Integrand size = 9, antiderivative size = 61 \[ \int \tanh ^p(a+2 \log (x)) \, dx=x \left (1-e^{2 a} x^4\right )^{-p} \left (-1+e^{2 a} x^4\right )^p \operatorname {AppellF1}\left (\frac {1}{4},-p,p,\frac {5}{4},e^{2 a} x^4,-e^{2 a} x^4\right ) \] Output:
x*(-1+exp(2*a)*x^4)^p*AppellF1(1/4,-p,p,5/4,exp(2*a)*x^4,-exp(2*a)*x^4)/(( 1-exp(2*a)*x^4)^p)
Leaf count is larger than twice the leaf count of optimal. \(171\) vs. \(2(61)=122\).
Time = 0.44 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.80 \[ \int \tanh ^p(a+2 \log (x)) \, dx=\frac {5 x \left (\frac {-1+e^{2 a} x^4}{1+e^{2 a} x^4}\right )^p \operatorname {AppellF1}\left (\frac {1}{4},-p,p,\frac {5}{4},e^{2 a} x^4,-e^{2 a} x^4\right )}{5 \operatorname {AppellF1}\left (\frac {1}{4},-p,p,\frac {5}{4},e^{2 a} x^4,-e^{2 a} x^4\right )-4 e^{2 a} p x^4 \left (\operatorname {AppellF1}\left (\frac {5}{4},1-p,p,\frac {9}{4},e^{2 a} x^4,-e^{2 a} x^4\right )+\operatorname {AppellF1}\left (\frac {5}{4},-p,1+p,\frac {9}{4},e^{2 a} x^4,-e^{2 a} x^4\right )\right )} \] Input:
Integrate[Tanh[a + 2*Log[x]]^p,x]
Output:
(5*x*((-1 + E^(2*a)*x^4)/(1 + E^(2*a)*x^4))^p*AppellF1[1/4, -p, p, 5/4, E^ (2*a)*x^4, -(E^(2*a)*x^4)])/(5*AppellF1[1/4, -p, p, 5/4, E^(2*a)*x^4, -(E^ (2*a)*x^4)] - 4*E^(2*a)*p*x^4*(AppellF1[5/4, 1 - p, p, 9/4, E^(2*a)*x^4, - (E^(2*a)*x^4)] + AppellF1[5/4, -p, 1 + p, 9/4, E^(2*a)*x^4, -(E^(2*a)*x^4) ]))
Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6067, 937, 936}
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.
\(\displaystyle \int \tanh ^p(a+2 \log (x)) \, dx\) |
\(\Big \downarrow \) 6067 |
\(\displaystyle \int \left (e^{2 a} x^4-1\right )^p \left (e^{2 a} x^4+1\right )^{-p}dx\) |
\(\Big \downarrow \) 937 |
\(\displaystyle \left (1-e^{2 a} x^4\right )^{-p} \left (e^{2 a} x^4-1\right )^p \int \left (1-e^{2 a} x^4\right )^p \left (e^{2 a} x^4+1\right )^{-p}dx\) |
\(\Big \downarrow \) 936 |
\(\displaystyle x \left (1-e^{2 a} x^4\right )^{-p} \left (e^{2 a} x^4-1\right )^p \operatorname {AppellF1}\left (\frac {1}{4},-p,p,\frac {5}{4},e^{2 a} x^4,-e^{2 a} x^4\right )\) |
Input:
Int[Tanh[a + 2*Log[x]]^p,x]
Output:
(x*(-1 + E^(2*a)*x^4)^p*AppellF1[1/4, -p, p, 5/4, E^(2*a)*x^4, -(E^(2*a)*x ^4)])/(1 - E^(2*a)*x^4)^p
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p, -q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c) ], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) Int[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, p, q }, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] && !(IntegerQ[p] || GtQ[a, 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]
\[\int \tanh \left (a +2 \ln \left (x \right )\right )^{p}d x\]
Input:
int(tanh(a+2*ln(x))^p,x)
Output:
int(tanh(a+2*ln(x))^p,x)
\[ \int \tanh ^p(a+2 \log (x)) \, dx=\int { \tanh \left (a + 2 \, \log \left (x\right )\right )^{p} \,d x } \] Input:
integrate(tanh(a+2*log(x))^p,x, algorithm="fricas")
Output:
integral(tanh(a + 2*log(x))^p, x)
\[ \int \tanh ^p(a+2 \log (x)) \, dx=\int \tanh ^{p}{\left (a + 2 \log {\left (x \right )} \right )}\, dx \] Input:
integrate(tanh(a+2*ln(x))**p,x)
Output:
Integral(tanh(a + 2*log(x))**p, x)
\[ \int \tanh ^p(a+2 \log (x)) \, dx=\int { \tanh \left (a + 2 \, \log \left (x\right )\right )^{p} \,d x } \] Input:
integrate(tanh(a+2*log(x))^p,x, algorithm="maxima")
Output:
integrate(tanh(a + 2*log(x))^p, x)
\[ \int \tanh ^p(a+2 \log (x)) \, dx=\int { \tanh \left (a + 2 \, \log \left (x\right )\right )^{p} \,d x } \] Input:
integrate(tanh(a+2*log(x))^p,x, algorithm="giac")
Output:
integrate(tanh(a + 2*log(x))^p, x)
Timed out. \[ \int \tanh ^p(a+2 \log (x)) \, dx=\int {\mathrm {tanh}\left (a+2\,\ln \left (x\right )\right )}^p \,d x \] Input:
int(tanh(a + 2*log(x))^p,x)
Output:
int(tanh(a + 2*log(x))^p, x)
\[ \int \tanh ^p(a+2 \log (x)) \, dx=\tanh \left (2 \,\mathrm {log}\left (x \right )+a \right )^{p} x -2 \left (\int \frac {\tanh \left (2 \,\mathrm {log}\left (x \right )+a \right )^{p}}{\tanh \left (2 \,\mathrm {log}\left (x \right )+a \right )}d x \right ) p +2 \left (\int \tanh \left (2 \,\mathrm {log}\left (x \right )+a \right )^{p} \tanh \left (2 \,\mathrm {log}\left (x \right )+a \right )d x \right ) p \] Input:
int(tanh(a+2*log(x))^p,x)
Output:
tanh(2*log(x) + a)**p*x - 2*int(tanh(2*log(x) + a)**p/tanh(2*log(x) + a),x )*p + 2*int(tanh(2*log(x) + a)**p*tanh(2*log(x) + a),x)*p