3.1.30 \(\int \frac {(a+b \log (c x^n)) \log (d (\frac {1}{d}+f x^2))}{x^2} \, dx\) [30]

Optimal. Leaf size=169 \[ 2 b \sqrt {d} \sqrt {f} n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )+2 \sqrt {d} \sqrt {f} \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log \left (1+d f x^2\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-i b \sqrt {d} \sqrt {f} n \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+i b \sqrt {d} \sqrt {f} n \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right ) \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2505, 211, 2423, 4940, 2438, 209} \begin {gather*} -i b \sqrt {d} \sqrt {f} n \text {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )+i b \sqrt {d} \sqrt {f} n \text {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )+2 \sqrt {d} \sqrt {f} \text {ArcTan}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}+2 b \sqrt {d} \sqrt {f} n \text {ArcTan}\left (\sqrt {d} \sqrt {f} x\right )-\frac {b n \log \left (d f x^2+1\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.07, size = 221, normalized size = 1.31 \begin {gather*} 2 a \sqrt {d} \sqrt {f} \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )+2 b \sqrt {d} \sqrt {f} \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (n-n \log (x)+\log \left (c x^n\right )\right )-\frac {a \log \left (1+d f x^2\right )}{x}-\frac {b \left (n+\log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}+2 b d f n \left (-\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 \sqrt {d} \sqrt {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 \sqrt {d} \sqrt {f}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.14, size = 547, normalized size = 3.24

method	result	size
risch	\(-\frac {b \ln \left (d f \,x^{2}+1\right ) \ln \left (x^{n}\right )}{x}-\frac {2 b d f \arctan \left (\frac {x d f}{\sqrt {d f}}\right ) n \ln \left (x \right )}{\sqrt {d f}}+\frac {2 b d f \arctan \left (\frac {x d f}{\sqrt {d f}}\right ) \ln \left (x^{n}\right )}{\sqrt {d f}}-\frac {b n \ln \left (d f \,x^{2}+1\right )}{x}+\frac {2 b n d f \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{\sqrt {d f}}-b n \sqrt {-d f}\, \ln \left (x \right ) \ln \left (1+x \sqrt {-d f}\right )+b n \sqrt {-d f}\, \ln \left (x \right ) \ln \left (1-x \sqrt {-d f}\right )-b n \sqrt {-d f}\, \dilog \left (1+x \sqrt {-d f}\right )+b n \sqrt {-d f}\, \dilog \left (1-x \sqrt {-d f}\right )-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (d f \,x^{2}+1\right )}{2 x}+\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d f \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{\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 ) d f \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{\sqrt {d f}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d f \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{\sqrt {d f}}-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} d f \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{\sqrt {d f}}-\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (d f \,x^{2}+1\right )}{2 x}+\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 ) \ln \left (d f \,x^{2}+1\right )}{2 x}+\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (d f \,x^{2}+1\right )}{2 x}-\frac {b \ln \left (c \right ) \ln \left (d f \,x^{2}+1\right )}{x}+\frac {2 b \ln \left (c \right ) d f \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{\sqrt {d f}}-\frac {\ln \left (d f \,x^{2}+1\right ) a}{x}+\frac {2 a d f \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{\sqrt {d f}}\)	\(547\)

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^2,x,method=_RETURNVERBOSE)

[Out]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \log {\left (d f x^{2} + 1 \right )}}{x^{2}}\, dx \end {gather*}

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**2,x)

[Out]

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

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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*(1/d+f*x^2))/x^2,x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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