Integrand size = 7, antiderivative size = 61 \[ \int \coth ^p(a+\log (x)) \, dx=x \left (-1-e^{2 a} x^2\right )^p \left (1+e^{2 a} x^2\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},p,-p,\frac {3}{2},e^{2 a} x^2,-e^{2 a} x^2\right ) \] Output:
x*(-1-exp(2*a)*x^2)^p*AppellF1(1/2,p,-p,3/2,exp(2*a)*x^2,-exp(2*a)*x^2)/(( 1+exp(2*a)*x^2)^p)
Leaf count is larger than twice the leaf count of optimal. \(171\) vs. \(2(61)=122\).
Time = 0.41 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.80 \[ \int \coth ^p(a+\log (x)) \, dx=\frac {3 x \left (\frac {1+e^{2 a} x^2}{-1+e^{2 a} x^2}\right )^p \operatorname {AppellF1}\left (\frac {1}{2},p,-p,\frac {3}{2},e^{2 a} x^2,-e^{2 a} x^2\right )}{3 \operatorname {AppellF1}\left (\frac {1}{2},p,-p,\frac {3}{2},e^{2 a} x^2,-e^{2 a} x^2\right )+2 e^{2 a} p x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},p,1-p,\frac {5}{2},e^{2 a} x^2,-e^{2 a} x^2\right )+\operatorname {AppellF1}\left (\frac {3}{2},1+p,-p,\frac {5}{2},e^{2 a} x^2,-e^{2 a} x^2\right )\right )} \] Input:
Integrate[Coth[a + Log[x]]^p,x]
Output:
(3*x*((1 + E^(2*a)*x^2)/(-1 + E^(2*a)*x^2))^p*AppellF1[1/2, p, -p, 3/2, E^ (2*a)*x^2, -(E^(2*a)*x^2)])/(3*AppellF1[1/2, p, -p, 3/2, E^(2*a)*x^2, -(E^ (2*a)*x^2)] + 2*E^(2*a)*p*x^2*(AppellF1[3/2, p, 1 - p, 5/2, E^(2*a)*x^2, - (E^(2*a)*x^2)] + AppellF1[3/2, 1 + p, -p, 5/2, E^(2*a)*x^2, -(E^(2*a)*x^2) ]))
Time = 0.26 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6068, 334, 333}
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.
\(\displaystyle \int \coth ^p(a+\log (x)) \, dx\) |
\(\Big \downarrow \) 6068 |
\(\displaystyle \int \left (-e^{2 a} x^2-1\right )^p \left (1-e^{2 a} x^2\right )^{-p}dx\) |
\(\Big \downarrow \) 334 |
\(\displaystyle \left (-e^{2 a} x^2-1\right )^p \left (e^{2 a} x^2+1\right )^{-p} \int \left (1-e^{2 a} x^2\right )^{-p} \left (e^{2 a} x^2+1\right )^pdx\) |
\(\Big \downarrow \) 333 |
\(\displaystyle x \left (-e^{2 a} x^2-1\right )^p \left (e^{2 a} x^2+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},p,-p,\frac {3}{2},e^{2 a} x^2,-e^{2 a} x^2\right )\) |
Input:
Int[Coth[a + Log[x]]^p,x]
Output:
(x*(-1 - E^(2*a)*x^2)^p*AppellF1[1/2, p, -p, 3/2, E^(2*a)*x^2, -(E^(2*a)*x ^2)])/(1 + E^(2*a)*x^2)^p
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[a^p*c^q*x*AppellF1[1/2, -p, -q, 3/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; F reeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[a^IntPart[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p]) Int[ (1 + b*(x^2/a))^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && !(IntegerQ[p] || GtQ[a, 0])
Int[Coth[((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]
\[\int \coth \left (a +\ln \left (x \right )\right )^{p}d x\]
Input:
int(coth(a+ln(x))^p,x)
Output:
int(coth(a+ln(x))^p,x)
\[ \int \coth ^p(a+\log (x)) \, dx=\int { \coth \left (a + \log \left (x\right )\right )^{p} \,d x } \] Input:
integrate(coth(a+log(x))^p,x, algorithm="fricas")
Output:
integral(coth(a + log(x))^p, x)
\[ \int \coth ^p(a+\log (x)) \, dx=\int \coth ^{p}{\left (a + \log {\left (x \right )} \right )}\, dx \] Input:
integrate(coth(a+ln(x))**p,x)
Output:
Integral(coth(a + log(x))**p, x)
\[ \int \coth ^p(a+\log (x)) \, dx=\int { \coth \left (a + \log \left (x\right )\right )^{p} \,d x } \] Input:
integrate(coth(a+log(x))^p,x, algorithm="maxima")
Output:
integrate(coth(a + log(x))^p, x)
\[ \int \coth ^p(a+\log (x)) \, dx=\int { \coth \left (a + \log \left (x\right )\right )^{p} \,d x } \] Input:
integrate(coth(a+log(x))^p,x, algorithm="giac")
Output:
integrate(coth(a + log(x))^p, x)
Timed out. \[ \int \coth ^p(a+\log (x)) \, dx=\int {\mathrm {coth}\left (a+\ln \left (x\right )\right )}^p \,d x \] Input:
int(coth(a + log(x))^p,x)
Output:
int(coth(a + log(x))^p, x)
\[ \int \coth ^p(a+\log (x)) \, dx=\coth \left (\mathrm {log}\left (x \right )+a \right )^{p} x -\left (\int \frac {\coth \left (\mathrm {log}\left (x \right )+a \right )^{p}}{\coth \left (\mathrm {log}\left (x \right )+a \right )}d x \right ) p +\left (\int \coth \left (\mathrm {log}\left (x \right )+a \right )^{p} \coth \left (\mathrm {log}\left (x \right )+a \right )d x \right ) p \] Input:
int(coth(a+log(x))^p,x)
Output:
coth(log(x) + a)**p*x - int(coth(log(x) + a)**p/coth(log(x) + a),x)*p + in t(coth(log(x) + a)**p*coth(log(x) + a),x)*p