Integrand size = 11, antiderivative size = 158 \[ \int \tanh ^p\left (a+\frac {\log (x)}{6}\right ) \, dx=-e^{-6 a} p \left (-1+e^{2 a} \sqrt [3]{x}\right )^{1+p} \left (1+e^{2 a} \sqrt [3]{x}\right )^{1-p}+e^{-4 a} \left (-1+e^{2 a} \sqrt [3]{x}\right )^{1+p} \left (1+e^{2 a} \sqrt [3]{x}\right )^{1-p} \sqrt [3]{x}+\frac {2^{-p} e^{-6 a} \left (1+2 p^2\right ) \left (-1+e^{2 a} \sqrt [3]{x}\right )^{1+p} \operatorname {Hypergeometric2F1}\left (p,1+p,2+p,\frac {1}{2} \left (1-e^{2 a} \sqrt [3]{x}\right )\right )}{1+p} \]
-p*(-1+exp(2*a)*x^(1/3))^(p+1)*(1+exp(2*a)*x^(1/3))^(1-p)/exp(6*a)+(-1+exp (2*a)*x^(1/3))^(p+1)*(1+exp(2*a)*x^(1/3))^(1-p)*x^(1/3)/exp(4*a)+(2*p^2+1) *(-1+exp(2*a)*x^(1/3))^(p+1)*hypergeom([p, p+1],[2+p],1/2-1/2*exp(2*a)*x^( 1/3))/(2^p)/exp(6*a)/(p+1)
Time = 1.12 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.89 \[ \int \tanh ^p\left (a+\frac {\log (x)}{6}\right ) \, dx=\frac {e^{-6 a} \left (-1+e^{2 a} \sqrt [3]{x}\right ) \left (\frac {-1+e^{2 a} \sqrt [3]{x}}{2+2 e^{2 a} \sqrt [3]{x}}\right )^p \left (2^p (1+p) \left (1+e^{2 a} \sqrt [3]{x}\right ) \left (-p+e^{2 a} \sqrt [3]{x}\right )+\left (1+2 p^2\right ) \left (1+e^{2 a} \sqrt [3]{x}\right )^p \operatorname {Hypergeometric2F1}\left (p,1+p,2+p,\frac {1}{2}-\frac {1}{2} e^{2 a} \sqrt [3]{x}\right )\right )}{1+p} \]
((-1 + E^(2*a)*x^(1/3))*((-1 + E^(2*a)*x^(1/3))/(2 + 2*E^(2*a)*x^(1/3)))^p *(2^p*(1 + p)*(1 + E^(2*a)*x^(1/3))*(-p + E^(2*a)*x^(1/3)) + (1 + 2*p^2)*( 1 + E^(2*a)*x^(1/3))^p*Hypergeometric2F1[p, 1 + p, 2 + p, 1/2 - (E^(2*a)*x ^(1/3))/2]))/(E^(6*a)*(1 + p))
Time = 0.30 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {6067, 900, 101, 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.
\(\displaystyle \int \tanh ^p\left (a+\frac {\log (x)}{6}\right ) \, dx\) |
\(\Big \downarrow \) 6067 |
\(\displaystyle \int \left (e^{2 a} \sqrt [3]{x}-1\right )^p \left (e^{2 a} \sqrt [3]{x}+1\right )^{-p}dx\) |
\(\Big \downarrow \) 900 |
\(\displaystyle 3 \int \left (e^{2 a} \sqrt [3]{x}-1\right )^p \left (e^{2 a} \sqrt [3]{x}+1\right )^{-p} x^{2/3}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 101 |
\(\displaystyle 3 \left (\frac {1}{3} e^{-4 a} \int \left (e^{2 a} \sqrt [3]{x}-1\right )^p \left (e^{2 a} \sqrt [3]{x}+1\right )^{-p} \left (1-2 e^{2 a} p \sqrt [3]{x}\right )d\sqrt [3]{x}+\frac {1}{3} e^{-4 a} \sqrt [3]{x} \left (e^{2 a} \sqrt [3]{x}-1\right )^{p+1} \left (e^{2 a} \sqrt [3]{x}+1\right )^{1-p}\right )\) |
\(\Big \downarrow \) 90 |
\(\displaystyle 3 \left (\frac {1}{3} e^{-4 a} \left (\left (2 p^2+1\right ) \int \left (e^{2 a} \sqrt [3]{x}-1\right )^p \left (e^{2 a} \sqrt [3]{x}+1\right )^{-p}d\sqrt [3]{x}-e^{-2 a} p \left (e^{2 a} \sqrt [3]{x}-1\right )^{p+1} \left (e^{2 a} \sqrt [3]{x}+1\right )^{1-p}\right )+\frac {1}{3} e^{-4 a} \sqrt [3]{x} \left (e^{2 a} \sqrt [3]{x}-1\right )^{p+1} \left (e^{2 a} \sqrt [3]{x}+1\right )^{1-p}\right )\) |
\(\Big \downarrow \) 79 |
\(\displaystyle 3 \left (\frac {1}{3} e^{-4 a} \left (\frac {e^{-2 a} 2^{-p} \left (2 p^2+1\right ) \left (e^{2 a} \sqrt [3]{x}-1\right )^{p+1} \operatorname {Hypergeometric2F1}\left (p,p+1,p+2,\frac {1}{2} \left (1-e^{2 a} \sqrt [3]{x}\right )\right )}{p+1}-e^{-2 a} p \left (e^{2 a} \sqrt [3]{x}-1\right )^{p+1} \left (e^{2 a} \sqrt [3]{x}+1\right )^{1-p}\right )+\frac {1}{3} e^{-4 a} \sqrt [3]{x} \left (e^{2 a} \sqrt [3]{x}-1\right )^{p+1} \left (e^{2 a} \sqrt [3]{x}+1\right )^{1-p}\right )\) |
3*(((-1 + E^(2*a)*x^(1/3))^(1 + p)*(1 + E^(2*a)*x^(1/3))^(1 - p)*x^(1/3))/ (3*E^(4*a)) + (-((p*(-1 + E^(2*a)*x^(1/3))^(1 + p)*(1 + E^(2*a)*x^(1/3))^( 1 - p))/E^(2*a)) + ((1 + 2*p^2)*(-1 + E^(2*a)*x^(1/3))^(1 + p)*Hypergeomet ric2F1[p, 1 + p, 2 + p, (1 - E^(2*a)*x^(1/3))/2])/(2^p*E^(2*a)*(1 + p)))/( 3*E^(4*a)))
3.2.67.3.1 Defintions of rubi rules used
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[((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]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
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]
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]
\[\int \tanh \left (a +\frac {\ln \left (x \right )}{6}\right )^{p}d x\]
\[ \int \tanh ^p\left (a+\frac {\log (x)}{6}\right ) \, dx=\int { \tanh \left (a + \frac {1}{6} \, \log \left (x\right )\right )^{p} \,d x } \]
\[ \int \tanh ^p\left (a+\frac {\log (x)}{6}\right ) \, dx=\int \tanh ^{p}{\left (a + \frac {\log {\left (x \right )}}{6} \right )}\, dx \]
\[ \int \tanh ^p\left (a+\frac {\log (x)}{6}\right ) \, dx=\int { \tanh \left (a + \frac {1}{6} \, \log \left (x\right )\right )^{p} \,d x } \]
\[ \int \tanh ^p\left (a+\frac {\log (x)}{6}\right ) \, dx=\int { \tanh \left (a + \frac {1}{6} \, \log \left (x\right )\right )^{p} \,d x } \]
Timed out. \[ \int \tanh ^p\left (a+\frac {\log (x)}{6}\right ) \, dx=\int {\mathrm {tanh}\left (a+\frac {\ln \left (x\right )}{6}\right )}^p \,d x \]