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

Optimal. Leaf size=182 \[ -\frac{i b n \text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}+\frac{i b n \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}+x \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d} \sqrt{f}}-2 x \left (a+b \log \left (c x^n\right )\right )-b n x \log \left (d f x^2+1\right )-\frac{2 b n \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}+4 b n x \]

[Out]

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

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Rubi [A]  time = 0.106502, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {2448, 321, 205, 2370, 4848, 2391, 203} \[ -\frac{i b n \text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}+\frac{i b n \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}+x \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d} \sqrt{f}}-2 x \left (a+b \log \left (c x^n\right )\right )-b n x \log \left (d f x^2+1\right )-\frac{2 b n \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}+4 b n x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2448

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 321

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 205

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

Rule 2370

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 4848

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]

Rule 2391

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 203

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

Rubi steps

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

Mathematica [A]  time = 0.090173, size = 254, normalized size = 1.4 \[ -2 b d f n \left (\frac{i \left (\text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )+\log (x) \log \left (1+i \sqrt{d} \sqrt{f} x\right )\right )}{2 d^{3/2} f^{3/2}}-\frac{i \left (\text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )+\log (x) \log \left (1-i \sqrt{d} \sqrt{f} x\right )\right )}{2 d^{3/2} f^{3/2}}+\frac{x (\log (x)-1)}{d f}\right )+a x \log \left (d f x^2+1\right )+\frac{2 a \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}-2 a x+b x \left (\log \left (c x^n\right )-n\right ) \log \left (d f x^2+1\right )+\frac{2 b \left (\log \left (c x^n\right )+n (-\log (x))-n\right ) \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}-2 b x \left (\log \left (c x^n\right )+n (-\log (x))-n\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.112, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \ln \left ( d \left ({d}^{-1}+f{x}^{2} \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right ) + a \log \left (d f x^{2} + 1\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [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((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., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + \frac{1}{d}\right )} d\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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