\(\int (e x)^m \tanh ^p(a+b \log (x)) \, dx\) [174]

Optimal result

Integrand size = 15, antiderivative size = 99 \[ \int (e x)^m \tanh ^p(a+b \log (x)) \, dx=\frac {(e x)^{1+m} \left (1-e^{2 a} x^{2 b}\right )^{-p} \left (-1+e^{2 a} x^{2 b}\right )^p \operatorname {AppellF1}\left (\frac {1+m}{2 b},-p,p,1+\frac {1+m}{2 b},e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right )}{e (1+m)} \] Output:

(e*x)^(1+m)*(-1+exp(2*a)*x^(2*b))^p*AppellF1(1/2*(1+m)/b,-p,p,1+1/2*(1+m)/ 
b,exp(2*a)*x^(2*b),-exp(2*a)*x^(2*b))/e/(1+m)/((1-exp(2*a)*x^(2*b))^p)
 

Mathematica [A] (warning: unable to verify)

Time = 0.49 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.27 \[ \int (e x)^m \tanh ^p(a+b \log (x)) \, dx=\frac {x (e x)^m \left (1-e^{2 a} x^{2 b}\right )^{-p} \left (\frac {-1+e^{2 a} x^{2 b}}{1+e^{2 a} x^{2 b}}\right )^p \left (1+e^{2 a} x^{2 b}\right )^p \operatorname {AppellF1}\left (\frac {1+m}{2 b},-p,p,1+\frac {1+m}{2 b},e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right )}{1+m} \] Input:

Integrate[(e*x)^m*Tanh[a + b*Log[x]]^p,x]
 

Output:

(x*(e*x)^m*((-1 + E^(2*a)*x^(2*b))/(1 + E^(2*a)*x^(2*b)))^p*(1 + E^(2*a)*x 
^(2*b))^p*AppellF1[(1 + m)/(2*b), -p, p, 1 + (1 + m)/(2*b), E^(2*a)*x^(2*b 
), -(E^(2*a)*x^(2*b))])/((1 + m)*(1 - E^(2*a)*x^(2*b))^p)
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6071, 1013, 1012}

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.

\(\displaystyle \int (e x)^m \tanh ^p(a+b \log (x)) \, dx\)

\(\Big \downarrow \) 6071

\(\displaystyle \int (e x)^m \left (e^{2 a} x^{2 b}-1\right )^p \left (e^{2 a} x^{2 b}+1\right )^{-p}dx\)

\(\Big \downarrow \) 1013

\(\displaystyle \left (1-e^{2 a} x^{2 b}\right )^{-p} \left (e^{2 a} x^{2 b}-1\right )^p \int (e x)^m \left (1-e^{2 a} x^{2 b}\right )^p \left (e^{2 a} x^{2 b}+1\right )^{-p}dx\)

\(\Big \downarrow \) 1012

\(\displaystyle \frac {(e x)^{m+1} \left (1-e^{2 a} x^{2 b}\right )^{-p} \left (e^{2 a} x^{2 b}-1\right )^p \operatorname {AppellF1}\left (\frac {m+1}{2 b},-p,p,\frac {m+1}{2 b}+1,e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right )}{e (m+1)}\)

Input:

Int[(e*x)^m*Tanh[a + b*Log[x]]^p,x]
 

Output:

((e*x)^(1 + m)*(-1 + E^(2*a)*x^(2*b))^p*AppellF1[(1 + m)/(2*b), -p, p, 1 + 
 (1 + m)/(2*b), E^(2*a)*x^(2*b), -(E^(2*a)*x^(2*b))])/(e*(1 + m)*(1 - E^(2 
*a)*x^(2*b))^p)
 

Defintions of rubi rules used

rule 1012

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m 
+ 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, 
 b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n 
 - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 1013

Int[((e_.)*(x_))^(m_.)*((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[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; 
 FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] & 
& NeQ[m, n - 1] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 6071

Int[((e_.)*(x_))^(m_.)*Tanh[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] 
 :> Int[(e*x)^m*((-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, e, m, p}, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((e*x)^m*tanh(b*log(x) + a)^p, x)
 

Sympy [F]

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

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

Output:

Integral((e*x)**m*tanh(a + b*log(x))**p, x)
 

Maxima [F]

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

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

Output:

integrate((e*x)^m*tanh(b*log(x) + a)^p, x)
 

Giac [F]

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

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

Output:

integrate((e*x)^m*tanh(b*log(x) + a)^p, x)
 

Mupad [F(-1)]

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

int(tanh(a + b*log(x))^p*(e*x)^m,x)
 

Output:

int(tanh(a + b*log(x))^p*(e*x)^m, x)
 

Reduce [F]

\[ \int (e x)^m \tanh ^p(a+b \log (x)) \, dx=\frac {e^{m} \left (x^{m} \tanh \left (\mathrm {log}\left (x \right ) b +a \right )^{p} x -\left (\int \frac {x^{m} \tanh \left (\mathrm {log}\left (x \right ) b +a \right )^{p}}{\tanh \left (\mathrm {log}\left (x \right ) b +a \right )}d x \right ) b p +\left (\int x^{m} \tanh \left (\mathrm {log}\left (x \right ) b +a \right )^{p} \tanh \left (\mathrm {log}\left (x \right ) b +a \right )d x \right ) b p \right )}{m +1} \] Input:

int((e*x)^m*tanh(a+b*log(x))^p,x)
 

Output:

(e**m*(x**m*tanh(log(x)*b + a)**p*x - int((x**m*tanh(log(x)*b + a)**p)/tan 
h(log(x)*b + a),x)*b*p + int(x**m*tanh(log(x)*b + a)**p*tanh(log(x)*b + a) 
,x)*b*p))/(m + 1)