Integrand size = 18, antiderivative size = 138 \[ \int \frac {x^3}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=-\frac {a \left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \operatorname {ExpIntegralEi}\left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{p}\right )}{2 b^2 p^2}+\frac {\left (a+b x^2\right )^2 \left (c \left (a+b x^2\right )^p\right )^{-2/p} \operatorname {ExpIntegralEi}\left (\frac {2 \log \left (c \left (a+b x^2\right )^p\right )}{p}\right )}{b^2 p^2}-\frac {x^2 \left (a+b x^2\right )}{2 b p \log \left (c \left (a+b x^2\right )^p\right )} \]
-1/2*a*(b*x^2+a)*Ei(ln(c*(b*x^2+a)^p)/p)/b^2/p^2/((c*(b*x^2+a)^p)^(1/p))+( b*x^2+a)^2*Ei(2*ln(c*(b*x^2+a)^p)/p)/b^2/p^2/((c*(b*x^2+a)^p)^(2/p))-1/2*x ^2*(b*x^2+a)/b/p/ln(c*(b*x^2+a)^p)
Time = 0.08 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.14 \[ \int \frac {x^3}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=-\frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-2/p} \left (b p x^2 \left (c \left (a+b x^2\right )^p\right )^{2/p}+a \left (c \left (a+b x^2\right )^p\right )^{\frac {1}{p}} \operatorname {ExpIntegralEi}\left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{p}\right ) \log \left (c \left (a+b x^2\right )^p\right )-2 \left (a+b x^2\right ) \operatorname {ExpIntegralEi}\left (\frac {2 \log \left (c \left (a+b x^2\right )^p\right )}{p}\right ) \log \left (c \left (a+b x^2\right )^p\right )\right )}{2 b^2 p^2 \log \left (c \left (a+b x^2\right )^p\right )} \]
-1/2*((a + b*x^2)*(b*p*x^2*(c*(a + b*x^2)^p)^(2/p) + a*(c*(a + b*x^2)^p)^p ^(-1)*ExpIntegralEi[Log[c*(a + b*x^2)^p]/p]*Log[c*(a + b*x^2)^p] - 2*(a + b*x^2)*ExpIntegralEi[(2*Log[c*(a + b*x^2)^p])/p]*Log[c*(a + b*x^2)^p]))/(b ^2*p^2*(c*(a + b*x^2)^p)^(2/p)*Log[c*(a + b*x^2)^p])
Time = 0.56 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.40, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {2904, 2847, 2836, 2737, 2609, 2846, 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.
\(\displaystyle \int \frac {x^3}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx\) |
\(\Big \downarrow \) 2904 |
\(\displaystyle \frac {1}{2} \int \frac {x^2}{\log ^2\left (c \left (b x^2+a\right )^p\right )}dx^2\) |
\(\Big \downarrow \) 2847 |
\(\displaystyle \frac {1}{2} \left (\frac {a \int \frac {1}{\log \left (c \left (b x^2+a\right )^p\right )}dx^2}{b p}+\frac {2 \int \frac {x^2}{\log \left (c \left (b x^2+a\right )^p\right )}dx^2}{p}-\frac {x^2 \left (a+b x^2\right )}{b p \log \left (c \left (a+b x^2\right )^p\right )}\right )\) |
\(\Big \downarrow \) 2836 |
\(\displaystyle \frac {1}{2} \left (\frac {a \int \frac {1}{\log \left (c \left (b x^2+a\right )^p\right )}d\left (b x^2+a\right )}{b^2 p}+\frac {2 \int \frac {x^2}{\log \left (c \left (b x^2+a\right )^p\right )}dx^2}{p}-\frac {x^2 \left (a+b x^2\right )}{b p \log \left (c \left (a+b x^2\right )^p\right )}\right )\) |
\(\Big \downarrow \) 2737 |
\(\displaystyle \frac {1}{2} \left (\frac {a \left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \int \frac {\left (c \left (b x^2+a\right )^p\right )^{\frac {1}{p}}}{x^2}d\log \left (c \left (b x^2+a\right )^p\right )}{b^2 p^2}+\frac {2 \int \frac {x^2}{\log \left (c \left (b x^2+a\right )^p\right )}dx^2}{p}-\frac {x^2 \left (a+b x^2\right )}{b p \log \left (c \left (a+b x^2\right )^p\right )}\right )\) |
\(\Big \downarrow \) 2609 |
\(\displaystyle \frac {1}{2} \left (\frac {2 \int \frac {x^2}{\log \left (c \left (b x^2+a\right )^p\right )}dx^2}{p}+\frac {a \left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \operatorname {ExpIntegralEi}\left (\frac {\log \left (c \left (b x^2+a\right )^p\right )}{p}\right )}{b^2 p^2}-\frac {x^2 \left (a+b x^2\right )}{b p \log \left (c \left (a+b x^2\right )^p\right )}\right )\) |
\(\Big \downarrow \) 2846 |
\(\displaystyle \frac {1}{2} \left (\frac {2 \int \left (\frac {b x^2+a}{b \log \left (c \left (b x^2+a\right )^p\right )}-\frac {a}{b \log \left (c \left (b x^2+a\right )^p\right )}\right )dx^2}{p}+\frac {a \left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \operatorname {ExpIntegralEi}\left (\frac {\log \left (c \left (b x^2+a\right )^p\right )}{p}\right )}{b^2 p^2}-\frac {x^2 \left (a+b x^2\right )}{b p \log \left (c \left (a+b x^2\right )^p\right )}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {a \left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \operatorname {ExpIntegralEi}\left (\frac {\log \left (c \left (b x^2+a\right )^p\right )}{p}\right )}{b^2 p^2}+\frac {2 \left (\frac {\left (a+b x^2\right )^2 \left (c \left (a+b x^2\right )^p\right )^{-2/p} \operatorname {ExpIntegralEi}\left (\frac {2 \log \left (c \left (b x^2+a\right )^p\right )}{p}\right )}{b^2 p}-\frac {a \left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \operatorname {ExpIntegralEi}\left (\frac {\log \left (c \left (b x^2+a\right )^p\right )}{p}\right )}{b^2 p}\right )}{p}-\frac {x^2 \left (a+b x^2\right )}{b p \log \left (c \left (a+b x^2\right )^p\right )}\right )\) |
((a*(a + b*x^2)*ExpIntegralEi[Log[c*(a + b*x^2)^p]/p])/(b^2*p^2*(c*(a + b* x^2)^p)^p^(-1)) + (2*(-((a*(a + b*x^2)*ExpIntegralEi[Log[c*(a + b*x^2)^p]/ p])/(b^2*p*(c*(a + b*x^2)^p)^p^(-1))) + ((a + b*x^2)^2*ExpIntegralEi[(2*Lo g[c*(a + b*x^2)^p])/p])/(b^2*p*(c*(a + b*x^2)^p)^(2/p))))/p - (x^2*(a + b* x^2))/(b*p*Log[c*(a + b*x^2)^p]))/2
3.2.9.3.1 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/(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[((f_.) + (g_.)*(x_))^(q_.)/((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.) ]*(b_.)), x_Symbol] :> Int[ExpandIntegrand[(f + g*x)^q/(a + b*Log[c*(d + e* x)^n]), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] & & IGtQ[q, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. )*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)*(f + g*x)^q*((a + b*Log[c*(d + e *x)^n])^(p + 1)/(b*e*n*(p + 1))), x] + (-Simp[(q + 1)/(b*n*(p + 1)) Int[( f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x] + Simp[q*((e*f - d*g) /(b*e*n*(p + 1))) Int[(f + g*x)^(q - 1)*(a + b*Log[c*(d + e*x)^n])^(p + 1 ), x], x]) /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && Lt Q[p, -1] && GtQ[q, 0]
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 = 0.85 (sec) , antiderivative size = 1487, normalized size of antiderivative = 10.78
-1/p/b*x^2*(b*x^2+a)/(I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2-I*P i*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)) -1/p^2*((b*x^2+a)^p)^(-2/p)*c^(-2/p)*exp(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,-2*ln(b*x^2+a)-(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)*x^4-2/p^2/b*((b*x^2+a)^p)^(-2/p)*c^(-2/p)*exp(I*Pi*c sgn(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,-2*ln(b*x^2+a)-(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)*c sgn(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)*a*x^2-1/p^2/b^2*((b*x^2+a )^p)^(-2/p)*c^(-2/p)*exp(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,-2*ln(b* x^2+a)-(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)*a^2+1/2/p^2/b*a*((b*x^2+a)^p)^(-1/p)*c^(-1/p)*exp(1/2*I*Pi*csgn(I...
Time = 0.31 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.02 \[ \int \frac {x^3}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=-\frac {{\left (a p \log \left (b x^{2} + a\right ) + a \log \left (c\right )\right )} c^{\left (\frac {1}{p}\right )} \operatorname {log\_integral}\left ({\left (b x^{2} + a\right )} c^{\left (\frac {1}{p}\right )}\right ) + {\left (b^{2} p x^{4} + a b p x^{2}\right )} c^{\frac {2}{p}} - 2 \, {\left (p \log \left (b x^{2} + a\right ) + \log \left (c\right )\right )} \operatorname {log\_integral}\left ({\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} c^{\frac {2}{p}}\right )}{2 \, {\left (b^{2} p^{3} \log \left (b x^{2} + a\right ) + b^{2} p^{2} \log \left (c\right )\right )} c^{\frac {2}{p}}} \]
-1/2*((a*p*log(b*x^2 + a) + a*log(c))*c^(1/p)*log_integral((b*x^2 + a)*c^( 1/p)) + (b^2*p*x^4 + a*b*p*x^2)*c^(2/p) - 2*(p*log(b*x^2 + a) + log(c))*lo g_integral((b^2*x^4 + 2*a*b*x^2 + a^2)*c^(2/p)))/((b^2*p^3*log(b*x^2 + a) + b^2*p^2*log(c))*c^(2/p))
\[ \int \frac {x^3}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=\int \frac {x^{3}}{\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}\, dx \]
\[ \int \frac {x^3}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=\int { \frac {x^{3}}{\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}} \,d x } \]
-1/2*(b*x^4 + a*x^2)/(b*p^2*log(b*x^2 + a) + b*p*log(c)) + integrate((2*b* x^3 + a*x)/(b*p^2*log(b*x^2 + a) + b*p*log(c)), x)
Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (136) = 272\).
Time = 0.30 (sec) , antiderivative size = 313, normalized size of antiderivative = 2.27 \[ \int \frac {x^3}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=\frac {1}{2} \, a {\left (\frac {{\left (b x^{2} + a\right )} p}{b^{2} p^{3} \log \left (b x^{2} + a\right ) + b^{2} p^{2} \log \left (c\right )} - \frac {p {\rm Ei}\left (\frac {\log \left (c\right )}{p} + \log \left (b x^{2} + a\right )\right ) \log \left (b x^{2} + a\right )}{{\left (b^{2} p^{3} \log \left (b x^{2} + a\right ) + b^{2} p^{2} \log \left (c\right )\right )} c^{\left (\frac {1}{p}\right )}} - \frac {{\rm Ei}\left (\frac {\log \left (c\right )}{p} + \log \left (b x^{2} + a\right )\right ) \log \left (c\right )}{{\left (b^{2} p^{3} \log \left (b x^{2} + a\right ) + b^{2} p^{2} \log \left (c\right )\right )} c^{\left (\frac {1}{p}\right )}}\right )} - \frac {\frac {{\left (b x^{2} + a\right )}^{2} p}{b p^{3} \log \left (b x^{2} + a\right ) + b p^{2} \log \left (c\right )} - \frac {2 \, p {\rm Ei}\left (\frac {2 \, \log \left (c\right )}{p} + 2 \, \log \left (b x^{2} + a\right )\right ) \log \left (b x^{2} + a\right )}{{\left (b p^{3} \log \left (b x^{2} + a\right ) + b p^{2} \log \left (c\right )\right )} c^{\frac {2}{p}}} - \frac {2 \, {\rm Ei}\left (\frac {2 \, \log \left (c\right )}{p} + 2 \, \log \left (b x^{2} + a\right )\right ) \log \left (c\right )}{{\left (b p^{3} \log \left (b x^{2} + a\right ) + b p^{2} \log \left (c\right )\right )} c^{\frac {2}{p}}}}{2 \, b} \]
1/2*a*((b*x^2 + a)*p/(b^2*p^3*log(b*x^2 + a) + b^2*p^2*log(c)) - p*Ei(log( c)/p + log(b*x^2 + a))*log(b*x^2 + a)/((b^2*p^3*log(b*x^2 + a) + b^2*p^2*l og(c))*c^(1/p)) - Ei(log(c)/p + log(b*x^2 + a))*log(c)/((b^2*p^3*log(b*x^2 + a) + b^2*p^2*log(c))*c^(1/p))) - 1/2*((b*x^2 + a)^2*p/(b*p^3*log(b*x^2 + a) + b*p^2*log(c)) - 2*p*Ei(2*log(c)/p + 2*log(b*x^2 + a))*log(b*x^2 + a )/((b*p^3*log(b*x^2 + a) + b*p^2*log(c))*c^(2/p)) - 2*Ei(2*log(c)/p + 2*lo g(b*x^2 + a))*log(c)/((b*p^3*log(b*x^2 + a) + b*p^2*log(c))*c^(2/p)))/b
Timed out. \[ \int \frac {x^3}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=\int \frac {x^3}{{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^2} \,d x \]