Integrand size = 9, antiderivative size = 79 \[ \int \coth ^p(a+b \log (x)) \, dx=x \left (-1-e^{2 a} x^{2 b}\right )^p \left (1+e^{2 a} x^{2 b}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2 b},p,-p,\frac {1}{2} \left (2+\frac {1}{b}\right ),e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right ) \] Output:
x*(-1-exp(2*a)*x^(2*b))^p*AppellF1(1/2/b,p,-p,1+1/2/b,exp(2*a)*x^(2*b),-ex p(2*a)*x^(2*b))/((1+exp(2*a)*x^(2*b))^p)
Leaf count is larger than twice the leaf count of optimal. \(259\) vs. \(2(79)=158\).
Time = 0.40 (sec) , antiderivative size = 259, normalized size of antiderivative = 3.28 \[ \int \coth ^p(a+b \log (x)) \, dx=\frac {(1+2 b) x \left (\frac {1+e^{2 a} x^{2 b}}{-1+e^{2 a} x^{2 b}}\right )^p \operatorname {AppellF1}\left (\frac {1}{2 b},p,-p,1+\frac {1}{2 b},e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right )}{2 b e^{2 a} p x^{2 b} \operatorname {AppellF1}\left (1+\frac {1}{2 b},p,1-p,2+\frac {1}{2 b},e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right )+2 b e^{2 a} p x^{2 b} \operatorname {AppellF1}\left (1+\frac {1}{2 b},1+p,-p,2+\frac {1}{2 b},e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right )+(1+2 b) \operatorname {AppellF1}\left (\frac {1}{2 b},p,-p,1+\frac {1}{2 b},e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right )} \] Input:
Integrate[Coth[a + b*Log[x]]^p,x]
Output:
((1 + 2*b)*x*((1 + E^(2*a)*x^(2*b))/(-1 + E^(2*a)*x^(2*b)))^p*AppellF1[1/( 2*b), p, -p, 1 + 1/(2*b), E^(2*a)*x^(2*b), -(E^(2*a)*x^(2*b))])/(2*b*E^(2* a)*p*x^(2*b)*AppellF1[1 + 1/(2*b), p, 1 - p, 2 + 1/(2*b), E^(2*a)*x^(2*b), -(E^(2*a)*x^(2*b))] + 2*b*E^(2*a)*p*x^(2*b)*AppellF1[1 + 1/(2*b), 1 + p, -p, 2 + 1/(2*b), E^(2*a)*x^(2*b), -(E^(2*a)*x^(2*b))] + (1 + 2*b)*AppellF1 [1/(2*b), p, -p, 1 + 1/(2*b), E^(2*a)*x^(2*b), -(E^(2*a)*x^(2*b))])
Time = 0.44 (sec) , antiderivative size = 79, 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 = {6068, 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.
\(\displaystyle \int \coth ^p(a+b \log (x)) \, dx\) |
\(\Big \downarrow \) 6068 |
\(\displaystyle \int \left (-e^{2 a} x^{2 b}-1\right )^p \left (1-e^{2 a} x^{2 b}\right )^{-p}dx\) |
\(\Big \downarrow \) 937 |
\(\displaystyle \left (-e^{2 a} x^{2 b}-1\right )^p \left (e^{2 a} x^{2 b}+1\right )^{-p} \int \left (1-e^{2 a} x^{2 b}\right )^{-p} \left (e^{2 a} x^{2 b}+1\right )^pdx\) |
\(\Big \downarrow \) 936 |
\(\displaystyle x \left (-e^{2 a} x^{2 b}-1\right )^p \left (e^{2 a} x^{2 b}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2 b},p,-p,\frac {1}{2} \left (2+\frac {1}{b}\right ),e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right )\) |
Input:
Int[Coth[a + b*Log[x]]^p,x]
Output:
(x*(-1 - E^(2*a)*x^(2*b))^p*AppellF1[1/(2*b), p, -p, (2 + b^(-1))/2, E^(2* a)*x^(2*b), -(E^(2*a)*x^(2*b))])/(1 + E^(2*a)*x^(2*b))^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[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]
\[\int \coth \left (a +b \ln \left (x \right )\right )^{p}d x\]
Input:
int(coth(a+b*ln(x))^p,x)
Output:
int(coth(a+b*ln(x))^p,x)
\[ \int \coth ^p(a+b \log (x)) \, dx=\int { \coth \left (b \log \left (x\right ) + a\right )^{p} \,d x } \] Input:
integrate(coth(a+b*log(x))^p,x, algorithm="fricas")
Output:
integral(coth(b*log(x) + a)^p, x)
\[ \int \coth ^p(a+b \log (x)) \, dx=\int \coth ^{p}{\left (a + b \log {\left (x \right )} \right )}\, dx \] Input:
integrate(coth(a+b*ln(x))**p,x)
Output:
Integral(coth(a + b*log(x))**p, x)
\[ \int \coth ^p(a+b \log (x)) \, dx=\int { \coth \left (b \log \left (x\right ) + a\right )^{p} \,d x } \] Input:
integrate(coth(a+b*log(x))^p,x, algorithm="maxima")
Output:
integrate(coth(b*log(x) + a)^p, x)
\[ \int \coth ^p(a+b \log (x)) \, dx=\int { \coth \left (b \log \left (x\right ) + a\right )^{p} \,d x } \] Input:
integrate(coth(a+b*log(x))^p,x, algorithm="giac")
Output:
integrate(coth(b*log(x) + a)^p, x)
Timed out. \[ \int \coth ^p(a+b \log (x)) \, dx=\int {\mathrm {coth}\left (a+b\,\ln \left (x\right )\right )}^p \,d x \] Input:
int(coth(a + b*log(x))^p,x)
Output:
int(coth(a + b*log(x))^p, x)
\[ \int \coth ^p(a+b \log (x)) \, dx=\coth \left (\mathrm {log}\left (x \right ) b +a \right )^{p} x -\left (\int \frac {\coth \left (\mathrm {log}\left (x \right ) b +a \right )^{p}}{\coth \left (\mathrm {log}\left (x \right ) b +a \right )}d x \right ) b p +\left (\int \coth \left (\mathrm {log}\left (x \right ) b +a \right )^{p} \coth \left (\mathrm {log}\left (x \right ) b +a \right )d x \right ) b p \] Input:
int(coth(a+b*log(x))^p,x)
Output:
coth(log(x)*b + a)**p*x - int(coth(log(x)*b + a)**p/coth(log(x)*b + a),x)* b*p + int(coth(log(x)*b + a)**p*coth(log(x)*b + a),x)*b*p