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

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

[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

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Rubi [A]  time = 0.378362, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, 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 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])

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 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 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 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 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 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 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]

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*}

Mathematica [A]  time = 0.106279, size = 228, normalized size = 1. \[ -\frac{\text{PolyLog}\left (2,\frac{\sqrt{b} (c+d x)}{\sqrt{b} c-\sqrt{-a} d}\right )}{d}-\frac{\text{PolyLog}\left (2,\frac{\sqrt{b} (c+d x)}{\sqrt{-a} d+\sqrt{b} c}\right )}{d}+\frac{2 \text{PolyLog}\left (2,\frac{c+d x}{c}\right )}{d}+\frac{\log \left (\frac{a}{x^2}+b\right ) \log (c+d x)}{d}-\frac{\log (c+d x) \log \left (\frac{d \left (\sqrt{-a}-\sqrt{b} x\right )}{\sqrt{-a} d+\sqrt{b} c}\right )}{d}-\frac{\log (c+d x) \log \left (-\frac{d \left (\sqrt{-a}+\sqrt{b} x\right )}{\sqrt{b} c-\sqrt{-a} d}\right )}{d}+\frac{2 \log \left (-\frac{d x}{c}\right ) \log (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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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Maple [A]  time = 0.112, size = 335, normalized size = 1.5 \begin{align*}{\frac{\ln \left ({x}^{-1} \right ) }{d}\ln \left ({ \left ( -{\frac{a}{x}}+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}} \right ) }+{\frac{\ln \left ({x}^{-1} \right ) }{d}\ln \left ({ \left ({\frac{a}{x}}+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}} \right ) }-{\frac{\ln \left ({x}^{-1} \right ) }{d}\ln \left ( b+{\frac{a}{{x}^{2}}} \right ) }+{\frac{1}{d}{\it dilog} \left ({ \left ( -{\frac{a}{x}}+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}} \right ) }+{\frac{1}{d}{\it dilog} \left ({ \left ({\frac{a}{x}}+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}} \right ) }+{\frac{1}{d}\ln \left ({\frac{c}{x}}+d \right ) \ln \left ( b+{\frac{a}{{x}^{2}}} \right ) }-{\frac{1}{d}\ln \left ({\frac{c}{x}}+d \right ) \ln \left ({ \left ( c\sqrt{-ab}-a \left ({\frac{c}{x}}+d \right ) +ad \right ) \left ( c\sqrt{-ab}+ad \right ) ^{-1}} \right ) }-{\frac{1}{d}\ln \left ({\frac{c}{x}}+d \right ) \ln \left ({ \left ( c\sqrt{-ab}+a \left ({\frac{c}{x}}+d \right ) -ad \right ) \left ( c\sqrt{-ab}-ad \right ) ^{-1}} \right ) }-{\frac{1}{d}{\it dilog} \left ({ \left ( c\sqrt{-ab}-a \left ({\frac{c}{x}}+d \right ) +ad \right ) \left ( c\sqrt{-ab}+ad \right ) ^{-1}} \right ) }-{\frac{1}{d}{\it dilog} \left ({ \left ( c\sqrt{-ab}+a \left ({\frac{c}{x}}+d \right ) -ad \right ) \left ( c\sqrt{-ab}-ad \right ) ^{-1}} \right ) } \end{align*}

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (\frac{b x^{2} + a}{x^{2}}\right )}{d x + c}\,{d x} \end{align*}

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

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

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (\frac{b x^{2} + a}{x^{2}}\right )}{d x + c}\,{d x} \end{align*}

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)