\(\int \tanh ^p(a+\frac {\log (x)}{4}) \, dx\) [176]

Optimal result

Integrand size = 11, antiderivative size = 106 \[ \int \tanh ^p\left (a+\frac {\log (x)}{4}\right ) \, dx=e^{-4 a} \left (-1+e^{2 a} \sqrt {x}\right )^{1+p} \left (1+e^{2 a} \sqrt {x}\right )^{1-p}-\frac {2^{1-p} e^{-4 a} p \left (-1+e^{2 a} \sqrt {x}\right )^{1+p} \operatorname {Hypergeometric2F1}\left (p,1+p,2+p,\frac {1}{2} \left (1-e^{2 a} \sqrt {x}\right )\right )}{1+p} \] Output:

(-1+exp(2*a)*x^(1/2))^(p+1)*(1+exp(2*a)*x^(1/2))^(1-p)/exp(4*a)-2^(1-p)*p* 
(-1+exp(2*a)*x^(1/2))^(p+1)*hypergeom([p, p+1],[2+p],1/2-1/2*exp(2*a)*x^(1 
/2))/exp(4*a)/(p+1)
 

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.14 \[ \int \tanh ^p\left (a+\frac {\log (x)}{4}\right ) \, dx=\frac {e^{-4 a} \left (-1+e^{2 a} \sqrt {x}\right ) \left (\frac {-1+e^{2 a} \sqrt {x}}{2+2 e^{2 a} \sqrt {x}}\right )^p \left (2^p (1+p) \left (1+e^{2 a} \sqrt {x}\right )-2 p \left (1+e^{2 a} \sqrt {x}\right )^p \operatorname {Hypergeometric2F1}\left (p,1+p,2+p,\frac {1}{2}-\frac {1}{2} e^{2 a} \sqrt {x}\right )\right )}{1+p} \] Input:

Integrate[Tanh[a + Log[x]/4]^p,x]
 

Output:

((-1 + E^(2*a)*Sqrt[x])*((-1 + E^(2*a)*Sqrt[x])/(2 + 2*E^(2*a)*Sqrt[x]))^p 
*(2^p*(1 + p)*(1 + E^(2*a)*Sqrt[x]) - 2*p*(1 + E^(2*a)*Sqrt[x])^p*Hypergeo 
metric2F1[p, 1 + p, 2 + p, 1/2 - (E^(2*a)*Sqrt[x])/2]))/(E^(4*a)*(1 + p))
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6067, 900, 90, 79}

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.

\(\displaystyle \int \tanh ^p\left (a+\frac {\log (x)}{4}\right ) \, dx\)

\(\Big \downarrow \) 6067

\(\displaystyle \int \left (e^{2 a} \sqrt {x}-1\right )^p \left (e^{2 a} \sqrt {x}+1\right )^{-p}dx\)

\(\Big \downarrow \) 900

\(\displaystyle 2 \int \left (e^{2 a} \sqrt {x}-1\right )^p \left (e^{2 a} \sqrt {x}+1\right )^{-p} \sqrt {x}d\sqrt {x}\)

\(\Big \downarrow \) 90

\(\displaystyle 2 \left (\frac {1}{2} e^{-4 a} \left (e^{2 a} \sqrt {x}-1\right )^{p+1} \left (e^{2 a} \sqrt {x}+1\right )^{1-p}-e^{-2 a} p \int \left (e^{2 a} \sqrt {x}-1\right )^p \left (e^{2 a} \sqrt {x}+1\right )^{-p}d\sqrt {x}\right )\)

\(\Big \downarrow \) 79

\(\displaystyle 2 \left (\frac {1}{2} e^{-4 a} \left (e^{2 a} \sqrt {x}-1\right )^{p+1} \left (e^{2 a} \sqrt {x}+1\right )^{1-p}-\frac {e^{-4 a} 2^{-p} p \left (e^{2 a} \sqrt {x}-1\right )^{p+1} \operatorname {Hypergeometric2F1}\left (p,p+1,p+2,\frac {1}{2} \left (1-e^{2 a} \sqrt {x}\right )\right )}{p+1}\right )\)

Input:

Int[Tanh[a + Log[x]/4]^p,x]
 

Output:

2*(((-1 + E^(2*a)*Sqrt[x])^(1 + p)*(1 + E^(2*a)*Sqrt[x])^(1 - p))/(2*E^(4* 
a)) - (p*(-1 + E^(2*a)*Sqrt[x])^(1 + p)*Hypergeometric2F1[p, 1 + p, 2 + p, 
 (1 - E^(2*a)*Sqrt[x])/2])/(2^p*E^(4*a)*(1 + p)))
 

Defintions of rubi rules used

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 900

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol 
] :> With[{g = Denominator[n]}, Simp[g   Subst[Int[x^(g - 1)*(a + b*x^(g*n) 
)^p*(c + d*x^(g*n))^q, x], x, x^(1/g)], x]] /; FreeQ[{a, b, c, d, p, q}, x] 
 && NeQ[b*c - a*d, 0] && FractionQ[n]
 

rule 6067

Int[Tanh[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Int[(-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, p}, x]
 

Maple [F]

\[\int \tanh \left (a +\frac {\ln \left (x \right )}{4}\right )^{p}d x\]

Input:

int(tanh(a+1/4*ln(x))^p,x)
 

Output:

int(tanh(a+1/4*ln(x))^p,x)
 

Fricas [F]

\[ \int \tanh ^p\left (a+\frac {\log (x)}{4}\right ) \, dx=\int { \tanh \left (a + \frac {1}{4} \, \log \left (x\right )\right )^{p} \,d x } \] Input:

integrate(tanh(a+1/4*log(x))^p,x, algorithm="fricas")
 

Output:

integral(tanh(a + 1/4*log(x))^p, x)
 

Sympy [F]

\[ \int \tanh ^p\left (a+\frac {\log (x)}{4}\right ) \, dx=\int \tanh ^{p}{\left (a + \frac {\log {\left (x \right )}}{4} \right )}\, dx \] Input:

integrate(tanh(a+1/4*ln(x))**p,x)
 

Output:

Integral(tanh(a + log(x)/4)**p, x)
 

Maxima [F]

\[ \int \tanh ^p\left (a+\frac {\log (x)}{4}\right ) \, dx=\int { \tanh \left (a + \frac {1}{4} \, \log \left (x\right )\right )^{p} \,d x } \] Input:

integrate(tanh(a+1/4*log(x))^p,x, algorithm="maxima")
 

Output:

integrate(tanh(a + 1/4*log(x))^p, x)
 

Giac [F]

\[ \int \tanh ^p\left (a+\frac {\log (x)}{4}\right ) \, dx=\int { \tanh \left (a + \frac {1}{4} \, \log \left (x\right )\right )^{p} \,d x } \] Input:

integrate(tanh(a+1/4*log(x))^p,x, algorithm="giac")
 

Output:

integrate(tanh(a + 1/4*log(x))^p, x)
 

Mupad [F(-1)]

Timed out. \[ \int \tanh ^p\left (a+\frac {\log (x)}{4}\right ) \, dx=\int {\mathrm {tanh}\left (a+\frac {\ln \left (x\right )}{4}\right )}^p \,d x \] Input:

int(tanh(a + log(x)/4)^p,x)
 

Output:

int(tanh(a + log(x)/4)^p, x)
 

Reduce [F]

\[ \int \tanh ^p\left (a+\frac {\log (x)}{4}\right ) \, dx=\tanh \left (\frac {\mathrm {log}\left (x \right )}{4}+a \right )^{p} x -\frac {\left (\int \frac {\tanh \left (\frac {\mathrm {log}\left (x \right )}{4}+a \right )^{p}}{\tanh \left (\frac {\mathrm {log}\left (x \right )}{4}+a \right )}d x \right ) p}{4}+\frac {\left (\int \tanh \left (\frac {\mathrm {log}\left (x \right )}{4}+a \right )^{p} \tanh \left (\frac {\mathrm {log}\left (x \right )}{4}+a \right )d x \right ) p}{4} \] Input:

int(tanh(a+1/4*log(x))^p,x)
 

Output:

(4*tanh((log(x) + 4*a)/4)**p*x - int(tanh((log(x) + 4*a)/4)**p/tanh((log(x 
) + 4*a)/4),x)*p + int(tanh((log(x) + 4*a)/4)**p*tanh((log(x) + 4*a)/4),x) 
*p)/4