\(\int \tanh ^p(a+b \log (x)) \, dx\) [163]

Optimal result

Integrand size = 9, antiderivative size = 79 \[ \int \tanh ^p(a+b \log (x)) \, dx=x \left (1-e^{2 a} x^{2 b}\right )^{-p} \left (-1+e^{2 a} x^{2 b}\right )^p \operatorname {AppellF1}\left (\frac {1}{2 b},-p,p,\frac {1}{2} \left (2+\frac {1}{b}\right ),e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right ) \]

[Out]

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),-exp(2*a)*x^(2*b))/((1-exp(2*a)*x^(2*b)
)^p)

Rubi [A] (verified)

Time = 0.03 (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 = {5652, 441, 440} \[ \int \tanh ^p(a+b \log (x)) \, dx=x \left (1-e^{2 a} x^{2 b}\right )^{-p} \left (e^{2 a} x^{2 b}-1\right )^p \operatorname {AppellF1}\left (\frac {1}{2 b},-p,p,\frac {1}{2} \left (2+\frac {1}{b}\right ),e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right ) \]

[In]

Int[Tanh[a + b*Log[x]]^p,x]

[Out]

(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

Rule 440

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])

Rule 441

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^F
racPart[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])

Rule 5652

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]

Rubi steps \begin{align*} \text {integral}& = \int \left (-1+e^{2 a} x^{2 b}\right )^p \left (1+e^{2 a} x^{2 b}\right )^{-p} \, dx \\ & = \left (\left (1-e^{2 a} x^{2 b}\right )^{-p} \left (-1+e^{2 a} x^{2 b}\right )^p\right ) \int \left (1-e^{2 a} x^{2 b}\right )^p \left (1+e^{2 a} x^{2 b}\right )^{-p} \, 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 ) \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(259\) vs. \(2(79)=158\).

Time = 0.48 (sec) , antiderivative size = 259, normalized size of antiderivative = 3.28 \[ \int \tanh ^p(a+b \log (x)) \, dx=\frac {(1+2 b) x \left (\frac {-1+e^{2 a} x^{2 b}}{1+e^{2 a} x^{2 b}}\right )^p \operatorname {AppellF1}\left (\frac {1}{2 b},-p,p,1+\frac {1}{2 b},e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right )}{-2 b e^{2 a} p x^{2 b} \operatorname {AppellF1}\left (1+\frac {1}{2 b},1-p,p,2+\frac {1}{2 b},e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right )-2 b e^{2 a} p x^{2 b} \operatorname {AppellF1}\left (1+\frac {1}{2 b},-p,1+p,2+\frac {1}{2 b},e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right )+(1+2 b) \operatorname {AppellF1}\left (\frac {1}{2 b},-p,p,1+\frac {1}{2 b},e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right )} \]

[In]

Integrate[Tanh[a + b*Log[x]]^p,x]

[Out]

((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), 1 - p, 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), -p, 1 + 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))])

Maple [F]

\[\int \tanh \left (a +b \ln \left (x \right )\right )^{p}d x\]

[In]

int(tanh(a+b*ln(x))^p,x)

[Out]

int(tanh(a+b*ln(x))^p,x)

Fricas [F]

\[ \int \tanh ^p(a+b \log (x)) \, dx=\int { \tanh \left (b \log \left (x\right ) + a\right )^{p} \,d x } \]

[In]

integrate(tanh(a+b*log(x))^p,x, algorithm="fricas")

[Out]

integral(tanh(b*log(x) + a)^p, x)

Sympy [F]

\[ \int \tanh ^p(a+b \log (x)) \, dx=\int \tanh ^{p}{\left (a + b \log {\left (x \right )} \right )}\, dx \]

[In]

integrate(tanh(a+b*ln(x))**p,x)

[Out]

Integral(tanh(a + b*log(x))**p, x)

Maxima [F]

\[ \int \tanh ^p(a+b \log (x)) \, dx=\int { \tanh \left (b \log \left (x\right ) + a\right )^{p} \,d x } \]

[In]

integrate(tanh(a+b*log(x))^p,x, algorithm="maxima")

[Out]

integrate(tanh(b*log(x) + a)^p, x)

Giac [F]

\[ \int \tanh ^p(a+b \log (x)) \, dx=\int { \tanh \left (b \log \left (x\right ) + a\right )^{p} \,d x } \]

[In]

integrate(tanh(a+b*log(x))^p,x, algorithm="giac")

[Out]

integrate(tanh(b*log(x) + a)^p, x)

Mupad [F(-1)]

Timed out. \[ \int \tanh ^p(a+b \log (x)) \, dx=\int {\mathrm {tanh}\left (a+b\,\ln \left (x\right )\right )}^p \,d x \]

[In]

int(tanh(a + b*log(x))^p,x)

[Out]

int(tanh(a + b*log(x))^p, x)