Integrand size = 16, antiderivative size = 114 \[ \int \frac {x}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=\frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \operatorname {ExpIntegralEi}\left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{p}\right )}{4 b p^3}-\frac {a+b x^2}{4 b p \log ^2\left (c \left (a+b x^2\right )^p\right )}-\frac {a+b x^2}{4 b p^2 \log \left (c \left (a+b x^2\right )^p\right )} \]
1/4*(b*x^2+a)*Ei(ln(c*(b*x^2+a)^p)/p)/b/p^3/((c*(b*x^2+a)^p)^(1/p))+1/4*(- b*x^2-a)/b/p/ln(c*(b*x^2+a)^p)^2+1/4*(-b*x^2-a)/b/p^2/ln(c*(b*x^2+a)^p)
Time = 0.04 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.99 \[ \int \frac {x}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=-\frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \left (-\operatorname {ExpIntegralEi}\left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{p}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )+p \left (c \left (a+b x^2\right )^p\right )^{\frac {1}{p}} \left (p+\log \left (c \left (a+b x^2\right )^p\right )\right )\right )}{4 b p^3 \log ^2\left (c \left (a+b x^2\right )^p\right )} \]
-1/4*((a + b*x^2)*(-(ExpIntegralEi[Log[c*(a + b*x^2)^p]/p]*Log[c*(a + b*x^ 2)^p]^2) + p*(c*(a + b*x^2)^p)^p^(-1)*(p + Log[c*(a + b*x^2)^p])))/(b*p^3* (c*(a + b*x^2)^p)^p^(-1)*Log[c*(a + b*x^2)^p]^2)
Time = 0.37 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2904, 2836, 2734, 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.
| \(\displaystyle \int \frac {x}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{\log ^3\left (c \left (b x^2+a\right )^p\right )}dx^2\) |
| \(\Big \downarrow \) 2836 |
| \(\displaystyle \frac {\int \frac {1}{\log ^3\left (c \left (b x^2+a\right )^p\right )}d\left (b x^2+a\right )}{2 b}\) |
| \(\Big \downarrow \) 2734 |
| \(\displaystyle \frac {\frac {\int \frac {1}{\log ^2\left (c \left (b x^2+a\right )^p\right )}d\left (b x^2+a\right )}{2 p}-\frac {a+b x^2}{2 p \log ^2\left (c \left (a+b x^2\right )^p\right )}}{2 b}\) |
| \(\Big \downarrow \) 2734 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {1}{\log \left (c \left (b x^2+a\right )^p\right )}d\left (b x^2+a\right )}{p}-\frac {a+b x^2}{p \log \left (c \left (a+b x^2\right )^p\right )}}{2 p}-\frac {a+b x^2}{2 p \log ^2\left (c \left (a+b x^2\right )^p\right )}}{2 b}\) |
| \(\Big \downarrow \) 2737 |
| \(\displaystyle \frac {\frac {\frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \int \frac {\left (c \left (b x^2+a\right )^p\right )^{\frac {1}{p}}}{x^2}d\log \left (c \left (b x^2+a\right )^p\right )}{p^2}-\frac {a+b x^2}{p \log \left (c \left (a+b x^2\right )^p\right )}}{2 p}-\frac {a+b x^2}{2 p \log ^2\left (c \left (a+b x^2\right )^p\right )}}{2 b}\) |
| \(\Big \downarrow \) 2609 |
| \(\displaystyle \frac {\frac {\frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \operatorname {ExpIntegralEi}\left (\frac {\log \left (c \left (b x^2+a\right )^p\right )}{p}\right )}{p^2}-\frac {a+b x^2}{p \log \left (c \left (a+b x^2\right )^p\right )}}{2 p}-\frac {a+b x^2}{2 p \log ^2\left (c \left (a+b x^2\right )^p\right )}}{2 b}\) |
((((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*p) - (a + b*x^2)/(2* p*Log[c*(a + b*x^2)^p]^2))/(2*b)
3.2.17.3.1 Defintions of rubi rules used
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]
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]
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]
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]
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])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.95 (sec) , antiderivative size = 716, normalized size of antiderivative = 6.28
| method | result | size |
| risch | \(-\frac {i \pi b \,x^{2} \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 b \,x^{2} \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 b \,x^{2} {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}+i \pi b \,x^{2} {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+i \pi a \,\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 a \,\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 a {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}+i \pi a {\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 ) b \,x^{2}+2 b \,x^{2} \ln \left (\left (b \,x^{2}+a \right )^{p}\right )+2 \ln \left (c \right ) a +2 a \ln \left (\left (b \,x^{2}+a \right )^{p}\right )+2 x^{2} p b +2 a p}{2 p^{2} {\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 )}^{2} b}-\frac {\left (b \,x^{2}+a \right ) c^{-\frac {1}{p}} {\left (\left (b \,x^{2}+a \right )^{p}\right )}^{-\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 {Ei}_{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 )}{4 p^{3} b}\) | \(716\) |
-1/2*(I*Pi*b*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2-I*Pi*b*x^2*cs gn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)-I*Pi*b*x^2*csgn(I*c*(b*x ^2+a)^p)^3+I*Pi*b*x^2*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)+I*Pi*a*csgn(I*(b*x ^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2-I*Pi*a*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^ 2+a)^p)*csgn(I*c)-I*Pi*a*csgn(I*c*(b*x^2+a)^p)^3+I*Pi*a*csgn(I*c*(b*x^2+a) ^p)^2*csgn(I*c)+2*ln(c)*b*x^2+2*b*x^2*ln((b*x^2+a)^p)+2*ln(c)*a+2*a*ln((b* x^2+a)^p)+2*x^2*p*b+2*a*p)/p^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/b-1/4/p^3/b*(b*x^2+a)*c^(-1/p)*((b*x^2+a)^p)^(-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)
Time = 0.33 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.38 \[ \int \frac {x}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=-\frac {{\left (b p^{2} x^{2} + a p^{2} + {\left (b p^{2} x^{2} + a p^{2}\right )} \log \left (b x^{2} + a\right ) + {\left (b p x^{2} + a p\right )} \log \left (c\right )\right )} c^{\left (\frac {1}{p}\right )} - {\left (p^{2} \log \left (b x^{2} + a\right )^{2} + 2 \, p \log \left (b x^{2} + a\right ) \log \left (c\right ) + \log \left (c\right )^{2}\right )} \operatorname {log\_integral}\left ({\left (b x^{2} + a\right )} c^{\left (\frac {1}{p}\right )}\right )}{4 \, {\left (b p^{5} \log \left (b x^{2} + a\right )^{2} + 2 \, b p^{4} \log \left (b x^{2} + a\right ) \log \left (c\right ) + b p^{3} \log \left (c\right )^{2}\right )} c^{\left (\frac {1}{p}\right )}} \]
-1/4*((b*p^2*x^2 + a*p^2 + (b*p^2*x^2 + a*p^2)*log(b*x^2 + a) + (b*p*x^2 + a*p)*log(c))*c^(1/p) - (p^2*log(b*x^2 + a)^2 + 2*p*log(b*x^2 + a)*log(c) + log(c)^2)*log_integral((b*x^2 + a)*c^(1/p)))/((b*p^5*log(b*x^2 + a)^2 + 2*b*p^4*log(b*x^2 + a)*log(c) + b*p^3*log(c)^2)*c^(1/p))
\[ \int \frac {x}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=\int \frac {x}{\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}}\, dx \]
\[ \int \frac {x}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=\int { \frac {x}{\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}} \,d x } \]
-1/4*(b*(p + log(c))*x^2 + a*(p + log(c)) + (b*p*x^2 + a*p)*log(b*x^2 + a) )/(b*p^4*log(b*x^2 + a)^2 + 2*b*p^3*log(b*x^2 + a)*log(c) + b*p^2*log(c)^2 ) + integrate(1/2*x/(p^3*log(b*x^2 + a) + p^2*log(c)), x)
Leaf count of result is larger than twice the leaf count of optimal. 406 vs. \(2 (108) = 216\).
Time = 0.34 (sec) , antiderivative size = 406, normalized size of antiderivative = 3.56 \[ \int \frac {x}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=-\frac {{\left (b x^{2} + a\right )} p^{2} \log \left (b x^{2} + a\right )}{4 \, {\left (b p^{5} \log \left (b x^{2} + a\right )^{2} + 2 \, b p^{4} \log \left (b x^{2} + a\right ) \log \left (c\right ) + b p^{3} \log \left (c\right )^{2}\right )}} + \frac {p^{2} {\rm Ei}\left (\frac {\log \left (c\right )}{p} + \log \left (b x^{2} + a\right )\right ) \log \left (b x^{2} + a\right )^{2}}{4 \, {\left (b p^{5} \log \left (b x^{2} + a\right )^{2} + 2 \, b p^{4} \log \left (b x^{2} + a\right ) \log \left (c\right ) + b p^{3} \log \left (c\right )^{2}\right )} c^{\left (\frac {1}{p}\right )}} - \frac {{\left (b x^{2} + a\right )} p^{2}}{4 \, {\left (b p^{5} \log \left (b x^{2} + a\right )^{2} + 2 \, b p^{4} \log \left (b x^{2} + a\right ) \log \left (c\right ) + b p^{3} \log \left (c\right )^{2}\right )}} - \frac {{\left (b x^{2} + a\right )} p \log \left (c\right )}{4 \, {\left (b p^{5} \log \left (b x^{2} + a\right )^{2} + 2 \, b p^{4} \log \left (b x^{2} + a\right ) \log \left (c\right ) + b p^{3} \log \left (c\right )^{2}\right )}} + \frac {p {\rm Ei}\left (\frac {\log \left (c\right )}{p} + \log \left (b x^{2} + a\right )\right ) \log \left (b x^{2} + a\right ) \log \left (c\right )}{2 \, {\left (b p^{5} \log \left (b x^{2} + a\right )^{2} + 2 \, b p^{4} \log \left (b x^{2} + a\right ) \log \left (c\right ) + b p^{3} \log \left (c\right )^{2}\right )} c^{\left (\frac {1}{p}\right )}} + \frac {{\rm Ei}\left (\frac {\log \left (c\right )}{p} + \log \left (b x^{2} + a\right )\right ) \log \left (c\right )^{2}}{4 \, {\left (b p^{5} \log \left (b x^{2} + a\right )^{2} + 2 \, b p^{4} \log \left (b x^{2} + a\right ) \log \left (c\right ) + b p^{3} \log \left (c\right )^{2}\right )} c^{\left (\frac {1}{p}\right )}} \]
-1/4*(b*x^2 + a)*p^2*log(b*x^2 + a)/(b*p^5*log(b*x^2 + a)^2 + 2*b*p^4*log( b*x^2 + a)*log(c) + b*p^3*log(c)^2) + 1/4*p^2*Ei(log(c)/p + log(b*x^2 + a) )*log(b*x^2 + a)^2/((b*p^5*log(b*x^2 + a)^2 + 2*b*p^4*log(b*x^2 + a)*log(c ) + b*p^3*log(c)^2)*c^(1/p)) - 1/4*(b*x^2 + a)*p^2/(b*p^5*log(b*x^2 + a)^2 + 2*b*p^4*log(b*x^2 + a)*log(c) + b*p^3*log(c)^2) - 1/4*(b*x^2 + a)*p*log (c)/(b*p^5*log(b*x^2 + a)^2 + 2*b*p^4*log(b*x^2 + a)*log(c) + b*p^3*log(c) ^2) + 1/2*p*Ei(log(c)/p + log(b*x^2 + a))*log(b*x^2 + a)*log(c)/((b*p^5*lo g(b*x^2 + a)^2 + 2*b*p^4*log(b*x^2 + a)*log(c) + b*p^3*log(c)^2)*c^(1/p)) + 1/4*Ei(log(c)/p + log(b*x^2 + a))*log(c)^2/((b*p^5*log(b*x^2 + a)^2 + 2* b*p^4*log(b*x^2 + a)*log(c) + b*p^3*log(c)^2)*c^(1/p))
Timed out. \[ \int \frac {x}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=\int \frac {x}{{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^3} \,d x \]