\(\int F^{c (a+b x)} (f \tanh (d+e x))^n \, dx\) [58]

Optimal result

Integrand size = 20, antiderivative size = 139 \[ \int F^{c (a+b x)} (f \tanh (d+e x))^n \, dx=-\frac {2^{-1-n} \left (e^{2 d+2 e x}\right )^{-\frac {b c \log (F)}{2 e}} \left (1-e^{2 d+2 e x}\right ) \left (1+e^{2 d+2 e x}\right )^n F^{c (a+b x)} \operatorname {AppellF1}\left (1+n,1-\frac {b c \log (F)}{2 e},n,2+n,1-e^{2 d+2 e x},\frac {1}{2} \left (1-e^{2 d+2 e x}\right )\right ) (f \tanh (d+e x))^n}{e (1+n)} \] Output:

-2^(-1-n)*(1-exp(2*e*x+2*d))*(1+exp(2*e*x+2*d))^n*F^(c*(b*x+a))*AppellF1(1 
+n,1-1/2*b*c*ln(F)/e,n,2+n,1-exp(2*e*x+2*d),1/2-1/2*exp(2*e*x+2*d))*(f*tan 
h(e*x+d))^n/e/(exp(2*e*x+2*d)^(1/2*b*c*ln(F)/e))/(1+n)
 

Mathematica [F]

\[ \int F^{c (a+b x)} (f \tanh (d+e x))^n \, dx=\int F^{c (a+b x)} (f \tanh (d+e x))^n \, dx \] Input:

Integrate[F^(c*(a + b*x))*(f*Tanh[d + e*x])^n,x]
 

Output:

Integrate[F^(c*(a + b*x))*(f*Tanh[d + e*x])^n, x]
 

Rubi [F]

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.

\(\displaystyle \int F^{c (a+b x)} (f \tanh (d+e x))^n \, dx\)

\(\Big \downarrow \) 7271

\(\displaystyle \tanh ^{-n}(d+e x) (f \tanh (d+e x))^n \int F^{c (a+b x)} \tanh ^n(d+e x)dx\)

\(\Big \downarrow \) 6030

\(\displaystyle \tanh ^{-n}(d+e x) (f \tanh (d+e x))^n \int F^{a c+b x c} \tanh ^n(d+e x)dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \tanh ^{-n}(d+e x) (f \tanh (d+e x))^n \int F^{a c+b x c} \tanh ^n(d+e x)dx\)

Input:

Int[F^(c*(a + b*x))*(f*Tanh[d + e*x])^n,x]
 

Output:

$Aborted
 

Maple [F]

\[\int F^{c \left (b x +a \right )} \left (f \tanh \left (e x +d \right )\right )^{n}d x\]

Input:

int(F^(c*(b*x+a))*(f*tanh(e*x+d))^n,x)
 

Output:

int(F^(c*(b*x+a))*(f*tanh(e*x+d))^n,x)
 

Fricas [F]

\[ \int F^{c (a+b x)} (f \tanh (d+e x))^n \, dx=\int { \left (f \tanh \left (e x + d\right )\right )^{n} F^{{\left (b x + a\right )} c} \,d x } \] Input:

integrate(F^(c*(b*x+a))*(f*tanh(e*x+d))^n,x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

integral((f*tanh(e*x + d))^n*F^(b*c*x + a*c), x)
 

Sympy [F]

\[ \int F^{c (a+b x)} (f \tanh (d+e x))^n \, dx=\int F^{c \left (a + b x\right )} \left (f \tanh {\left (d + e x \right )}\right )^{n}\, dx \] Input:

integrate(F**(c*(b*x+a))*(f*tanh(e*x+d))**n,x)
 

Output:

Integral(F**(c*(a + b*x))*(f*tanh(d + e*x))**n, x)
 

Maxima [F]

\[ \int F^{c (a+b x)} (f \tanh (d+e x))^n \, dx=\int { \left (f \tanh \left (e x + d\right )\right )^{n} F^{{\left (b x + a\right )} c} \,d x } \] Input:

integrate(F^(c*(b*x+a))*(f*tanh(e*x+d))^n,x, algorithm="maxima")
 

Output:

integrate((f*tanh(e*x + d))^n*F^((b*x + a)*c), x)
 

Giac [F]

\[ \int F^{c (a+b x)} (f \tanh (d+e x))^n \, dx=\int { \left (f \tanh \left (e x + d\right )\right )^{n} F^{{\left (b x + a\right )} c} \,d x } \] Input:

integrate(F^(c*(b*x+a))*(f*tanh(e*x+d))^n,x, algorithm="giac")
 

Output:

integrate((f*tanh(e*x + d))^n*F^((b*x + a)*c), x)
 

Mupad [F(-1)]

Timed out. \[ \int F^{c (a+b x)} (f \tanh (d+e x))^n \, dx=\int F^{c\,\left (a+b\,x\right )}\,{\left (f\,\mathrm {tanh}\left (d+e\,x\right )\right )}^n \,d x \] Input:

int(F^(c*(a + b*x))*(f*tanh(d + e*x))^n,x)
 

Output:

int(F^(c*(a + b*x))*(f*tanh(d + e*x))^n, x)
 

Reduce [F]

\[ \int F^{c (a+b x)} (f \tanh (d+e x))^n \, dx=f^{a c +n} \left (\int f^{b c x} \tanh \left (e x +d \right )^{n}d x \right ) \] Input:

int(F^(c*(b*x+a))*(f*tanh(e*x+d))^n,x)
 

Output:

f**(a*c + n)*int(f**(b*c*x)*tanh(d + e*x)**n,x)