+e)^m)-2*b*m*n*arctan(x*f
^(1/2)/e^(1/2))*e^(1/2)/f^(1/2)+2*m*arctan(x*f^(1/2)/e^(1/2))*(a+b*ln(c*x^n))*e^(1/2)/f^(1/2)-I*b*m*n*polylog(
2,-I*x*f^(1/2)/e^(1/2))*e^(1/2)/f^(1/2)+I*b*m*n*polylog(2,I*x*f^(1/2)/e^(1/2))*e^(1/2)/f^(1/2)

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Rubi [A]
time = 0.09, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2498, 327, 211, 2417, 4940, 2438} \begin {gather*} -\frac {i b \sqrt {e} m n \text {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+\frac {i b \sqrt {e} m n \text {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+\frac {2 \sqrt {e} m \text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {f}}+x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-2 m x \left (a+b \log \left (c x^n\right )\right )-\frac {2 b \sqrt {e} m n \text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}-b n x \log \left (d \left (e+f x^2\right )^m\right )+4 b m n x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4*b*m*n*x - (2*b*Sqrt[e]*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/Sqrt[f] - 2*m*x*(a + b*Log[c*x^n]) + (2*Sqrt[e]*m*Ar
cTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*Log[c*x^n]))/Sqrt[f] - b*n*x*Log[d*(e + f*x^2)^m] + x*(a + b*Log[c*x^n])*Log[
d*(e + f*x^2)^m] - (I*b*Sqrt[e]*m*n*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]])/Sqrt[f] + (I*b*Sqrt[e]*m*n*PolyLog[2
, (I*Sqrt[f]*x)/Sqrt[e]])/Sqrt[f]

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 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2417

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> With[
{u = IntHide[Log[d*(e + f*x^m)^r], x]}, Dist[(a + b*Log[c*x^n])^p, u, x] - Dist[b*n*p, Int[Dist[(a + b*Log[c*x
^n])^(p - 1)/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] && RationalQ[m] && (EqQ[
p, 1] || (FractionQ[m] && IntegerQ[1/m]) || (EqQ[r, 1] && EqQ[m, 1] && EqQ[d*e, 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 2498

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

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

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Mathematica [A]
time = 0.06, size = 332, normalized size = 1.71 \begin {gather*} \frac {-2 a \sqrt {f} m x+4 b \sqrt {f} m n x+2 a \sqrt {e} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )-2 b \sqrt {e} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )-2 b \sqrt {e} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log (x)-2 b \sqrt {f} m x \log \left (c x^n\right )+2 b \sqrt {e} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right )+i b \sqrt {e} m n \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-i b \sqrt {e} m n \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+a \sqrt {f} x \log \left (d \left (e+f x^2\right )^m\right )-b \sqrt {f} n x \log \left (d \left (e+f x^2\right )^m\right )+b \sqrt {f} x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-i b \sqrt {e} m n \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+i b \sqrt {e} m n \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*Sqrt[f]*m*x + 4*b*Sqrt[f]*m*n*x + 2*a*Sqrt[e]*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]] - 2*b*Sqrt[e]*m*n*ArcTan[(Sq
rt[f]*x)/Sqrt[e]] - 2*b*Sqrt[e]*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] - 2*b*Sqrt[f]*m*x*Log[c*x^n] + 2*b*Sqrt
[e]*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] + I*b*Sqrt[e]*m*n*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - I*b*Sqr
t[e]*m*n*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + a*Sqrt[f]*x*Log[d*(e + f*x^2)^m] - b*Sqrt[f]*n*x*Log[d*(e + f
*x^2)^m] + b*Sqrt[f]*x*Log[c*x^n]*Log[d*(e + f*x^2)^m] - I*b*Sqrt[e]*m*n*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]]
+ I*b*Sqrt[e]*m*n*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/Sqrt[f]

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.16, size = 2001, normalized size = 10.31

method	result	size
risch	\(\text {Expression too large to display}\)	\(2001\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x*a*m+1/4*Pi^2*x*b*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/4*Pi^2
*x*b*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)*csgn(I*c)*csgn(I*c*x^n)^2-1/2*I*ln(c)*Pi*x*b*csgn(I*d
)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)-1/2*I*Pi*ln(x^n)*x*b*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2
+e)^m)+1/4*Pi^2*x*b*csgn(I*d*(f*x^2+e)^m)^3*csgn(I*x^n)*csgn(I*c*x^n)^2+4*b*m*n*x+1/4*Pi^2*x*b*csgn(I*d)*csgn(
I*d*(f*x^2+e)^m)^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*m*x*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-2*m*ln(c
)*b*x+ln(d)*ln(c)*b*x-ln(d)*b*n*x-I*m*e/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*b*Pi*csgn(I*c*x^n)^3+1/4*Pi^2*x*b*
csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)*csgn(I*x^n)*csgn(I*c*x^n)^2+ln(d)*a*x-I*m*x*b*Pi*csgn(I*x^
n)*csgn(I*c*x^n)^2+1/2*I*ln(d)*Pi*b*x*csgn(I*c)*csgn(I*c*x^n)^2+1/4*Pi^2*x*b*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2
*csgn(I*c*x^n)^3+2*a*m*e/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))-2*m*e/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*b*n+m*b
*n*e/(-e*f)^(1/2)*dilog((-f*x+(-e*f)^(1/2))/(-e*f)^(1/2))-m*b*n*e/(-e*f)^(1/2)*dilog((f*x+(-e*f)^(1/2))/(-e*f)
^(1/2))+2*m*e/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*b*ln(c)-2*m*b*e/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*n*ln(x)+
m*b*n*e/(-e*f)^(1/2)*ln(x)*ln((-f*x+(-e*f)^(1/2))/(-e*f)^(1/2))-m*b*n*e/(-e*f)^(1/2)*ln(x)*ln((f*x+(-e*f)^(1/2
))/(-e*f)^(1/2))-2*m*b*ln(x^n)*x+ln(x^n)*ln(d)*x*b-1/4*Pi^2*x*b*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+
e)^m)*csgn(I*c*x^n)^3-1/4*Pi^2*x*b*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2*csgn(I*c)*csgn(I*c*x^n)^2-1/4*Pi^2*x*b*cs
gn(I*d*(f*x^2+e)^m)^3*csgn(I*c*x^n)^3-1/2*I*Pi*x*a*csgn(I*d*(f*x^2+e)^m)^3+I*m*x*b*Pi*csgn(I*c*x^n)^3+(b*x*ln(
x^n)+1/2*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn
(I*c*x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln(c)-2*b*n+2*a)*x)*ln((f*x^2+e)^m)+1/2*I*Pi*x*b*n*csgn(I*d)*csgn(I*(f*
x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)+1/2*I*Pi*ln(x^n)*x*b*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2-1/2*I*Pi*x*b*n*csgn(I*d
)*csgn(I*d*(f*x^2+e)^m)^2-I*m*x*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/2*I*ln(d)*Pi*b*x*csgn(I*c)*csgn(I*x^n)*csgn(I
*c*x^n)-1/4*Pi^2*x*b*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2*csgn(I*x^n)*csgn(I*c*x^n)^2-1/4*Pi^2*x*b*csgn
(I*d*(f*x^2+e)^m)^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*ln(c)*Pi*x*b*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2
+e)^m)^2-1/2*I*ln(c)*Pi*x*b*csgn(I*d*(f*x^2+e)^m)^3-1/2*I*Pi*ln(x^n)*x*b*csgn(I*d*(f*x^2+e)^m)^3+1/2*I*Pi*x*b*
n*csgn(I*d*(f*x^2+e)^m)^3+1/2*I*Pi*x*a*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2+1/2*I*Pi*x*a*csgn(I*(f*x^2+e)^m)*csgn
(I*d*(f*x^2+e)^m)^2-1/4*Pi^2*x*b*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)*csgn(I*c)*csgn(I*x^n)*csg
n(I*c*x^n)+I*m*e/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+2*m*b*e/(e*f)^(1/2)*arctan
(x*f/(e*f)^(1/2))*ln(x^n)-1/4*Pi^2*x*b*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2*csgn(I*x^n)*csgn(I*c*x^n)^2-1/4*Pi^2*
x*b*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2*csgn(I*c)*csgn(I*c*x^n)^2+1/2*I*Pi*ln(x^n)*x*b*csgn(I*(f*x^2+e
)^m)*csgn(I*d*(f*x^2+e)^m)^2-1/2*I*ln(d)*Pi*b*x*csgn(I*c*x^n)^3+I*m*e/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*b*Pi
*csgn(I*x^n)*csgn(I*c*x^n)^2+1/4*Pi^2*x*b*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2*csgn(I*c*x^n)^3-1/2*I*Pi
*x*b*n*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2-1/2*I*Pi*x*a*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+
e)^m)+1/2*I*ln(c)*Pi*x*b*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2+1/4*Pi^2*x*b*csgn(I*d*(f*x^2+e)^m)^3*csgn(I*c)*csgn
(I*c*x^n)^2-I*m*e/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*ln(d)*Pi*
b*x*csgn(I*x^n)*csgn(I*c*x^n)^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((a+b*log(c*x^n))*log(d*(f*x^2+e)^m),x, algorithm="maxima")

[Out]

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

[Out]

integral((b*log(c*x^n) + a)*log((f*x^2 + e)^m*d), 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((a+b*ln(c*x**n))*ln(d*(f*x**2+e)**m),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((a+b*log(c*x^n))*log(d*(f*x^2+e)^m),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \ln \left (d\,{\left (f\,x^2+e\right )}^m\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(log(d*(e + f*x^2)^m)*(a + b*log(c*x^n)),x)

[Out]

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

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