3.397 \(\int \frac {\log (\frac {a+b x^2}{x^2})}{c+d x} \, dx\)

Optimal. Leaf size=227 \[ -\frac {\text {Li}_2\left (\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}\right )}{d}-\frac {\text {Li}_2\left (\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {-a} d}\right )}{d}+\frac {\log \left (\frac {a}{x^2}+b\right ) \log (c+d x)}{d}-\frac {\log (c+d x) \log \left (\frac {d \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {-a} d+\sqrt {b} c}\right )}{d}-\frac {\log (c+d x) \log \left (-\frac {d \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} c-\sqrt {-a} d}\right )}{d}+\frac {2 \text {Li}_2\left (\frac {d x}{c}+1\right )}{d}+\frac {2 \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.38, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2465, 2462, 260, 2416, 2394, 2315, 2393, 2391} \[ -\frac {\text {PolyLog}\left (2,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}\right )}{d}-\frac {\text {PolyLog}\left (2,\frac {\sqrt {b} (c+d x)}{\sqrt {-a} d+\sqrt {b} c}\right )}{d}+\frac {2 \text {PolyLog}\left (2,\frac {d x}{c}+1\right )}{d}+\frac {\log \left (\frac {a}{x^2}+b\right ) \log (c+d x)}{d}-\frac {\log (c+d x) \log \left (\frac {d \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {-a} d+\sqrt {b} c}\right )}{d}-\frac {\log (c+d x) \log \left (-\frac {d \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} c-\sqrt {-a} d}\right )}{d}+\frac {2 \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[Log[(a + b*x^2)/x^2]/(c + d*x),x]

[Out]

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

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

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 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2462

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

Rule 2465

Int[((a_.) + Log[(c_.)*(v_)^(p_.)]*(b_.))^(q_.)*(u_)^(r_.), x_Symbol] :> Int[ExpandToSum[u, x]^r*(a + b*Log[c*
ExpandToSum[v, x]^p])^q, x] /; FreeQ[{a, b, c, p, q, r}, x] && LinearQ[u, x] && BinomialQ[v, x] &&  !(LinearMa
tchQ[u, x] && BinomialMatchQ[v, x])

Rubi steps

\begin {align*} \int \frac {\log \left (\frac {a+b x^2}{x^2}\right )}{c+d x} \, dx &=\int \frac {\log \left (b+\frac {a}{x^2}\right )}{c+d x} \, dx\\ &=\frac {\log \left (b+\frac {a}{x^2}\right ) \log (c+d x)}{d}+\frac {(2 a) \int \frac {\log (c+d x)}{\left (b+\frac {a}{x^2}\right ) x^3} \, dx}{d}\\ &=\frac {\log \left (b+\frac {a}{x^2}\right ) \log (c+d x)}{d}+\frac {(2 a) \int \left (\frac {\log (c+d x)}{a x}-\frac {b x \log (c+d x)}{a \left (a+b x^2\right )}\right ) \, dx}{d}\\ &=\frac {\log \left (b+\frac {a}{x^2}\right ) \log (c+d x)}{d}+\frac {2 \int \frac {\log (c+d x)}{x} \, dx}{d}-\frac {(2 b) \int \frac {x \log (c+d x)}{a+b x^2} \, dx}{d}\\ &=\frac {\log \left (b+\frac {a}{x^2}\right ) \log (c+d x)}{d}+\frac {2 \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{d}-2 \int \frac {\log \left (-\frac {d x}{c}\right )}{c+d x} \, dx-\frac {(2 b) \int \left (-\frac {\log (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\log (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{d}\\ &=\frac {\log \left (b+\frac {a}{x^2}\right ) \log (c+d x)}{d}+\frac {2 \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{d}+\frac {2 \text {Li}_2\left (1+\frac {d x}{c}\right )}{d}+\frac {\sqrt {b} \int \frac {\log (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{d}-\frac {\sqrt {b} \int \frac {\log (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{d}\\ &=\frac {\log \left (b+\frac {a}{x^2}\right ) \log (c+d x)}{d}+\frac {2 \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{d}-\frac {\log \left (\frac {d \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} c+\sqrt {-a} d}\right ) \log (c+d x)}{d}-\frac {\log \left (-\frac {d \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} c-\sqrt {-a} d}\right ) \log (c+d x)}{d}+\frac {2 \text {Li}_2\left (1+\frac {d x}{c}\right )}{d}+\int \frac {\log \left (\frac {d \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} c+\sqrt {-a} d}\right )}{c+d x} \, dx+\int \frac {\log \left (\frac {d \left (\sqrt {-a}+\sqrt {b} x\right )}{-\sqrt {b} c+\sqrt {-a} d}\right )}{c+d x} \, dx\\ &=\frac {\log \left (b+\frac {a}{x^2}\right ) \log (c+d x)}{d}+\frac {2 \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{d}-\frac {\log \left (\frac {d \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} c+\sqrt {-a} d}\right ) \log (c+d x)}{d}-\frac {\log \left (-\frac {d \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} c-\sqrt {-a} d}\right ) \log (c+d x)}{d}+\frac {2 \text {Li}_2\left (1+\frac {d x}{c}\right )}{d}+\frac {\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {b} x}{-\sqrt {b} c+\sqrt {-a} d}\right )}{x} \, dx,x,c+d x\right )}{d}+\frac {\operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {b} x}{\sqrt {b} c+\sqrt {-a} d}\right )}{x} \, dx,x,c+d x\right )}{d}\\ &=\frac {\log \left (b+\frac {a}{x^2}\right ) \log (c+d x)}{d}+\frac {2 \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{d}-\frac {\log \left (\frac {d \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} c+\sqrt {-a} d}\right ) \log (c+d x)}{d}-\frac {\log \left (-\frac {d \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} c-\sqrt {-a} d}\right ) \log (c+d x)}{d}-\frac {\text {Li}_2\left (\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}\right )}{d}-\frac {\text {Li}_2\left (\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {-a} d}\right )}{d}+\frac {2 \text {Li}_2\left (1+\frac {d x}{c}\right )}{d}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 228, normalized size = 1.00 \[ -\frac {\text {Li}_2\left (\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}\right )}{d}-\frac {\text {Li}_2\left (\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {-a} d}\right )}{d}+\frac {\log \left (\frac {a}{x^2}+b\right ) \log (c+d x)}{d}-\frac {\log (c+d x) \log \left (\frac {d \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {-a} d+\sqrt {b} c}\right )}{d}-\frac {\log (c+d x) \log \left (-\frac {d \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} c-\sqrt {-a} d}\right )}{d}+\frac {2 \text {Li}_2\left (\frac {c+d x}{c}\right )}{d}+\frac {2 \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Log[b + a/x^2]*Log[c + d*x])/d + (2*Log[-((d*x)/c)]*Log[c + d*x])/d - (Log[(d*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b
]*c + Sqrt[-a]*d)]*Log[c + d*x])/d - (Log[-((d*(Sqrt[-a] + Sqrt[b]*x))/(Sqrt[b]*c - Sqrt[-a]*d))]*Log[c + d*x]
)/d + (2*PolyLog[2, (c + d*x)/c])/d - PolyLog[2, (Sqrt[b]*(c + d*x))/(Sqrt[b]*c - Sqrt[-a]*d)]/d - PolyLog[2,
(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[-a]*d)]/d

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fricas [F]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (\frac {b x^{2} + a}{x^{2}}\right )}{d x + c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(log((b*x^2 + a)/x^2)/(d*x + c), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (\frac {b x^{2} + a}{x^{2}}\right )}{d x + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.14, size = 335, normalized size = 1.48 \[ \frac {\ln \left (\frac {1}{x}\right ) \ln \left (\frac {-\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{d}+\frac {\ln \left (\frac {1}{x}\right ) \ln \left (\frac {\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{d}-\frac {\ln \left (\frac {1}{x}\right ) \ln \left (b +\frac {a}{x^{2}}\right )}{d}-\frac {\ln \left (\frac {a d -\left (d +\frac {c}{x}\right ) a +\sqrt {-a b}\, c}{a d +\sqrt {-a b}\, c}\right ) \ln \left (d +\frac {c}{x}\right )}{d}-\frac {\ln \left (\frac {-a d +\left (d +\frac {c}{x}\right ) a +\sqrt {-a b}\, c}{-a d +\sqrt {-a b}\, c}\right ) \ln \left (d +\frac {c}{x}\right )}{d}+\frac {\ln \left (b +\frac {a}{x^{2}}\right ) \ln \left (d +\frac {c}{x}\right )}{d}+\frac {\dilog \left (\frac {-\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{d}+\frac {\dilog \left (\frac {\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{d}-\frac {\dilog \left (\frac {a d -\left (d +\frac {c}{x}\right ) a +\sqrt {-a b}\, c}{a d +\sqrt {-a b}\, c}\right )}{d}-\frac {\dilog \left (\frac {-a d +\left (d +\frac {c}{x}\right ) a +\sqrt {-a b}\, c}{-a d +\sqrt {-a b}\, c}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*ln(d+c/x)*ln(b+a/x^2)-1/d*ln(d+c/x)*ln((c*(-a*b)^(1/2)-a*(d+c/x)+a*d)/(c*(-a*b)^(1/2)+a*d))-1/d*ln(d+c/x)*
ln((c*(-a*b)^(1/2)+a*(d+c/x)-a*d)/(c*(-a*b)^(1/2)-a*d))-1/d*dilog((c*(-a*b)^(1/2)-a*(d+c/x)+a*d)/(c*(-a*b)^(1/
2)+a*d))-1/d*dilog((c*(-a*b)^(1/2)+a*(d+c/x)-a*d)/(c*(-a*b)^(1/2)-a*d))-1/d*ln(1/x)*ln(b+a/x^2)+1/d*ln(1/x)*ln
((-a/x+(-a*b)^(1/2))/(-a*b)^(1/2))+1/d*ln(1/x)*ln((a/x+(-a*b)^(1/2))/(-a*b)^(1/2))+1/d*dilog((-a/x+(-a*b)^(1/2
))/(-a*b)^(1/2))+1/d*dilog((a/x+(-a*b)^(1/2))/(-a*b)^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (\frac {b x^{2} + a}{x^{2}}\right )}{d x + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (\frac {b\,x^2+a}{x^2}\right )}{c+d\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log((a + b*x^2)/x^2)/(c + d*x),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (\frac {a}{x^{2}} + b \right )}}{c + d x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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