\(\int \frac {1}{\sqrt {d x} (a+b \log (c x^n))^2} \, dx\) [110]

Optimal result

Integrand size = 20, antiderivative size = 98 \[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {e^{-\frac {a}{2 b n}} \sqrt {d x} \left (c x^n\right )^{\left .-\frac {1}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{2 b^2 d n^2}-\frac {\sqrt {d x}}{b d n \left (a+b \log \left (c x^n\right )\right )} \]

[Out]

1/2*Ei(1/2*(a+b*ln(c*x^n))/b/n)*(d*x)^(1/2)/b^2/d/exp(1/2*a/b/n)/n^2/((c*x^n)^(1/2/n))-(d*x)^(1/2)/b/d/n/(a+b*
ln(c*x^n))

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2343, 2347, 2209} \[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {\sqrt {d x} e^{-\frac {a}{2 b n}} \left (c x^n\right )^{\left .-\frac {1}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{2 b^2 d n^2}-\frac {\sqrt {d x}}{b d n \left (a+b \log \left (c x^n\right )\right )} \]

[In]

Int[1/(Sqrt[d*x]*(a + b*Log[c*x^n])^2),x]

[Out]

(Sqrt[d*x]*ExpIntegralEi[(a + b*Log[c*x^n])/(2*b*n)])/(2*b^2*d*E^(a/(2*b*n))*n^2*(c*x^n)^(1/(2*n))) - Sqrt[d*x
]/(b*d*n*(a + b*Log[c*x^n]))

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 2343

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

Rule 2347

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

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {x \left (e^{-\frac {a}{2 b n}} \left (c x^n\right )^{\left .-\frac {1}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c x^n\right )}{2 b n}\right )-\frac {2 b n}{a+b \log \left (c x^n\right )}\right )}{2 b^2 n^2 \sqrt {d x}} \]

[In]

Integrate[1/(Sqrt[d*x]*(a + b*Log[c*x^n])^2),x]

[Out]

(x*(ExpIntegralEi[(a + b*Log[c*x^n])/(2*b*n)]/(E^(a/(2*b*n))*(c*x^n)^(1/(2*n))) - (2*b*n)/(a + b*Log[c*x^n])))
/(2*b^2*n^2*Sqrt[d*x])

Maple [C] (warning: unable to verify)

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

Time = 0.82 (sec) , antiderivative size = 427, normalized size of antiderivative = 4.36

method	result	size
risch	\(-\frac {2 x}{b n \sqrt {d x}\, \left (2 a +2 b \ln \left (c \right )+2 b \ln \left ({\mathrm e}^{n \ln \left (x \right )}\right )-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{3}\right )}-\frac {{\mathrm e}^{\frac {i \left (b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )-b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}-b \pi \,\operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+b \pi \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{3}+2 i b n \left (\ln \left (x \right )-\ln \left (d x \right )\right )+2 i b \ln \left (c \right )+2 i b \left (\ln \left ({\mathrm e}^{n \ln \left (x \right )}\right )-n \ln \left (x \right )\right )+2 i a \right )}{4 b n}} \operatorname {Ei}_{1}\left (-\frac {\ln \left (d x \right )}{2}+\frac {i \left (b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )-b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}-b \pi \,\operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+b \pi \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{3}+2 i b n \left (\ln \left (x \right )-\ln \left (d x \right )\right )+2 i b \ln \left (c \right )+2 i b \left (\ln \left ({\mathrm e}^{n \ln \left (x \right )}\right )-n \ln \left (x \right )\right )+2 i a \right )}{4 b n}\right )}{2 d \,b^{2} n^{2}}\)	\(427\)

[In]

int(1/(d*x)^(1/2)/(a+b*ln(c*x^n))^2,x,method=_RETURNVERBOSE)

[Out]

-2/b/n*x/(d*x)^(1/2)/(2*a+2*b*ln(c)+2*b*ln(exp(n*ln(x)))-I*b*Pi*csgn(I*c)*csgn(I*exp(n*ln(x)))*csgn(I*c*exp(n*
ln(x)))+I*b*Pi*csgn(I*c)*csgn(I*c*exp(n*ln(x)))^2+I*b*Pi*csgn(I*exp(n*ln(x)))*csgn(I*c*exp(n*ln(x)))^2-I*b*Pi*
csgn(I*c*exp(n*ln(x)))^3)-1/2/d/b^2/n^2*exp(1/4*I*(b*Pi*csgn(I*c)*csgn(I*exp(n*ln(x)))*csgn(I*c*exp(n*ln(x)))-
b*Pi*csgn(I*c)*csgn(I*c*exp(n*ln(x)))^2-b*Pi*csgn(I*exp(n*ln(x)))*csgn(I*c*exp(n*ln(x)))^2+b*Pi*csgn(I*c*exp(n
*ln(x)))^3+2*I*b*n*(ln(x)-ln(d*x))+2*I*b*ln(c)+2*I*b*(ln(exp(n*ln(x)))-n*ln(x))+2*I*a)/b/n)*Ei(1,-1/2*ln(d*x)+
1/4*I*(b*Pi*csgn(I*c)*csgn(I*exp(n*ln(x)))*csgn(I*c*exp(n*ln(x)))-b*Pi*csgn(I*c)*csgn(I*c*exp(n*ln(x)))^2-b*Pi
*csgn(I*exp(n*ln(x)))*csgn(I*c*exp(n*ln(x)))^2+b*Pi*csgn(I*c*exp(n*ln(x)))^3+2*I*b*n*(ln(x)-ln(d*x))+2*I*b*ln(
c)+2*I*b*(ln(exp(n*ln(x)))-n*ln(x))+2*I*a)/b/n)

Fricas [F]

\[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {1}{\sqrt {d x} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(d*x)^(1/2)/(a+b*log(c*x^n))^2,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {1}{\sqrt {d x} \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}\, dx \]

[In]

integrate(1/(d*x)**(1/2)/(a+b*ln(c*x**n))**2,x)

[Out]

Integral(1/(sqrt(d*x)*(a + b*log(c*x**n))**2), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {1}{\sqrt {d x} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(d*x)^(1/2)/(a+b*log(c*x^n))^2,x, algorithm="maxima")

[Out]

4*b*n*integrate(1/((b^3*sqrt(d)*log(c)^3 + b^3*sqrt(d)*log(x^n)^3 + 3*a*b^2*sqrt(d)*log(c)^2 + 3*a^2*b*sqrt(d)
*log(c) + a^3*sqrt(d) + 3*(b^3*sqrt(d)*log(c) + a*b^2*sqrt(d))*log(x^n)^2 + 3*(b^3*sqrt(d)*log(c)^2 + 2*a*b^2*
sqrt(d)*log(c) + a^2*b*sqrt(d))*log(x^n))*sqrt(x)), x) + 2*sqrt(x)/(b^2*sqrt(d)*log(c)^2 + b^2*sqrt(d)*log(x^n
)^2 + 2*a*b*sqrt(d)*log(c) + a^2*sqrt(d) + 2*(b^2*sqrt(d)*log(c) + a*b*sqrt(d))*log(x^n))

Giac [F]

\[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {1}{\sqrt {d x} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(d*x)^(1/2)/(a+b*log(c*x^n))^2,x, algorithm="giac")

[Out]

integrate(1/(sqrt(d*x)*(b*log(c*x^n) + a)^2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {1}{\sqrt {d\,x}\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]

[In]

int(1/((d*x)^(1/2)*(a + b*log(c*x^n))^2),x)

[Out]

int(1/((d*x)^(1/2)*(a + b*log(c*x^n))^2), x)