\(\int \frac {x}{\log ^2(c (a+b x^2)^p)} \, dx\) [110]

Optimal result

Integrand size = 16, antiderivative size = 83 \[ \int \frac {x}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=\frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \operatorname {ExpIntegralEi}\left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{p}\right )}{2 b p^2}-\frac {a+b x^2}{2 b p \log \left (c \left (a+b x^2\right )^p\right )} \] Output:

1/2*(b*x^2+a)*Ei(ln(c*(b*x^2+a)^p)/p)/b/p^2/((c*(b*x^2+a)^p)^(1/p))-1/2*(b 
*x^2+a)/b/p/ln(c*(b*x^2+a)^p)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.17 \[ \int \frac {x}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=-\frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \left (p \left (c \left (a+b x^2\right )^p\right )^{\frac {1}{p}}-\operatorname {ExpIntegralEi}\left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{p}\right ) \log \left (c \left (a+b x^2\right )^p\right )\right )}{2 b p^2 \log \left (c \left (a+b x^2\right )^p\right )} \] Input:

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

Output:

-1/2*((a + b*x^2)*(p*(c*(a + b*x^2)^p)^p^(-1) - ExpIntegralEi[Log[c*(a + b 
*x^2)^p]/p]*Log[c*(a + b*x^2)^p]))/(b*p^2*(c*(a + b*x^2)^p)^p^(-1)*Log[c*( 
a + b*x^2)^p])
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2904, 2836, 2734, 2737, 2609}

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.

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

\(\Big \downarrow \) 2904

\(\displaystyle \frac {1}{2} \int \frac {1}{\log ^2\left (c \left (b x^2+a\right )^p\right )}dx^2\)

\(\Big \downarrow \) 2836

\(\displaystyle \frac {\int \frac {1}{\log ^2\left (c \left (b x^2+a\right )^p\right )}d\left (b x^2+a\right )}{2 b}\)

\(\Big \downarrow \) 2734

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

\(\Big \downarrow \) 2737

\(\displaystyle \frac {\frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \int \frac {\left (c \left (b x^2+a\right )^p\right )^{\frac {1}{p}}}{x^2}d\log \left (c \left (b x^2+a\right )^p\right )}{p^2}-\frac {a+b x^2}{p \log \left (c \left (a+b x^2\right )^p\right )}}{2 b}\)

\(\Big \downarrow \) 2609

\(\displaystyle \frac {\frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \operatorname {ExpIntegralEi}\left (\frac {\log \left (c \left (b x^2+a\right )^p\right )}{p}\right )}{p^2}-\frac {a+b x^2}{p \log \left (c \left (a+b x^2\right )^p\right )}}{2 b}\)

Input:

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

Output:

(((a + b*x^2)*ExpIntegralEi[Log[c*(a + b*x^2)^p]/p])/(p^2*(c*(a + b*x^2)^p 
)^p^(-1)) - (a + b*x^2)/(p*Log[c*(a + b*x^2)^p]))/(2*b)
 

Defintions of rubi rules used

rule 2609

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Si 
mp[(F^(g*(e - c*(f/d)))/d)*ExpIntegralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; F 
reeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

rule 2734

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[x*((a + b 
*Log[c*x^n])^(p + 1)/(b*n*(p + 1))), x] - Simp[1/(b*n*(p + 1))   Int[(a + b 
*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] && Int 
egerQ[2*p]
 

rule 2737

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[x/(n*(c*x 
^n)^(1/n))   Subst[Int[E^(x/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ 
[{a, b, c, n, p}, x]
 

rule 2836

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

rule 2904

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L 
og[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, 
 x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) & 
&  !(EqQ[q, 1] && ILtQ[n, 0] && IGtQ[m, 0])
 

Maple [C] (warning: unable to verify)

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

Time = 2.82 (sec) , antiderivative size = 421, normalized size of antiderivative = 5.07

method	result	size
risch	\(-\frac {b \,x^{2}+a}{\left (i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}+i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+2 \ln \left (c \right )+2 \ln \left (\left (b \,x^{2}+a \right )^{p}\right )\right ) p b}-\frac {\left (b \,x^{2}+a \right ) {\left (\left (b \,x^{2}+a \right )^{p}\right )}^{-\frac {1}{p}} c^{-\frac {1}{p}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \left (-\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )+\operatorname {csgn}\left (i c \right )\right ) \left (-\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )+\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right )\right )}{2 p}} \operatorname {expIntegral}_{1}\left (-\ln \left (b \,x^{2}+a \right )-\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}+i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+2 \ln \left (c \right )+2 \ln \left (\left (b \,x^{2}+a \right )^{p}\right )-2 p \ln \left (b \,x^{2}+a \right )}{2 p}\right )}{2 p^{2} b}\)	\(421\)

Input:

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

Output:

-1/(I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2-I*Pi*csgn(I*(b*x^2+a) 
^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)-I*Pi*csgn(I*c*(b*x^2+a)^p)^3+I*Pi*csgn 
(I*c*(b*x^2+a)^p)^2*csgn(I*c)+2*ln(c)+2*ln((b*x^2+a)^p))/p/b*(b*x^2+a)-1/2 
/p^2/b*(b*x^2+a)*((b*x^2+a)^p)^(-1/p)*c^(-1/p)*exp(1/2*I*Pi*csgn(I*c*(b*x^ 
2+a)^p)*(-csgn(I*c*(b*x^2+a)^p)+csgn(I*c))*(-csgn(I*c*(b*x^2+a)^p)+csgn(I* 
(b*x^2+a)^p))/p)*Ei(1,-ln(b*x^2+a)-1/2*(I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c* 
(b*x^2+a)^p)^2-I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)-I* 
Pi*csgn(I*c*(b*x^2+a)^p)^3+I*Pi*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)+2*ln(c)+ 
2*ln((b*x^2+a)^p)-2*p*ln(b*x^2+a))/p)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.94 \[ \int \frac {x}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=-\frac {{\left (b p x^{2} + a p\right )} c^{\left (\frac {1}{p}\right )} - {\left (p \log \left (b x^{2} + a\right ) + \log \left (c\right )\right )} \operatorname {log\_integral}\left ({\left (b x^{2} + a\right )} c^{\left (\frac {1}{p}\right )}\right )}{2 \, {\left (b p^{3} \log \left (b x^{2} + a\right ) + b p^{2} \log \left (c\right )\right )} c^{\left (\frac {1}{p}\right )}} \] Input:

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

Output:

-1/2*((b*p*x^2 + a*p)*c^(1/p) - (p*log(b*x^2 + a) + log(c))*log_integral(( 
b*x^2 + a)*c^(1/p)))/((b*p^3*log(b*x^2 + a) + b*p^2*log(c))*c^(1/p))
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/2*(b*x^2 + a)/(b*p^2*log(b*x^2 + a) + b*p*log(c)) + integrate(x/(p^2*lo 
g(b*x^2 + a) + p*log(c)), x)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.70 \[ \int \frac {x}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=-\frac {{\left (b x^{2} + a\right )} p}{2 \, {\left (b p^{3} \log \left (b x^{2} + a\right ) + b p^{2} \log \left (c\right )\right )}} + \frac {p {\rm Ei}\left (\frac {\log \left (c\right )}{p} + \log \left (b x^{2} + a\right )\right ) \log \left (b x^{2} + a\right )}{2 \, {\left (b p^{3} \log \left (b x^{2} + a\right ) + b p^{2} \log \left (c\right )\right )} c^{\left (\frac {1}{p}\right )}} + \frac {{\rm Ei}\left (\frac {\log \left (c\right )}{p} + \log \left (b x^{2} + a\right )\right ) \log \left (c\right )}{2 \, {\left (b p^{3} \log \left (b x^{2} + a\right ) + b p^{2} \log \left (c\right )\right )} c^{\left (\frac {1}{p}\right )}} \] Input:

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

Output:

-1/2*(b*x^2 + a)*p/(b*p^3*log(b*x^2 + a) + b*p^2*log(c)) + 1/2*p*Ei(log(c) 
/p + log(b*x^2 + a))*log(b*x^2 + a)/((b*p^3*log(b*x^2 + a) + b*p^2*log(c)) 
*c^(1/p)) + 1/2*Ei(log(c)/p + log(b*x^2 + a))*log(c)/((b*p^3*log(b*x^2 + a 
) + b*p^2*log(c))*c^(1/p))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=\frac {2 \left (\int \frac {x^{3}}{{\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right )}^{2} a +{\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right )}^{2} b \,x^{2}}d x \right ) \mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right ) b^{2} p -a}{2 \,\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right ) b p} \] Input:

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

Output:

(2*int(x**3/(log((a + b*x**2)**p*c)**2*a + log((a + b*x**2)**p*c)**2*b*x** 
2),x)*log((a + b*x**2)**p*c)*b**2*p - a)/(2*log((a + b*x**2)**p*c)*b*p)