Integrand size = 11, antiderivative size = 51 \[ \int \tanh ^p\left (a+\frac {\log (x)}{2}\right ) \, dx=\frac {2^{-p} e^{-2 a} \left (-1+e^{2 a} x\right )^{1+p} \operatorname {Hypergeometric2F1}\left (p,1+p,2+p,\frac {1}{2} \left (1-e^{2 a} x\right )\right )}{1+p} \] Output:
(-1+exp(2*a)*x)^(p+1)*hypergeom([p, p+1],[2+p],1/2-1/2*exp(2*a)*x)/(2^p)/e xp(2*a)/(p+1)
Time = 0.55 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.49 \[ \int \tanh ^p\left (a+\frac {\log (x)}{2}\right ) \, dx=\frac {2^{-p} e^{-2 a} \left (\frac {-1+e^{2 a} x}{1+e^{2 a} x}\right )^{1+p} \left (1+e^{2 a} x\right )^{1+p} \operatorname {Hypergeometric2F1}\left (p,1+p,2+p,\frac {1}{2}-\frac {1}{2} e^{2 a} x\right )}{1+p} \] Input:
Integrate[Tanh[a + Log[x]/2]^p,x]
Output:
(((-1 + E^(2*a)*x)/(1 + E^(2*a)*x))^(1 + p)*(1 + E^(2*a)*x)^(1 + p)*Hyperg eometric2F1[p, 1 + p, 2 + p, 1/2 - (E^(2*a)*x)/2])/(2^p*E^(2*a)*(1 + p))
Time = 0.24 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6067, 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.
\(\displaystyle \int \tanh ^p\left (a+\frac {\log (x)}{2}\right ) \, dx\) |
\(\Big \downarrow \) 6067 |
\(\displaystyle \int \left (e^{2 a} x-1\right )^p \left (e^{2 a} x+1\right )^{-p}dx\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {e^{-2 a} 2^{-p} \left (e^{2 a} x-1\right )^{p+1} \operatorname {Hypergeometric2F1}\left (p,p+1,p+2,\frac {1}{2} \left (1-e^{2 a} x\right )\right )}{p+1}\) |
Input:
Int[Tanh[a + Log[x]/2]^p,x]
Output:
((-1 + E^(2*a)*x)^(1 + p)*Hypergeometric2F1[p, 1 + p, 2 + p, (1 - E^(2*a)* x)/2])/(2^p*E^(2*a)*(1 + p))
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]))
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]
\[\int \tanh \left (a +\frac {\ln \left (x \right )}{2}\right )^{p}d x\]
Input:
int(tanh(a+1/2*ln(x))^p,x)
Output:
int(tanh(a+1/2*ln(x))^p,x)
\[ \int \tanh ^p\left (a+\frac {\log (x)}{2}\right ) \, dx=\int { \tanh \left (a + \frac {1}{2} \, \log \left (x\right )\right )^{p} \,d x } \] Input:
integrate(tanh(a+1/2*log(x))^p,x, algorithm="fricas")
Output:
integral(tanh(a + 1/2*log(x))^p, x)
\[ \int \tanh ^p\left (a+\frac {\log (x)}{2}\right ) \, dx=\int \tanh ^{p}{\left (a + \frac {\log {\left (x \right )}}{2} \right )}\, dx \] Input:
integrate(tanh(a+1/2*ln(x))**p,x)
Output:
Integral(tanh(a + log(x)/2)**p, x)
\[ \int \tanh ^p\left (a+\frac {\log (x)}{2}\right ) \, dx=\int { \tanh \left (a + \frac {1}{2} \, \log \left (x\right )\right )^{p} \,d x } \] Input:
integrate(tanh(a+1/2*log(x))^p,x, algorithm="maxima")
Output:
integrate(tanh(a + 1/2*log(x))^p, x)
\[ \int \tanh ^p\left (a+\frac {\log (x)}{2}\right ) \, dx=\int { \tanh \left (a + \frac {1}{2} \, \log \left (x\right )\right )^{p} \,d x } \] Input:
integrate(tanh(a+1/2*log(x))^p,x, algorithm="giac")
Output:
integrate(tanh(a + 1/2*log(x))^p, x)
Timed out. \[ \int \tanh ^p\left (a+\frac {\log (x)}{2}\right ) \, dx=\int {\mathrm {tanh}\left (a+\frac {\ln \left (x\right )}{2}\right )}^p \,d x \] Input:
int(tanh(a + log(x)/2)^p,x)
Output:
int(tanh(a + log(x)/2)^p, x)
\[ \int \tanh ^p\left (a+\frac {\log (x)}{2}\right ) \, dx=\tanh \left (\frac {\mathrm {log}\left (x \right )}{2}+a \right )^{p} x -\frac {\left (\int \frac {\tanh \left (\frac {\mathrm {log}\left (x \right )}{2}+a \right )^{p}}{\tanh \left (\frac {\mathrm {log}\left (x \right )}{2}+a \right )}d x \right ) p}{2}+\frac {\left (\int \tanh \left (\frac {\mathrm {log}\left (x \right )}{2}+a \right )^{p} \tanh \left (\frac {\mathrm {log}\left (x \right )}{2}+a \right )d x \right ) p}{2} \] Input:
int(tanh(a+1/2*log(x))^p,x)
Output:
(2*tanh((log(x) + 2*a)/2)**p*x - int(tanh((log(x) + 2*a)/2)**p/tanh((log(x ) + 2*a)/2),x)*p + int(tanh((log(x) + 2*a)/2)**p*tanh((log(x) + 2*a)/2),x) *p)/2