\(\int \frac {\tanh ^2(d (a+b \log (c x^n)))}{x^3} \, dx\) [195]

Optimal result

Integrand size = 19, antiderivative size = 136 \[ \int \frac {\tanh ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {2-b d n}{2 b d n x^2}+\frac {1-e^{2 a d} \left (c x^n\right )^{2 b d}}{b d n x^2 \left (1+e^{2 a d} \left (c x^n\right )^{2 b d}\right )}-\frac {2 \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{b d n},1-\frac {1}{b d n},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n x^2} \] Output:

1/2*(-b*d*n+2)/b/d/n/x^2+(1-exp(2*a*d)*(c*x^n)^(2*b*d))/b/d/n/x^2/(1+exp(2 
*a*d)*(c*x^n)^(2*b*d))-2*hypergeom([1, -1/b/d/n],[1-1/b/d/n],-exp(2*a*d)*( 
c*x^n)^(2*b*d))/b/d/n/x^2
 

Mathematica [A] (verified)

Time = 2.26 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.17 \[ \int \frac {\tanh ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=-\frac {2 e^{2 d \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (1,1-\frac {1}{b d n},2-\frac {1}{b d n},-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+(-1+b d n) \left (b d n+2 \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{b d n},1-\frac {1}{b d n},-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+2 \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )}{2 b d n (-1+b d n) x^2} \] Input:

Integrate[Tanh[d*(a + b*Log[c*x^n])]^2/x^3,x]
 

Output:

-1/2*(2*E^(2*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - 1/(b*d*n), 2 - 
 1/(b*d*n), -E^(2*d*(a + b*Log[c*x^n]))] + (-1 + b*d*n)*(b*d*n + 2*Hyperge 
ometric2F1[1, -(1/(b*d*n)), 1 - 1/(b*d*n), -E^(2*d*(a + b*Log[c*x^n]))] + 
2*Tanh[d*(a + b*Log[c*x^n])]))/(b*d*n*(-1 + b*d*n)*x^2)
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.35, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6073, 6071, 1004, 27, 959, 888}

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.

\(\displaystyle \int \frac {\tanh ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx\)

\(\Big \downarrow \) 6073

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

\(\Big \downarrow \) 6071

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

\(\Big \downarrow \) 1004

\(\displaystyle \frac {\left (c x^n\right )^{2/n} \left (\frac {\left (c x^n\right )^{-2/n} \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )}-\frac {e^{-2 a d} \int -\frac {2 \left (c x^n\right )^{-1-\frac {2}{n}} \left (\frac {e^{2 a d} (b d n+2)}{n}-\frac {e^{4 a d} (2-b d n) \left (c x^n\right )^{2 b d}}{n}\right )}{e^{2 a d} \left (c x^n\right )^{2 b d}+1}d\left (c x^n\right )}{2 b d}\right )}{n x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (c x^n\right )^{2/n} \left (\frac {e^{-2 a d} \int \frac {\left (c x^n\right )^{-1-\frac {2}{n}} \left (\frac {e^{2 a d} (b d n+2)}{n}-\frac {e^{4 a d} (2-b d n) \left (c x^n\right )^{2 b d}}{n}\right )}{e^{2 a d} \left (c x^n\right )^{2 b d}+1}d\left (c x^n\right )}{b d}+\frac {\left (c x^n\right )^{-2/n} \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )}\right )}{n x^2}\)

\(\Big \downarrow \) 959

\(\displaystyle \frac {\left (c x^n\right )^{2/n} \left (\frac {e^{-2 a d} \left (\frac {4 e^{2 a d} \int \frac {\left (c x^n\right )^{-1-\frac {2}{n}}}{e^{2 a d} \left (c x^n\right )^{2 b d}+1}d\left (c x^n\right )}{n}+\frac {1}{2} e^{2 a d} (2-b d n) \left (c x^n\right )^{-2/n}\right )}{b d}+\frac {\left (c x^n\right )^{-2/n} \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )}\right )}{n x^2}\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {\left (c x^n\right )^{2/n} \left (\frac {e^{-2 a d} \left (\frac {1}{2} e^{2 a d} (2-b d n) \left (c x^n\right )^{-2/n}-2 e^{2 a d} \left (c x^n\right )^{-2/n} \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{b d n},1-\frac {1}{b d n},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )\right )}{b d}+\frac {\left (c x^n\right )^{-2/n} \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )}\right )}{n x^2}\)

Input:

Int[Tanh[d*(a + b*Log[c*x^n])]^2/x^3,x]
 

Output:

((c*x^n)^(2/n)*((1 - E^(2*a*d)*(c*x^n)^(2*b*d))/(b*d*(c*x^n)^(2/n)*(1 + E^ 
(2*a*d)*(c*x^n)^(2*b*d))) + ((E^(2*a*d)*(2 - b*d*n))/(2*(c*x^n)^(2/n)) - ( 
2*E^(2*a*d)*Hypergeometric2F1[1, -(1/(b*d*n)), 1 - 1/(b*d*n), -(E^(2*a*d)* 
(c*x^n)^(2*b*d))])/(c*x^n)^(2/n))/(b*d*E^(2*a*d))))/(n*x^2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 959

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p 
+ 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p 
 + 1) + 1))   Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, 
 n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
 

rule 1004

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[(-(c*b - a*d))*(e*x)^(m + 1)*(a + b*x^n)^(p + 1) 
*((c + d*x^n)^(q - 1)/(a*b*e*n*(p + 1))), x] + Simp[1/(a*b*n*(p + 1))   Int 
[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c 
*b - a*d)*(m + 1)) + d*(c*b*n*(p + 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^ 
n, x], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && Lt 
Q[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
 

rule 6071

Int[((e_.)*(x_))^(m_.)*Tanh[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] 
 :> Int[(e*x)^m*((-1 + E^(2*a*d)*x^(2*b*d))^p/(1 + E^(2*a*d)*x^(2*b*d))^p), 
 x] /; FreeQ[{a, b, d, e, m, p}, x]
 

rule 6073

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

Maple [F]

\[\int \frac {{\tanh \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}^{2}}{x^{3}}d x\]

Input:

int(tanh(d*(a+b*ln(c*x^n)))^2/x^3,x)
 

Output:

int(tanh(d*(a+b*ln(c*x^n)))^2/x^3,x)
 

Fricas [F]

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

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

Output:

integral(tanh(b*d*log(c*x^n) + a*d)^2/x^3, x)
 

Sympy [F]

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

integrate(tanh(d*(a+b*ln(c*x**n)))**2/x**3,x)
 

Output:

Integral(tanh(a*d + b*d*log(c*x**n))**2/x**3, x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

int(tanh(d*(a + b*log(c*x^n)))^2/x^3,x)
 

Output:

int(tanh(d*(a + b*log(c*x^n)))^2/x^3, x)
 

Reduce [F]

\[ \int \frac {\tanh ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {-8 e^{2 a d} c^{2 b d} \left (\int \frac {x^{2 b d n}}{x^{4 b d n} e^{4 a d} c^{4 b d} x^{3}+2 x^{2 b d n} e^{2 a d} c^{2 b d} x^{3}+x^{3}}d x \right ) x^{2}-1}{2 x^{2}} \] Input:

int(tanh(d*(a+b*log(c*x^n)))^2/x^3,x)
                                                                                    
                                                                                    
 

Output:

( - 8*e**(2*a*d)*c**(2*b*d)*int(x**(2*b*d*n)/(x**(4*b*d*n)*e**(4*a*d)*c**( 
4*b*d)*x**3 + 2*x**(2*b*d*n)*e**(2*a*d)*c**(2*b*d)*x**3 + x**3),x)*x**2 - 
1)/(2*x**2)