\(\int \frac {\log (c (a+b x^2)^n)}{a+b x^2} \, dx\) [357]

Optimal result

Integrand size = 22, antiderivative size = 163 \[ \int \frac {\log \left (c \left (a+b x^2\right )^n\right )}{a+b x^2} \, dx=\frac {i n \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {a} \sqrt {b}}+\frac {2 n \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {a} \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^n\right )}{\sqrt {a} \sqrt {b}}+\frac {i n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {a} \sqrt {b}} \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.79 \[ \int \frac {\log \left (c \left (a+b x^2\right )^n\right )}{a+b x^2} \, dx=\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (i n \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )+2 n \log \left (\frac {2 i}{i-\frac {\sqrt {b} x}{\sqrt {a}}}\right )+\log \left (c \left (a+b x^2\right )^n\right )\right )+i n \operatorname {PolyLog}\left (2,\frac {i \sqrt {a}+\sqrt {b} x}{-i \sqrt {a}+\sqrt {b} x}\right )}{\sqrt {a} \sqrt {b}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2920, 27, 5455, 27, 5379, 27, 2849, 2752}

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.

\(\displaystyle \int \frac {\log \left (c \left (a+b x^2\right )^n\right )}{a+b x^2} \, dx\)

\(\Big \downarrow \) 2920

\(\displaystyle \frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^n\right )}{\sqrt {a} \sqrt {b}}-2 b n \int \frac {x \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \left (b x^2+a\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^n\right )}{\sqrt {a} \sqrt {b}}-\frac {2 \sqrt {b} n \int \frac {x \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b x^2+a}dx}{\sqrt {a}}\)

\(\Big \downarrow \) 5455

\(\displaystyle \frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^n\right )}{\sqrt {a} \sqrt {b}}-\frac {2 \sqrt {b} n \left (-\frac {\int \frac {\sqrt {a} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{i \sqrt {a}-\sqrt {b} x}dx}{\sqrt {a} \sqrt {b}}-\frac {i \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{2 b}\right )}{\sqrt {a}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^n\right )}{\sqrt {a} \sqrt {b}}-\frac {2 \sqrt {b} n \left (-\frac {\int \frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{i \sqrt {a}-\sqrt {b} x}dx}{\sqrt {b}}-\frac {i \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{2 b}\right )}{\sqrt {a}}\)

\(\Big \downarrow \) 5379

\(\displaystyle \frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^n\right )}{\sqrt {a} \sqrt {b}}-\frac {2 \sqrt {b} n \left (-\frac {\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {b}}-\frac {\int \frac {a \log \left (\frac {2 \sqrt {a}}{i \sqrt {b} x+\sqrt {a}}\right )}{b x^2+a}dx}{\sqrt {a}}}{\sqrt {b}}-\frac {i \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{2 b}\right )}{\sqrt {a}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^n\right )}{\sqrt {a} \sqrt {b}}-\frac {2 \sqrt {b} n \left (-\frac {\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {b}}-\sqrt {a} \int \frac {\log \left (\frac {2 \sqrt {a}}{i \sqrt {b} x+\sqrt {a}}\right )}{b x^2+a}dx}{\sqrt {b}}-\frac {i \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{2 b}\right )}{\sqrt {a}}\)

\(\Big \downarrow \) 2849

\(\displaystyle \frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^n\right )}{\sqrt {a} \sqrt {b}}-\frac {2 \sqrt {b} n \left (-\frac {\frac {i \sqrt {a} \int \frac {\log \left (\frac {2 \sqrt {a}}{i \sqrt {b} x+\sqrt {a}}\right )}{1-\frac {2 \sqrt {a}}{i \sqrt {b} x+\sqrt {a}}}d\frac {1}{i \sqrt {b} x+\sqrt {a}}}{\sqrt {b}}+\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {b}}}{\sqrt {b}}-\frac {i \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{2 b}\right )}{\sqrt {a}}\)

\(\Big \downarrow \) 2752

\(\displaystyle \frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^n\right )}{\sqrt {a} \sqrt {b}}-\frac {2 \sqrt {b} n \left (-\frac {\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {b}}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {a}}{i \sqrt {b} x+\sqrt {a}}\right )}{2 \sqrt {b}}}{\sqrt {b}}-\frac {i \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{2 b}\right )}{\sqrt {a}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2752

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

rule 2849

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

rule 2920

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.) 
*(x_)^2), x_Symbol] :> With[{u = IntHide[1/(f + g*x^2), x]}, Simp[u*(a + b* 
Log[c*(d + e*x^n)^p]), x] - Simp[b*e*n*p   Int[u*(x^(n - 1)/(d + e*x^n)), x 
], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]
 

rule 5379

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] 
 :> Simp[(-(a + b*ArcTan[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c*( 
p/e)   Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 + c^2*x^2)) 
, x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0 
]
 

rule 5455

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), 
x_Symbol] :> Simp[(-I)*((a + b*ArcTan[c*x])^(p + 1)/(b*e*(p + 1))), x] - Si 
mp[1/(c*d)   Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b, c, 
 d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 2.51 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.62

method	result	size
risch	\(\frac {\left (\ln \left (\left (b \,x^{2}+a \right )^{n}\right )-n \ln \left (b \,x^{2}+a \right )\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}+\frac {n \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (b \,x^{2}+a \right )-b \left (\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{\underline {\hspace {1.25 ex}}\alpha b}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{a}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{a}\right )}{\underline {\hspace {1.25 ex}}\alpha }\right )}{4 b}+\frac {\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{n}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{n}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{n}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{n}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{n}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}\)	\(264\)

Input:

int(ln(c*(b*x^2+a)^n)/(b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

(ln((b*x^2+a)^n)-n*ln(b*x^2+a))/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))+1/4*n/ 
b*sum(1/_alpha*(2*ln(x-_alpha)*ln(b*x^2+a)-b*(1/_alpha/b*ln(x-_alpha)^2+2* 
_alpha/a*ln(x-_alpha)*ln(1/2*(x+_alpha)/_alpha)+2*_alpha/a*dilog(1/2*(x+_a 
lpha)/_alpha))),_alpha=RootOf(_Z^2*b+a))+(1/2*I*Pi*csgn(I*(b*x^2+a)^n)*csg 
n(I*c*(b*x^2+a)^n)^2-1/2*I*Pi*csgn(I*(b*x^2+a)^n)*csgn(I*c*(b*x^2+a)^n)*cs 
gn(I*c)-1/2*I*Pi*csgn(I*c*(b*x^2+a)^n)^3+1/2*I*Pi*csgn(I*c*(b*x^2+a)^n)^2* 
csgn(I*c)+ln(c))/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))
 

Fricas [F]

\[ \int \frac {\log \left (c \left (a+b x^2\right )^n\right )}{a+b x^2} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{n} c\right )}{b x^{2} + a} \,d x } \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {\log \left (c \left (a+b x^2\right )^n\right )}{a+b x^2} \, dx=\int \frac {\log {\left (c \left (a + b x^{2}\right )^{n} \right )}}{a + b x^{2}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\log \left (c \left (a+b x^2\right )^n\right )}{a+b x^2} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{n} c\right )}{b x^{2} + a} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\log \left (c \left (a+b x^2\right )^n\right )}{a+b x^2} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{n} c\right )}{b x^{2} + a} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (c \left (a+b x^2\right )^n\right )}{a+b x^2} \, dx=\int \frac {\ln \left (c\,{\left (b\,x^2+a\right )}^n\right )}{b\,x^2+a} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\log \left (c \left (a+b x^2\right )^n\right )}{a+b x^2} \, dx=\int \frac {\mathrm {log}\left (\left (b \,x^{2}+a \right )^{n} c \right )}{b \,x^{2}+a}d x \] Input:

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

Output:

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