\(\int \frac {x}{\log ^3(c (a+b x^2)^p)} \, dx\) [117]

Optimal result

Integrand size = 16, antiderivative size = 114 \[ \int \frac {x}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=\frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \operatorname {ExpIntegralEi}\left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{p}\right )}{4 b p^3}-\frac {a+b x^2}{4 b p \log ^2\left (c \left (a+b x^2\right )^p\right )}-\frac {a+b x^2}{4 b p^2 \log \left (c \left (a+b x^2\right )^p\right )} \]

[Out]

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)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2504, 2436, 2334, 2337, 2209} \[ \int \frac {x}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=\frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \operatorname {ExpIntegralEi}\left (\frac {\log \left (c \left (b x^2+a\right )^p\right )}{p}\right )}{4 b p^3}-\frac {a+b x^2}{4 b p^2 \log \left (c \left (a+b x^2\right )^p\right )}-\frac {a+b x^2}{4 b p \log ^2\left (c \left (a+b x^2\right )^p\right )} \]

[In]

Int[x/Log[c*(a + b*x^2)^p]^3,x]

[Out]

((a + b*x^2)*ExpIntegralEi[Log[c*(a + b*x^2)^p]/p])/(4*b*p^3*(c*(a + b*x^2)^p)^p^(-1)) - (a + b*x^2)/(4*b*p*Lo
g[c*(a + b*x^2)^p]^2) - (a + b*x^2)/(4*b*p^2*Log[c*(a + b*x^2)^p])

Rule 2209

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

Rule 2334

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] - Dist[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] &&
IntegerQ[2*p]

Rule 2337

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[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 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[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 2504

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[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])

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\log ^3\left (c (a+b x)^p\right )} \, dx,x,x^2\right ) \\ & = \frac {\text {Subst}\left (\int \frac {1}{\log ^3\left (c x^p\right )} \, dx,x,a+b x^2\right )}{2 b} \\ & = -\frac {a+b x^2}{4 b p \log ^2\left (c \left (a+b x^2\right )^p\right )}+\frac {\text {Subst}\left (\int \frac {1}{\log ^2\left (c x^p\right )} \, dx,x,a+b x^2\right )}{4 b p} \\ & = -\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 )}+\frac {\text {Subst}\left (\int \frac {1}{\log \left (c x^p\right )} \, dx,x,a+b x^2\right )}{4 b p^2} \\ & = -\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 )}+\frac {\left (\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{p}}}{x} \, dx,x,\log \left (c \left (a+b x^2\right )^p\right )\right )}{4 b p^3} \\ & = \frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \text {Ei}\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 )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.99 \[ \int \frac {x}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=-\frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \left (-\operatorname {ExpIntegralEi}\left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{p}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )+p \left (c \left (a+b x^2\right )^p\right )^{\frac {1}{p}} \left (p+\log \left (c \left (a+b x^2\right )^p\right )\right )\right )}{4 b p^3 \log ^2\left (c \left (a+b x^2\right )^p\right )} \]

[In]

Integrate[x/Log[c*(a + b*x^2)^p]^3,x]

[Out]

-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)

Maple [C] (warning: unable to verify)

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}\)	\(716\)

[In]

int(x/ln(c*(b*x^2+a)^p)^3,x,method=_RETURNVERBOSE)

[Out]

-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*csgn(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)*(-cs
gn(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*csg
n(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)

none

Time = 0.33 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.38 \[ \int \frac {x}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=-\frac {{\left (b p^{2} x^{2} + a p^{2} + {\left (b p^{2} x^{2} + a p^{2}\right )} \log \left (b x^{2} + a\right ) + {\left (b p x^{2} + a p\right )} \log \left (c\right )\right )} c^{\left (\frac {1}{p}\right )} - {\left (p^{2} \log \left (b x^{2} + a\right )^{2} + 2 \, p \log \left (b x^{2} + a\right ) \log \left (c\right ) + \log \left (c\right )^{2}\right )} \operatorname {log\_integral}\left ({\left (b x^{2} + a\right )} c^{\left (\frac {1}{p}\right )}\right )}{4 \, {\left (b p^{5} \log \left (b x^{2} + a\right )^{2} + 2 \, b p^{4} \log \left (b x^{2} + a\right ) \log \left (c\right ) + b p^{3} \log \left (c\right )^{2}\right )} c^{\left (\frac {1}{p}\right )}} \]

[In]

integrate(x/log(c*(b*x^2+a)^p)^3,x, algorithm="fricas")

[Out]

-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))

Sympy [F]

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

[In]

integrate(x/ln(c*(b*x**2+a)**p)**3,x)

[Out]

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

Maxima [F]

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

[In]

integrate(x/log(c*(b*x^2+a)^p)^3,x, algorithm="maxima")

[Out]

-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)

Giac [B] (verification not implemented)

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 \[ \int \frac {x}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=-\frac {{\left (b x^{2} + a\right )} p^{2} \log \left (b x^{2} + a\right )}{4 \, {\left (b p^{5} \log \left (b x^{2} + a\right )^{2} + 2 \, b p^{4} \log \left (b x^{2} + a\right ) \log \left (c\right ) + b p^{3} \log \left (c\right )^{2}\right )}} + \frac {p^{2} {\rm Ei}\left (\frac {\log \left (c\right )}{p} + \log \left (b x^{2} + a\right )\right ) \log \left (b x^{2} + a\right )^{2}}{4 \, {\left (b p^{5} \log \left (b x^{2} + a\right )^{2} + 2 \, b p^{4} \log \left (b x^{2} + a\right ) \log \left (c\right ) + b p^{3} \log \left (c\right )^{2}\right )} c^{\left (\frac {1}{p}\right )}} - \frac {{\left (b x^{2} + a\right )} p^{2}}{4 \, {\left (b p^{5} \log \left (b x^{2} + a\right )^{2} + 2 \, b p^{4} \log \left (b x^{2} + a\right ) \log \left (c\right ) + b p^{3} \log \left (c\right )^{2}\right )}} - \frac {{\left (b x^{2} + a\right )} p \log \left (c\right )}{4 \, {\left (b p^{5} \log \left (b x^{2} + a\right )^{2} + 2 \, b p^{4} \log \left (b x^{2} + a\right ) \log \left (c\right ) + b p^{3} \log \left (c\right )^{2}\right )}} + \frac {p {\rm Ei}\left (\frac {\log \left (c\right )}{p} + \log \left (b x^{2} + a\right )\right ) \log \left (b x^{2} + a\right ) \log \left (c\right )}{2 \, {\left (b p^{5} \log \left (b x^{2} + a\right )^{2} + 2 \, b p^{4} \log \left (b x^{2} + a\right ) \log \left (c\right ) + b p^{3} \log \left (c\right )^{2}\right )} c^{\left (\frac {1}{p}\right )}} + \frac {{\rm Ei}\left (\frac {\log \left (c\right )}{p} + \log \left (b x^{2} + a\right )\right ) \log \left (c\right )^{2}}{4 \, {\left (b p^{5} \log \left (b x^{2} + a\right )^{2} + 2 \, b p^{4} \log \left (b x^{2} + a\right ) \log \left (c\right ) + b p^{3} \log \left (c\right )^{2}\right )} c^{\left (\frac {1}{p}\right )}} \]

[In]

integrate(x/log(c*(b*x^2+a)^p)^3,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=\int \frac {x}{{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^3} \,d x \]

[In]

int(x/log(c*(a + b*x^2)^p)^3,x)

[Out]

int(x/log(c*(a + b*x^2)^p)^3, x)