Integrand size = 20, antiderivative size = 227 \[ \int \frac {\log \left (\frac {a+b x^2}{x^2}\right )}{c+d x} \, dx=\frac {\log \left (b+\frac {a}{x^2}\right ) \log (c+d x)}{d}+\frac {2 \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{d}-\frac {\log \left (\frac {d \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} c+\sqrt {-a} d}\right ) \log (c+d x)}{d}-\frac {\log \left (-\frac {d \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} c-\sqrt {-a} d}\right ) \log (c+d x)}{d}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}\right )}{d}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {-a} d}\right )}{d}+\frac {2 \operatorname {PolyLog}\left (2,1+\frac {d x}{c}\right )}{d} \] Output:
ln(b+a/x^2)*ln(d*x+c)/d+2*ln(-d*x/c)*ln(d*x+c)/d-ln(d*((-a)^(1/2)-b^(1/2)* x)/(b^(1/2)*c+(-a)^(1/2)*d))*ln(d*x+c)/d-ln(-d*((-a)^(1/2)+b^(1/2)*x)/(b^( 1/2)*c-(-a)^(1/2)*d))*ln(d*x+c)/d-polylog(2,b^(1/2)*(d*x+c)/(b^(1/2)*c-(-a )^(1/2)*d))/d-polylog(2,b^(1/2)*(d*x+c)/(b^(1/2)*c+(-a)^(1/2)*d))/d+2*poly log(2,1+d*x/c)/d
Time = 0.12 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (\frac {a+b x^2}{x^2}\right )}{c+d x} \, dx=\frac {\log \left (b+\frac {a}{x^2}\right ) \log (c+d x)}{d}+\frac {2 \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{d}-\frac {\log \left (\frac {d \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} c+\sqrt {-a} d}\right ) \log (c+d x)}{d}-\frac {\log \left (-\frac {d \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} c-\sqrt {-a} d}\right ) \log (c+d x)}{d}+\frac {2 \operatorname {PolyLog}\left (2,\frac {c+d x}{c}\right )}{d}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}\right )}{d}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {-a} d}\right )}{d} \] Input:
Integrate[Log[(a + b*x^2)/x^2]/(c + d*x),x]
Output:
(Log[b + a/x^2]*Log[c + d*x])/d + (2*Log[-((d*x)/c)]*Log[c + d*x])/d - (Lo g[(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
Time = 1.14 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2915, 2912, 2005, 2863, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\log \left (\frac {a+b x^2}{x^2}\right )}{c+d x} \, dx\) |
| \(\Big \downarrow \) 2915 |
| \(\displaystyle \int \frac {\log \left (\frac {a}{x^2}+b\right )}{c+d x}dx\) |
| \(\Big \downarrow \) 2912 |
| \(\displaystyle \frac {2 a \int \frac {\log (c+d x)}{\left (\frac {a}{x^2}+b\right ) x^3}dx}{d}+\frac {\log \left (\frac {a}{x^2}+b\right ) \log (c+d x)}{d}\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle \frac {2 a \int \frac {\log (c+d x)}{x \left (b x^2+a\right )}dx}{d}+\frac {\log \left (\frac {a}{x^2}+b\right ) \log (c+d x)}{d}\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \frac {2 a \int \left (\frac {\log (c+d x)}{a x}-\frac {b x \log (c+d x)}{a \left (b x^2+a\right )}\right )dx}{d}+\frac {\log \left (\frac {a}{x^2}+b\right ) \log (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 a \left (-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}\right )}{2 a}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {-a} d}\right )}{2 a}-\frac {\log (c+d x) \log \left (\frac {d \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {-a} d+\sqrt {b} c}\right )}{2 a}-\frac {\log (c+d x) \log \left (-\frac {d \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} c-\sqrt {-a} d}\right )}{2 a}+\frac {\operatorname {PolyLog}\left (2,\frac {d x}{c}+1\right )}{a}+\frac {\log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a}\right )}{d}+\frac {\log \left (\frac {a}{x^2}+b\right ) \log (c+d x)}{d}\) |
Input:
Int[Log[(a + b*x^2)/x^2]/(c + d*x),x]
Output:
(Log[b + a/x^2]*Log[c + d*x])/d + (2*a*((Log[-((d*x)/c)]*Log[c + d*x])/a - (Log[(d*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*c + Sqrt[-a]*d)]*Log[c + d*x])/( 2*a) - (Log[-((d*(Sqrt[-a] + Sqrt[b]*x))/(Sqrt[b]*c - Sqrt[-a]*d))]*Log[c + d*x])/(2*a) - PolyLog[2, (Sqrt[b]*(c + d*x))/(Sqrt[b]*c - Sqrt[-a]*d)]/( 2*a) - PolyLog[2, (Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[-a]*d)]/(2*a) + Po lyLog[2, 1 + (d*x)/c]/a))/d
Int[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg Q[n]
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]
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 ] - Simp[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]
Int[((a_.) + Log[(c_.)*(v_)^(p_.)]*(b_.))^(q_.)*(u_)^(r_.), x_Symbol] :> In t[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] && !(LinearMatchQ[ u, x] && BinomialMatchQ[v, x])
Time = 4.63 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.01
| method | result | size |
| parts | \(\frac {\ln \left (\frac {b \,x^{2}+a}{x^{2}}\right ) \ln \left (d x +c \right )}{d}-\frac {2 \left (-\left (\operatorname {dilog}\left (-\frac {d x}{c}\right )+\ln \left (d x +c \right ) \ln \left (-\frac {d x}{c}\right )\right ) d^{3}-\left (-\frac {\ln \left (d x +c \right ) \left (\ln \left (\frac {d \sqrt {-a b}+b c -b \left (d x +c \right )}{d \sqrt {-a b}+b c}\right )+\ln \left (\frac {d \sqrt {-a b}-b c +b \left (d x +c \right )}{d \sqrt {-a b}-b c}\right )\right )}{2 b}-\frac {\operatorname {dilog}\left (\frac {d \sqrt {-a b}+b c -b \left (d x +c \right )}{d \sqrt {-a b}+b c}\right )+\operatorname {dilog}\left (\frac {d \sqrt {-a b}-b c +b \left (d x +c \right )}{d \sqrt {-a b}-b c}\right )}{2 b}\right ) b \,d^{3}\right )}{d^{4}}\) | \(230\) |
| derivativedivides | \(-\frac {\ln \left (\frac {1}{x}\right ) \ln \left (b +\frac {a}{x^{2}}\right )-2 a \left (\frac {\ln \left (\frac {1}{x}\right ) \left (\ln \left (\frac {-\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )+\ln \left (\frac {\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )\right )}{2 a}+\frac {\operatorname {dilog}\left (\frac {-\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )+\operatorname {dilog}\left (\frac {\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{2 a}\right )}{d}+\frac {\left (\frac {\ln \left (\frac {c}{x}+d \right ) \ln \left (b +\frac {a}{x^{2}}\right )}{c}-\frac {2 a \left (\frac {\ln \left (\frac {c}{x}+d \right ) \left (\ln \left (\frac {c \sqrt {-a b}+d a -a \left (\frac {c}{x}+d \right )}{c \sqrt {-a b}+d a}\right )+\ln \left (\frac {c \sqrt {-a b}-d a +a \left (\frac {c}{x}+d \right )}{c \sqrt {-a b}-d a}\right )\right )}{2 a}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-a b}+d a -a \left (\frac {c}{x}+d \right )}{c \sqrt {-a b}+d a}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-a b}-d a +a \left (\frac {c}{x}+d \right )}{c \sqrt {-a b}-d a}\right )}{2 a}\right )}{c}\right ) c}{d}\) | \(330\) |
| default | \(-\frac {\ln \left (\frac {1}{x}\right ) \ln \left (b +\frac {a}{x^{2}}\right )-2 a \left (\frac {\ln \left (\frac {1}{x}\right ) \left (\ln \left (\frac {-\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )+\ln \left (\frac {\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )\right )}{2 a}+\frac {\operatorname {dilog}\left (\frac {-\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )+\operatorname {dilog}\left (\frac {\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{2 a}\right )}{d}+\frac {\left (\frac {\ln \left (\frac {c}{x}+d \right ) \ln \left (b +\frac {a}{x^{2}}\right )}{c}-\frac {2 a \left (\frac {\ln \left (\frac {c}{x}+d \right ) \left (\ln \left (\frac {c \sqrt {-a b}+d a -a \left (\frac {c}{x}+d \right )}{c \sqrt {-a b}+d a}\right )+\ln \left (\frac {c \sqrt {-a b}-d a +a \left (\frac {c}{x}+d \right )}{c \sqrt {-a b}-d a}\right )\right )}{2 a}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-a b}+d a -a \left (\frac {c}{x}+d \right )}{c \sqrt {-a b}+d a}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-a b}-d a +a \left (\frac {c}{x}+d \right )}{c \sqrt {-a b}-d a}\right )}{2 a}\right )}{c}\right ) c}{d}\) | \(330\) |
| risch | \(\frac {\ln \left (\frac {1}{x}\right ) \ln \left (\frac {-\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{d}+\frac {\ln \left (\frac {1}{x}\right ) \ln \left (\frac {\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{d}-\frac {\ln \left (\frac {1}{x}\right ) \ln \left (b +\frac {a}{x^{2}}\right )}{d}+\frac {\operatorname {dilog}\left (\frac {-\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{d}+\frac {\operatorname {dilog}\left (\frac {\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{d}+\frac {\ln \left (\frac {c}{x}+d \right ) \ln \left (b +\frac {a}{x^{2}}\right )}{d}-\frac {\ln \left (\frac {c}{x}+d \right ) \ln \left (\frac {c \sqrt {-a b}+d a -a \left (\frac {c}{x}+d \right )}{c \sqrt {-a b}+d a}\right )}{d}-\frac {\ln \left (\frac {c}{x}+d \right ) \ln \left (\frac {c \sqrt {-a b}-d a +a \left (\frac {c}{x}+d \right )}{c \sqrt {-a b}-d a}\right )}{d}-\frac {\operatorname {dilog}\left (\frac {c \sqrt {-a b}+d a -a \left (\frac {c}{x}+d \right )}{c \sqrt {-a b}+d a}\right )}{d}-\frac {\operatorname {dilog}\left (\frac {c \sqrt {-a b}-d a +a \left (\frac {c}{x}+d \right )}{c \sqrt {-a b}-d a}\right )}{d}\) | \(335\) |
Input:
int(ln((b*x^2+a)/x^2)/(d*x+c),x,method=_RETURNVERBOSE)
Output:
ln((b*x^2+a)/x^2)/d*ln(d*x+c)-2/d^4*(-(dilog(-d*x/c)+ln(d*x+c)*ln(-d*x/c)) *d^3-(-1/2*ln(d*x+c)*(ln((d*(-a*b)^(1/2)+b*c-b*(d*x+c))/(d*(-a*b)^(1/2)+b* c))+ln((d*(-a*b)^(1/2)-b*c+b*(d*x+c))/(d*(-a*b)^(1/2)-b*c)))/b-1/2*(dilog( (d*(-a*b)^(1/2)+b*c-b*(d*x+c))/(d*(-a*b)^(1/2)+b*c))+dilog((d*(-a*b)^(1/2) -b*c+b*(d*x+c))/(d*(-a*b)^(1/2)-b*c)))/b)*b*d^3)
\[ \int \frac {\log \left (\frac {a+b x^2}{x^2}\right )}{c+d x} \, dx=\int { \frac {\log \left (\frac {b x^{2} + a}{x^{2}}\right )}{d x + c} \,d x } \] Input:
integrate(log((b*x^2+a)/x^2)/(d*x+c),x, algorithm="fricas")
Output:
integral(log((b*x^2 + a)/x^2)/(d*x + c), x)
\[ \int \frac {\log \left (\frac {a+b x^2}{x^2}\right )}{c+d x} \, dx=\int \frac {\log {\left (\frac {a}{x^{2}} + b \right )}}{c + d x}\, dx \] Input:
integrate(ln((b*x**2+a)/x**2)/(d*x+c),x)
Output:
Integral(log(a/x**2 + b)/(c + d*x), x)
\[ \int \frac {\log \left (\frac {a+b x^2}{x^2}\right )}{c+d x} \, dx=\int { \frac {\log \left (\frac {b x^{2} + a}{x^{2}}\right )}{d x + c} \,d x } \] Input:
integrate(log((b*x^2+a)/x^2)/(d*x+c),x, algorithm="maxima")
Output:
integrate(log((b*x^2 + a)/x^2)/(d*x + c), x)
\[ \int \frac {\log \left (\frac {a+b x^2}{x^2}\right )}{c+d x} \, dx=\int { \frac {\log \left (\frac {b x^{2} + a}{x^{2}}\right )}{d x + c} \,d x } \] Input:
integrate(log((b*x^2+a)/x^2)/(d*x+c),x, algorithm="giac")
Output:
integrate(log((b*x^2 + a)/x^2)/(d*x + c), x)
Timed out. \[ \int \frac {\log \left (\frac {a+b x^2}{x^2}\right )}{c+d x} \, dx=\int \frac {\ln \left (\frac {b\,x^2+a}{x^2}\right )}{c+d\,x} \,d x \] Input:
int(log((a + b*x^2)/x^2)/(c + d*x),x)
Output:
int(log((a + b*x^2)/x^2)/(c + d*x), x)
\[ \int \frac {\log \left (\frac {a+b x^2}{x^2}\right )}{c+d x} \, dx=\int \frac {\mathrm {log}\left (\frac {b \,x^{2}+a}{x^{2}}\right )}{d x +c}d x \] Input:
int(log((b*x^2+a)/x^2)/(d*x+c),x)
Output:
int(log((a + b*x**2)/x**2)/(c + d*x),x)