3.1.92 \(\int \frac {\log (d (a+c x^2)^n)}{a e+c e x^2} \, dx\) [92]

3.1.92.1 Optimal result

Integrand size = 25, antiderivative size = 175 \[ \int \frac {\log \left (d \left (a+c x^2\right )^n\right )}{a e+c e x^2} \, dx=\frac {i n \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )^2}{\sqrt {a} \sqrt {c} e}+\frac {2 n \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {c} x}\right )}{\sqrt {a} \sqrt {c} e}+\frac {\arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \log \left (d \left (a+c x^2\right )^n\right )}{\sqrt {a} \sqrt {c} e}+\frac {i n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {c} x}\right )}{\sqrt {a} \sqrt {c} e} \]

output

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

3.1.92.2 Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.75 \[ \int \frac {\log \left (d \left (a+c x^2\right )^n\right )}{a e+c e x^2} \, dx=\frac {\arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (i n \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )+2 n \log \left (\frac {2 i}{i-\frac {\sqrt {c} x}{\sqrt {a}}}\right )+\log \left (d \left (a+c x^2\right )^n\right )\right )+i n \operatorname {PolyLog}\left (2,\frac {i \sqrt {a}+\sqrt {c} x}{-i \sqrt {a}+\sqrt {c} x}\right )}{\sqrt {a} \sqrt {c} e} \]

input

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

output

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

3.1.92.3 Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, 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.

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

\(\Big \downarrow \) 2920

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5455

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5379

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

input

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

output

(ArcTan[(Sqrt[c]*x)/Sqrt[a]]*Log[d*(a + c*x^2)^n])/(Sqrt[a]*Sqrt[c]*e) - ( 
2*Sqrt[c]*n*(((-1/2*I)*ArcTan[(Sqrt[c]*x)/Sqrt[a]]^2)/c - ((ArcTan[(Sqrt[c 
]*x)/Sqrt[a]]*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[c]*x)])/Sqrt[c] + ((I/2)*P 
olyLog[2, 1 - (2*Sqrt[a])/(Sqrt[a] + I*Sqrt[c]*x)])/Sqrt[c])/Sqrt[c]))/(Sq 
rt[a]*e)
 

3.1.92.3.1 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]
 

3.1.92.4 Maple [C] (warning: unable to verify)

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

Time = 1.60 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.67

method	result	size
risch	\(-\frac {\arctan \left (\frac {x c}{\sqrt {c a}}\right ) n \ln \left (c \,x^{2}+a \right )}{e \sqrt {c a}}+\frac {\arctan \left (\frac {x c}{\sqrt {c a}}\right ) \ln \left (\left (c \,x^{2}+a \right )^{n}\right )}{e \sqrt {c a}}+\frac {n \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (c \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (c \,x^{2}+a \right )-c \left (\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{\underline {\hspace {1.25 ex}}\alpha c}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{a}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{a}\right )}{\underline {\hspace {1.25 ex}}\alpha }\right )}{4 e c}+\frac {\left (i \pi \,\operatorname {csgn}\left (i \left (c \,x^{2}+a \right )^{n}\right ) {\operatorname {csgn}\left (i d \left (c \,x^{2}+a \right )^{n}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i \left (c \,x^{2}+a \right )^{n}\right ) \operatorname {csgn}\left (i d \left (c \,x^{2}+a \right )^{n}\right ) \operatorname {csgn}\left (i d \right )-i \pi {\operatorname {csgn}\left (i d \left (c \,x^{2}+a \right )^{n}\right )}^{3}+i \pi {\operatorname {csgn}\left (i d \left (c \,x^{2}+a \right )^{n}\right )}^{2} \operatorname {csgn}\left (i d \right )+2 \ln \left (d \right )\right ) \arctan \left (\frac {x c}{\sqrt {c a}}\right )}{2 e \sqrt {c a}}\)	\(292\)

input

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

output

-1/e/(c*a)^(1/2)*arctan(x*c/(c*a)^(1/2))*n*ln(c*x^2+a)+1/e/(c*a)^(1/2)*arc 
tan(x*c/(c*a)^(1/2))*ln((c*x^2+a)^n)+1/4/e*n/c*sum(1/_alpha*(2*ln(x-_alpha 
)*ln(c*x^2+a)-c*(1/_alpha/c*ln(x-_alpha)^2+2*_alpha/a*ln(x-_alpha)*ln(1/2* 
(x+_alpha)/_alpha)+2*_alpha/a*dilog(1/2*(x+_alpha)/_alpha))),_alpha=RootOf 
(_Z^2*c+a))+1/2*(I*Pi*csgn(I*(c*x^2+a)^n)*csgn(I*d*(c*x^2+a)^n)^2-I*Pi*csg 
n(I*(c*x^2+a)^n)*csgn(I*d*(c*x^2+a)^n)*csgn(I*d)-I*Pi*csgn(I*d*(c*x^2+a)^n 
)^3+I*Pi*csgn(I*d*(c*x^2+a)^n)^2*csgn(I*d)+2*ln(d))/e/(c*a)^(1/2)*arctan(x 
*c/(c*a)^(1/2))
 

3.1.92.5 Fricas [F]

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

input

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

output

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

3.1.92.6 Sympy [F]

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

input

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

output

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

3.1.92.7 Maxima [F]

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

input

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

output

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

3.1.92.8 Giac [F]

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

input

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

output

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

3.1.92.9 Mupad [F(-1)]

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

input

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

output

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