\(\int F^{c (a+b x)} \sqrt {f \text {sech}(d+e x)} \sqrt {g \tanh (d+e x)} \, dx\) [156]

Optimal result

Integrand size = 34, antiderivative size = 147 \[ \int F^{c (a+b x)} \sqrt {f \text {sech}(d+e x)} \sqrt {g \tanh (d+e x)} \, dx=-\frac {\left (e^{2 d+2 e x}\right )^{-\frac {e+2 b c \log (F)}{4 e}} \left (1-e^{2 d+2 e x}\right ) \left (1+e^{2 d+2 e x}\right ) F^{c (a+b x)} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4} \left (3-\frac {2 b c \log (F)}{e}\right ),1,\frac {5}{2},1-e^{2 d+2 e x},\frac {1}{2} \left (1-e^{2 d+2 e x}\right )\right ) \sqrt {f \text {sech}(d+e x)} \sqrt {g \tanh (d+e x)}}{6 e} \] Output:

-1/6*(1-exp(2*e*x+2*d))*(1+exp(2*e*x+2*d))*F^(c*(b*x+a))*AppellF1(3/2,3/4- 
1/2*b*c*ln(F)/e,1,5/2,1-exp(2*e*x+2*d),1/2-1/2*exp(2*e*x+2*d))*(f*sech(e*x 
+d))^(1/2)*(g*tanh(e*x+d))^(1/2)/e/(exp(2*e*x+2*d)^(1/4*(e+2*b*c*ln(F))/e) 
)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 7271

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

\(\Big \downarrow \) 7271

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

int(F^(c*(b*x+a))*(f*sech(e*x+d))^(1/2)*(g*tanh(e*x+d))^(1/2),x)
 

Output:

int(F^(c*(b*x+a))*(f*sech(e*x+d))^(1/2)*(g*tanh(e*x+d))^(1/2),x)
 

Fricas [F]

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

integrate(F^(c*(b*x+a))*(f*sech(e*x+d))^(1/2)*(g*tanh(e*x+d))^(1/2),x, alg 
orithm="fricas")
                                                                                    
                                                                                    
 

Output:

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

Sympy [F]

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

integrate(F**(c*(b*x+a))*(f*sech(e*x+d))**(1/2)*(g*tanh(e*x+d))**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate(F^(c*(b*x+a))*(f*sech(e*x+d))^(1/2)*(g*tanh(e*x+d))^(1/2),x, alg 
orithm="maxima")
 

Output:

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

Giac [F]

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

integrate(F^(c*(b*x+a))*(f*sech(e*x+d))^(1/2)*(g*tanh(e*x+d))^(1/2),x, alg 
orithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int(F^(c*(a + b*x))*(g*tanh(d + e*x))^(1/2)*(f/cosh(d + e*x))^(1/2),x)
 

Output:

int(F^(c*(a + b*x))*(g*tanh(d + e*x))^(1/2)*(f/cosh(d + e*x))^(1/2), x)
 

Reduce [F]

\[ \int F^{c (a+b x)} \sqrt {f \text {sech}(d+e x)} \sqrt {g \tanh (d+e x)} \, dx=\frac {\sqrt {g}\, f^{a c +\frac {1}{2}} \left (2 f^{b c x} \sqrt {\tanh \left (e x +d \right )}\, \sqrt {\mathrm {sech}\left (e x +d \right )}-\left (\int \frac {f^{b c x} \sqrt {\tanh \left (e x +d \right )}\, \sqrt {\mathrm {sech}\left (e x +d \right )}}{\tanh \left (e x +d \right )}d x \right ) e +2 \left (\int f^{b c x} \sqrt {\tanh \left (e x +d \right )}\, \sqrt {\mathrm {sech}\left (e x +d \right )}\, \tanh \left (e x +d \right )d x \right ) e \right )}{2 \,\mathrm {log}\left (f \right ) b c} \] Input:

int(F^(c*(b*x+a))*(f*sech(e*x+d))^(1/2)*(g*tanh(e*x+d))^(1/2),x)
 

Output:

(sqrt(g)*f**((2*a*c + 1)/2)*(2*f**(b*c*x)*sqrt(tanh(d + e*x))*sqrt(sech(d 
+ e*x)) - int((f**(b*c*x)*sqrt(tanh(d + e*x))*sqrt(sech(d + e*x)))/tanh(d 
+ e*x),x)*e + 2*int(f**(b*c*x)*sqrt(tanh(d + e*x))*sqrt(sech(d + e*x))*tan 
h(d + e*x),x)*e))/(2*log(f)*b*c)