\(\int \frac {\tanh ^5(a+b \log (c x^n))}{x} \, dx\) [198]

Optimal result

Integrand size = 17, antiderivative size = 66 \[ \int \frac {\tanh ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\log \left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}-\frac {\tanh ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n} \] Output:

ln(cosh(a+b*ln(c*x^n)))/b/n-1/2*tanh(a+b*ln(c*x^n))^2/b/n-1/4*tanh(a+b*ln( 
c*x^n))^4/b/n
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.83 \[ \int \frac {\tanh ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {4 \log \left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )+4 \text {sech}^2\left (a+b \log \left (c x^n\right )\right )-\text {sech}^4\left (a+b \log \left (c x^n\right )\right )}{4 b n} \] Input:

Integrate[Tanh[a + b*Log[c*x^n]]^5/x,x]
 

Output:

(4*Log[Cosh[a + b*Log[c*x^n]]] + 4*Sech[a + b*Log[c*x^n]]^2 - Sech[a + b*L 
og[c*x^n]]^4)/(4*b*n)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.42 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.08, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {3039, 3042, 26, 3954, 26, 3042, 26, 3954, 26, 3042, 26, 3956}

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.

\(\displaystyle \int \frac {\tanh ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx\)

\(\Big \downarrow \) 3039

\(\displaystyle \frac {\int \tanh ^5\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int -i \tan \left (i a+i b \log \left (c x^n\right )\right )^5d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i \int \tan \left (i a+i b \log \left (c x^n\right )\right )^5d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 3954

\(\displaystyle -\frac {i \left (-\int -i \tanh ^3\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )-\frac {i \tanh ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}\right )}{n}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i \left (i \int \tanh ^3\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )-\frac {i \tanh ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}\right )}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {i \left (i \int i \tan \left (i a+i b \log \left (c x^n\right )\right )^3d\log \left (c x^n\right )-\frac {i \tanh ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}\right )}{n}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i \left (-\int \tan \left (i a+i b \log \left (c x^n\right )\right )^3d\log \left (c x^n\right )-\frac {i \tanh ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}\right )}{n}\)

\(\Big \downarrow \) 3954

\(\displaystyle -\frac {i \left (\int i \tanh \left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )-\frac {i \tanh ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}-\frac {i \tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b}\right )}{n}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i \left (i \int \tanh \left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )-\frac {i \tanh ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}-\frac {i \tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b}\right )}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {i \left (i \int -i \tan \left (i a+i b \log \left (c x^n\right )\right )d\log \left (c x^n\right )-\frac {i \tanh ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}-\frac {i \tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b}\right )}{n}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i \left (\int \tan \left (i a+i b \log \left (c x^n\right )\right )d\log \left (c x^n\right )-\frac {i \tanh ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}-\frac {i \tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b}\right )}{n}\)

\(\Big \downarrow \) 3956

\(\displaystyle -\frac {i \left (-\frac {i \tanh ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}-\frac {i \tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b}+\frac {i \log \left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b}\right )}{n}\)

Input:

Int[Tanh[a + b*Log[c*x^n]]^5/x,x]
 

Output:

((-I)*((I*Log[Cosh[a + b*Log[c*x^n]]])/b - ((I/2)*Tanh[a + b*Log[c*x^n]]^2 
)/b - ((I/4)*Tanh[a + b*Log[c*x^n]]^4)/b))/n
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 3039

Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst 
[[3]]   Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /;  !FalseQ[lst]] /; 
NonsumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3954

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d 
*x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2   Int[(b*Tan[c + d*x])^(n - 2), x] 
, x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
 

rule 3956

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d 
*x], x]]/d, x] /; FreeQ[{c, d}, x]
 

Maple [A] (verified)

Time = 1.54 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.94

method	result	size
parallelrisch	\(-\frac {{\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{4}+4 \ln \left (x \right ) b n +2 {\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}+4 \ln \left (1-\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{4 n b}\)	\(62\)
derivativedivides	\(\frac {-\frac {{\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{4}}{4}-\frac {{\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{2}-\frac {\ln \left (\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )-1\right )}{2}-\frac {\ln \left (\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )+1\right )}{2}}{n b}\)	\(71\)
default	\(\frac {-\frac {{\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{4}}{4}-\frac {{\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{2}-\frac {\ln \left (\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )-1\right )}{2}-\frac {\ln \left (\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )+1\right )}{2}}{n b}\)	\(71\)
risch	\(\ln \left (x \right )-\frac {2 a}{n b}-\frac {2 \ln \left (c \right )}{n}-\frac {2 \ln \left (x^{n}\right )}{n}+\frac {i \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{n}-\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{n}-\frac {i \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{n}+\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{n}+\frac {4 \left (x^{n}\right )^{2 b} c^{2 b} \left ({\mathrm e}^{3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{-3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{-3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{6 a} \left (x^{n}\right )^{4 b} c^{4 b}+{\mathrm e}^{2 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{-2 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{-2 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{2 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{4 a} \left (x^{n}\right )^{2 b} c^{2 b}+{\mathrm e}^{i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 a}\right )}{b n {\left (\left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{2 a} {\mathrm e}^{i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}+1\right )}^{4}}+\frac {\ln \left (\left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{2 a} {\mathrm e}^{i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}+1\right )}{b n}\)	\(657\)

Input:

int(tanh(a+b*ln(c*x^n))^5/x,x,method=_RETURNVERBOSE)
 

Output:

-1/4*(tanh(a+b*ln(c*x^n))^4+4*ln(x)*b*n+2*tanh(a+b*ln(c*x^n))^2+4*ln(1-tan 
h(a+b*ln(c*x^n))))/n/b
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1568 vs. \(2 (62) = 124\).

Time = 0.11 (sec) , antiderivative size = 1568, normalized size of antiderivative = 23.76 \[ \int \frac {\tanh ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Too large to display} \] Input:

integrate(tanh(a+b*log(c*x^n))^5/x,x, algorithm="fricas")
 

Output:

-(b*n*cosh(b*n*log(x) + b*log(c) + a)^8*log(x) + 8*b*n*cosh(b*n*log(x) + b 
*log(c) + a)*log(x)*sinh(b*n*log(x) + b*log(c) + a)^7 + b*n*log(x)*sinh(b* 
n*log(x) + b*log(c) + a)^8 + 4*(b*n*log(x) - 1)*cosh(b*n*log(x) + b*log(c) 
 + a)^6 + 4*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^2*log(x) + b*n*log(x) - 
 1)*sinh(b*n*log(x) + b*log(c) + a)^6 + 8*(7*b*n*cosh(b*n*log(x) + b*log(c 
) + a)^3*log(x) + 3*(b*n*log(x) - 1)*cosh(b*n*log(x) + b*log(c) + a))*sinh 
(b*n*log(x) + b*log(c) + a)^5 + 2*(3*b*n*log(x) - 2)*cosh(b*n*log(x) + b*l 
og(c) + a)^4 + 2*(35*b*n*cosh(b*n*log(x) + b*log(c) + a)^4*log(x) + 30*(b* 
n*log(x) - 1)*cosh(b*n*log(x) + b*log(c) + a)^2 + 3*b*n*log(x) - 2)*sinh(b 
*n*log(x) + b*log(c) + a)^4 + 8*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^5*l 
og(x) + 10*(b*n*log(x) - 1)*cosh(b*n*log(x) + b*log(c) + a)^3 + (3*b*n*log 
(x) - 2)*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^ 
3 + 4*(b*n*log(x) - 1)*cosh(b*n*log(x) + b*log(c) + a)^2 + b*n*log(x) + 4* 
(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^6*log(x) + 15*(b*n*log(x) - 1)*cosh 
(b*n*log(x) + b*log(c) + a)^4 + 3*(3*b*n*log(x) - 2)*cosh(b*n*log(x) + b*l 
og(c) + a)^2 + b*n*log(x) - 1)*sinh(b*n*log(x) + b*log(c) + a)^2 - (cosh(b 
*n*log(x) + b*log(c) + a)^8 + 8*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*l 
og(x) + b*log(c) + a)^7 + sinh(b*n*log(x) + b*log(c) + a)^8 + 4*(7*cosh(b* 
n*log(x) + b*log(c) + a)^2 + 1)*sinh(b*n*log(x) + b*log(c) + a)^6 + 4*cosh 
(b*n*log(x) + b*log(c) + a)^6 + 8*(7*cosh(b*n*log(x) + b*log(c) + a)^3 ...
 

Sympy [A] (verification not implemented)

Time = 3.89 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.32 \[ \int \frac {\tanh ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\begin {cases} \log {\left (x \right )} \tanh ^{5}{\left (a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\left (x \right )} \tanh ^{5}{\left (a + b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\frac {\log {\left (c x^{n} \right )}}{n} - \frac {\log {\left (\tanh {\left (a + b \log {\left (c x^{n} \right )} \right )} + 1 \right )}}{b n} - \frac {\tanh ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b n} - \frac {\tanh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{2 b n} & \text {otherwise} \end {cases} \] Input:

integrate(tanh(a+b*ln(c*x**n))**5/x,x)
 

Output:

Piecewise((log(x)*tanh(a)**5, Eq(b, 0) & (Eq(b, 0) | Eq(n, 0))), (log(x)*t 
anh(a + b*log(c))**5, Eq(n, 0)), (log(c*x**n)/n - log(tanh(a + b*log(c*x** 
n)) + 1)/(b*n) - tanh(a + b*log(c*x**n))**4/(4*b*n) - tanh(a + b*log(c*x** 
n))**2/(2*b*n), True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 829 vs. \(2 (62) = 124\).

Time = 0.17 (sec) , antiderivative size = 829, normalized size of antiderivative = 12.56 \[ \int \frac {\tanh ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Too large to display} \] Input:

integrate(tanh(a+b*log(c*x^n))^5/x,x, algorithm="maxima")
 

Output:

1/24*(48*c^(6*b)*e^(6*b*log(x^n) + 6*a) + 108*c^(4*b)*e^(4*b*log(x^n) + 4* 
a) + 88*c^(2*b)*e^(2*b*log(x^n) + 2*a) + 25)/(b*c^(8*b)*n*e^(8*b*log(x^n) 
+ 8*a) + 4*b*c^(6*b)*n*e^(6*b*log(x^n) + 6*a) + 6*b*c^(4*b)*n*e^(4*b*log(x 
^n) + 4*a) + 4*b*c^(2*b)*n*e^(2*b*log(x^n) + 2*a) + b*n) - 1/24*(12*c^(6*b 
)*e^(6*b*log(x^n) + 6*a) + 42*c^(4*b)*e^(4*b*log(x^n) + 4*a) + 52*c^(2*b)* 
e^(2*b*log(x^n) + 2*a) + 25)/(b*c^(8*b)*n*e^(8*b*log(x^n) + 8*a) + 4*b*c^( 
6*b)*n*e^(6*b*log(x^n) + 6*a) + 6*b*c^(4*b)*n*e^(4*b*log(x^n) + 4*a) + 4*b 
*c^(2*b)*n*e^(2*b*log(x^n) + 2*a) + b*n) + 5/8*(4*c^(6*b)*e^(6*b*log(x^n) 
+ 6*a) + 6*c^(4*b)*e^(4*b*log(x^n) + 4*a) + 4*c^(2*b)*e^(2*b*log(x^n) + 2* 
a) + 1)/(b*c^(8*b)*n*e^(8*b*log(x^n) + 8*a) + 4*b*c^(6*b)*n*e^(6*b*log(x^n 
) + 6*a) + 6*b*c^(4*b)*n*e^(4*b*log(x^n) + 4*a) + 4*b*c^(2*b)*n*e^(2*b*log 
(x^n) + 2*a) + b*n) - 5/12*(6*c^(4*b)*e^(4*b*log(x^n) + 4*a) + 4*c^(2*b)*e 
^(2*b*log(x^n) + 2*a) + 1)/(b*c^(8*b)*n*e^(8*b*log(x^n) + 8*a) + 4*b*c^(6* 
b)*n*e^(6*b*log(x^n) + 6*a) + 6*b*c^(4*b)*n*e^(4*b*log(x^n) + 4*a) + 4*b*c 
^(2*b)*n*e^(2*b*log(x^n) + 2*a) + b*n) + 5/12*(4*c^(2*b)*e^(2*b*log(x^n) + 
 2*a) + 1)/(b*c^(8*b)*n*e^(8*b*log(x^n) + 8*a) + 4*b*c^(6*b)*n*e^(6*b*log( 
x^n) + 6*a) + 6*b*c^(4*b)*n*e^(4*b*log(x^n) + 4*a) + 4*b*c^(2*b)*n*e^(2*b* 
log(x^n) + 2*a) + b*n) - 5/8/(b*c^(8*b)*n*e^(8*b*log(x^n) + 8*a) + 4*b*c^( 
6*b)*n*e^(6*b*log(x^n) + 6*a) + 6*b*c^(4*b)*n*e^(4*b*log(x^n) + 4*a) + 4*b 
*c^(2*b)*n*e^(2*b*log(x^n) + 2*a) + b*n) + log((c^(2*b)*e^(2*b*log(x^n)...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (62) = 124\).

Time = 0.22 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.59 \[ \int \frac {\tanh ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\log \left (x^{b n}\right )}{b n} + \frac {\log \left (\sqrt {2 \, x^{2 \, b n} {\left | c \right |}^{2 \, b} \cos \left (\pi b \mathrm {sgn}\left (c\right ) - \pi b\right ) e^{\left (2 \, a\right )} + x^{4 \, b n} {\left | c \right |}^{4 \, b} e^{\left (4 \, a\right )} + 1}\right )}{b n} - \frac {25 \, c^{8 \, b} x^{8 \, b n} e^{\left (8 \, a\right )} + 52 \, c^{6 \, b} x^{6 \, b n} e^{\left (6 \, a\right )} + 102 \, c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} + 52 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 25}{12 \, {\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1\right )}^{4} b n} \] Input:

integrate(tanh(a+b*log(c*x^n))^5/x,x, algorithm="giac")
 

Output:

-log(x^(b*n))/(b*n) + log(sqrt(2*x^(2*b*n)*abs(c)^(2*b)*cos(pi*b*sgn(c) - 
pi*b)*e^(2*a) + x^(4*b*n)*abs(c)^(4*b)*e^(4*a) + 1))/(b*n) - 1/12*(25*c^(8 
*b)*x^(8*b*n)*e^(8*a) + 52*c^(6*b)*x^(6*b*n)*e^(6*a) + 102*c^(4*b)*x^(4*b* 
n)*e^(4*a) + 52*c^(2*b)*x^(2*b*n)*e^(2*a) + 25)/((c^(2*b)*x^(2*b*n)*e^(2*a 
) + 1)^4*b*n)
 

Mupad [B] (verification not implemented)

Time = 2.22 (sec) , antiderivative size = 227, normalized size of antiderivative = 3.44 \[ \int \frac {\tanh ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {8}{b\,n+3\,b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+3\,b\,n\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}+b\,n\,{\mathrm {e}}^{6\,a}\,{\left (c\,x^n\right )}^{6\,b}}-\ln \left (x\right )+\frac {4}{b\,n+b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}}-\frac {4}{b\,n+4\,b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+6\,b\,n\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}+4\,b\,n\,{\mathrm {e}}^{6\,a}\,{\left (c\,x^n\right )}^{6\,b}+b\,n\,{\mathrm {e}}^{8\,a}\,{\left (c\,x^n\right )}^{8\,b}}-\frac {8}{b\,n+2\,b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+b\,n\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}}+\frac {\ln \left ({\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+1\right )}{b\,n} \] Input:

int(tanh(a + b*log(c*x^n))^5/x,x)
 

Output:

8/(b*n + 3*b*n*exp(2*a)*(c*x^n)^(2*b) + 3*b*n*exp(4*a)*(c*x^n)^(4*b) + b*n 
*exp(6*a)*(c*x^n)^(6*b)) - log(x) + 4/(b*n + b*n*exp(2*a)*(c*x^n)^(2*b)) - 
 4/(b*n + 4*b*n*exp(2*a)*(c*x^n)^(2*b) + 6*b*n*exp(4*a)*(c*x^n)^(4*b) + 4* 
b*n*exp(6*a)*(c*x^n)^(6*b) + b*n*exp(8*a)*(c*x^n)^(8*b)) - 8/(b*n + 2*b*n* 
exp(2*a)*(c*x^n)^(2*b) + b*n*exp(4*a)*(c*x^n)^(4*b)) + log(exp(2*a)*(c*x^n 
)^(2*b) + 1)/(b*n)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 385, normalized size of antiderivative = 5.83 \[ \int \frac {\tanh ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {x^{8 b n} e^{8 a} c^{8 b} \mathrm {log}\left (x^{2 b n} e^{2 a} c^{2 b}+1\right )-x^{8 b n} e^{8 a} c^{8 b} \mathrm {log}\left (x \right ) b n -x^{8 b n} e^{8 a} c^{8 b}+4 x^{6 b n} e^{6 a} c^{6 b} \mathrm {log}\left (x^{2 b n} e^{2 a} c^{2 b}+1\right )-4 x^{6 b n} e^{6 a} c^{6 b} \mathrm {log}\left (x \right ) b n +6 x^{4 b n} e^{4 a} c^{4 b} \mathrm {log}\left (x^{2 b n} e^{2 a} c^{2 b}+1\right )-6 x^{4 b n} e^{4 a} c^{4 b} \mathrm {log}\left (x \right ) b n -2 x^{4 b n} e^{4 a} c^{4 b}+4 x^{2 b n} e^{2 a} c^{2 b} \mathrm {log}\left (x^{2 b n} e^{2 a} c^{2 b}+1\right )-4 x^{2 b n} e^{2 a} c^{2 b} \mathrm {log}\left (x \right ) b n +\mathrm {log}\left (x^{2 b n} e^{2 a} c^{2 b}+1\right )-\mathrm {log}\left (x \right ) b n -1}{b n \left (x^{8 b n} e^{8 a} c^{8 b}+4 x^{6 b n} e^{6 a} c^{6 b}+6 x^{4 b n} e^{4 a} c^{4 b}+4 x^{2 b n} e^{2 a} c^{2 b}+1\right )} \] Input:

int(tanh(a+b*log(c*x^n))^5/x,x)
 

Output:

(x**(8*b*n)*e**(8*a)*c**(8*b)*log(x**(2*b*n)*e**(2*a)*c**(2*b) + 1) - x**( 
8*b*n)*e**(8*a)*c**(8*b)*log(x)*b*n - x**(8*b*n)*e**(8*a)*c**(8*b) + 4*x** 
(6*b*n)*e**(6*a)*c**(6*b)*log(x**(2*b*n)*e**(2*a)*c**(2*b) + 1) - 4*x**(6* 
b*n)*e**(6*a)*c**(6*b)*log(x)*b*n + 6*x**(4*b*n)*e**(4*a)*c**(4*b)*log(x** 
(2*b*n)*e**(2*a)*c**(2*b) + 1) - 6*x**(4*b*n)*e**(4*a)*c**(4*b)*log(x)*b*n 
 - 2*x**(4*b*n)*e**(4*a)*c**(4*b) + 4*x**(2*b*n)*e**(2*a)*c**(2*b)*log(x** 
(2*b*n)*e**(2*a)*c**(2*b) + 1) - 4*x**(2*b*n)*e**(2*a)*c**(2*b)*log(x)*b*n 
 + log(x**(2*b*n)*e**(2*a)*c**(2*b) + 1) - log(x)*b*n - 1)/(b*n*(x**(8*b*n 
)*e**(8*a)*c**(8*b) + 4*x**(6*b*n)*e**(6*a)*c**(6*b) + 6*x**(4*b*n)*e**(4* 
a)*c**(4*b) + 4*x**(2*b*n)*e**(2*a)*c**(2*b) + 1))