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

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {x \left (e^{-\frac {a}{2 b n}} \left (c x^n\right )^{\left .-\frac {1}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c x^n\right )}{2 b n}\right )-\frac {2 b n}{a+b \log \left (c x^n\right )}\right )}{2 b^2 n^2 \sqrt {d x}} \] Input:

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

Output:

(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])
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2743, 2747, 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.

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

\(\Big \downarrow \) 2743

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

\(\Big \downarrow \) 2747

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

\(\Big \downarrow \) 2609

\(\displaystyle \frac {\sqrt {d x} e^{-\frac {a}{2 b n}} \left (c x^n\right )^{\left .-\frac {1}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{2 b^2 d n^2}-\frac {\sqrt {d x}}{b d n \left (a+b \log \left (c x^n\right )\right )}\)

Input:

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

Output:

(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]))
 

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 2743

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] - 
Simp[(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 2747

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

Maple [C] (warning: unable to verify)

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

Time = 2.59 (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 \pi b \,\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 \pi b \,\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 ) \operatorname {csgn}\left (i c \right )-i \pi b \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{3}+i \pi b \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2} \operatorname {csgn}\left (i c \right )\right )}-\frac {{\mathrm e}^{\frac {i \left (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 ) \operatorname {csgn}\left (i c \right )-b \pi \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2} \operatorname {csgn}\left (i c \right )-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 \left (\ln \left (x \right )-\ln \left (d x \right )\right ) b n +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 n b}} \operatorname {expIntegral}_{1}\left (-\frac {\ln \left (d x \right )}{2}+\frac {i \left (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 ) \operatorname {csgn}\left (i c \right )-b \pi \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2} \operatorname {csgn}\left (i c \right )-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 \left (\ln \left (x \right )-\ln \left (d x \right )\right ) b n +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 n b}\right )}{2 d \,b^{2} n^{2}}\)	\(427\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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]

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

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

Output:

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

Maxima [F]

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

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

Output:

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*sq 
rt(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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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