\(\int F^{c (a+b x)} (f \text {sech}(d+e x))^p (g \tanh (d+e x))^q \, dx\) [177]

Optimal result

Integrand size = 30, antiderivative size = 166 \[ \int F^{c (a+b x)} (f \text {sech}(d+e x))^p (g \tanh (d+e x))^q \, dx=-\frac {2^{-1-p-q} \left (e^{2 d+2 e x}\right )^{-\frac {e p+b c \log (F)}{2 e}} \left (1-e^{2 d+2 e x}\right ) \left (1+e^{2 d+2 e x}\right )^{p+q} F^{c (a+b x)} \operatorname {AppellF1}\left (1+q,\frac {1}{2} \left (2-p-\frac {b c \log (F)}{e}\right ),p+q,2+q,1-e^{2 d+2 e x},\frac {1}{2} \left (1-e^{2 d+2 e x}\right )\right ) (f \text {sech}(d+e x))^p (g \tanh (d+e x))^q}{e (1+q)} \] Output:

-2^(-1-p-q)*(1-exp(2*e*x+2*d))*(1+exp(2*e*x+2*d))^(p+q)*F^(c*(b*x+a))*Appe 
llF1(1+q,1-1/2*p-1/2*b*c*ln(F)/e,p+q,2+q,1-exp(2*e*x+2*d),1/2-1/2*exp(2*e* 
x+2*d))*(f*sech(e*x+d))^p*(g*tanh(e*x+d))^q/e/(exp(2*e*x+2*d)^(1/2*(e*p+b* 
c*ln(F))/e))/(1+q)
 

Mathematica [F]

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

Integrate[F^(c*(a + b*x))*(f*Sech[d + e*x])^p*(g*Tanh[d + e*x])^q,x]
 

Output:

Integrate[F^(c*(a + b*x))*(f*Sech[d + e*x])^p*(g*Tanh[d + e*x])^q, 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.

\(\displaystyle \int F^{c (a+b x)} (f \text {sech}(d+e x))^p (g \tanh (d+e x))^q \, dx\)

\(\Big \downarrow \) 7271

\(\displaystyle \text {sech}^{-p}(d+e x) (f \text {sech}(d+e x))^p \int F^{c (a+b x)} \text {sech}^p(d+e x) (g \tanh (d+e x))^qdx\)

\(\Big \downarrow \) 7271

\(\displaystyle \text {sech}^{-p}(d+e x) \tanh ^{-q}(d+e x) (f \text {sech}(d+e x))^p (g \tanh (d+e x))^q \int F^{c (a+b x)} \text {sech}^p(d+e x) \tanh ^q(d+e x)dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \text {sech}^{-p}(d+e x) \tanh ^{-q}(d+e x) (f \text {sech}(d+e x))^p (g \tanh (d+e x))^q \int F^{a c+b x c} \text {sech}^p(d+e x) \tanh ^q(d+e x)dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \text {sech}^{-p}(d+e x) \tanh ^{-q}(d+e x) (f \text {sech}(d+e x))^p (g \tanh (d+e x))^q \int F^{a c+b x c} \text {sech}^p(d+e x) \tanh ^q(d+e x)dx\)

Input:

Int[F^(c*(a + b*x))*(f*Sech[d + e*x])^p*(g*Tanh[d + e*x])^q,x]
 

Output:

$Aborted
 

Maple [F]

\[\int F^{c \left (b x +a \right )} \left (f \,\operatorname {sech}\left (e x +d \right )\right )^{p} \left (g \tanh \left (e x +d \right )\right )^{q}d x\]

Input:

int(F^(c*(b*x+a))*(f*sech(e*x+d))^p*(g*tanh(e*x+d))^q,x)
 

Output:

int(F^(c*(b*x+a))*(f*sech(e*x+d))^p*(g*tanh(e*x+d))^q,x)
 

Fricas [F]

\[ \int F^{c (a+b x)} (f \text {sech}(d+e x))^p (g \tanh (d+e x))^q \, dx=\int { \left (f \operatorname {sech}\left (e x + d\right )\right )^{p} \left (g \tanh \left (e x + d\right )\right )^{q} F^{{\left (b x + a\right )} c} \,d x } \] Input:

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

Output:

integral((f*sech(e*x + d))^p*(g*tanh(e*x + d))^q*F^(b*c*x + a*c), x)
 

Sympy [F]

\[ \int F^{c (a+b x)} (f \text {sech}(d+e x))^p (g \tanh (d+e x))^q \, dx=\int F^{c \left (a + b x\right )} \left (f \operatorname {sech}{\left (d + e x \right )}\right )^{p} \left (g \tanh {\left (d + e x \right )}\right )^{q}\, dx \] Input:

integrate(F**(c*(b*x+a))*(f*sech(e*x+d))**p*(g*tanh(e*x+d))**q,x)
 

Output:

Integral(F**(c*(a + b*x))*(f*sech(d + e*x))**p*(g*tanh(d + e*x))**q, x)
 

Maxima [F]

\[ \int F^{c (a+b x)} (f \text {sech}(d+e x))^p (g \tanh (d+e x))^q \, dx=\int { \left (f \operatorname {sech}\left (e x + d\right )\right )^{p} \left (g \tanh \left (e x + d\right )\right )^{q} F^{{\left (b x + a\right )} c} \,d x } \] Input:

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

Output:

integrate((f*sech(e*x + d))^p*(g*tanh(e*x + d))^q*F^((b*x + a)*c), x)
 

Giac [F]

\[ \int F^{c (a+b x)} (f \text {sech}(d+e x))^p (g \tanh (d+e x))^q \, dx=\int { \left (f \operatorname {sech}\left (e x + d\right )\right )^{p} \left (g \tanh \left (e x + d\right )\right )^{q} F^{{\left (b x + a\right )} c} \,d x } \] Input:

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

Output:

integrate((f*sech(e*x + d))^p*(g*tanh(e*x + d))^q*F^((b*x + a)*c), x)
 

Mupad [F(-1)]

Timed out. \[ \int F^{c (a+b x)} (f \text {sech}(d+e x))^p (g \tanh (d+e x))^q \, dx=\int F^{c\,\left (a+b\,x\right )}\,{\left (g\,\mathrm {tanh}\left (d+e\,x\right )\right )}^q\,{\left (\frac {f}{\mathrm {cosh}\left (d+e\,x\right )}\right )}^p \,d x \] Input:

int(F^(c*(a + b*x))*(g*tanh(d + e*x))^q*(f/cosh(d + e*x))^p,x)
 

Output:

int(F^(c*(a + b*x))*(g*tanh(d + e*x))^q*(f/cosh(d + e*x))^p, x)
 

Reduce [F]

\[ \int F^{c (a+b x)} (f \text {sech}(d+e x))^p (g \tanh (d+e x))^q \, dx=g^{q} f^{a c +p} \left (\int f^{b c x} \tanh \left (e x +d \right )^{q} \mathrm {sech}\left (e x +d \right )^{p}d x \right ) \] Input:

int(F^(c*(b*x+a))*(f*sech(e*x+d))^p*(g*tanh(e*x+d))^q,x)
 

Output:

g**q*f**(a*c + p)*int(f**(b*c*x)*tanh(d + e*x)**q*sech(d + e*x)**p,x)