Integrand size = 30, antiderivative size = 168 \[ \int F^{c (a+b x)} (f \text {csch}(d+e x))^p (g \tanh (d+e x))^q \, dx=-\frac {2^{-1-q} \left (e^{2 d+2 e x}\right )^{-\frac {e p+b c \log (F)}{2 e}} \left (1-e^{2 d+2 e x}\right ) \left (1+e^{2 d+2 e x}\right )^q F^{c (a+b x)} \operatorname {AppellF1}\left (1-p+q,\frac {1}{2} \left (2-p-\frac {b c \log (F)}{e}\right ),q,2-p+q,1-e^{2 d+2 e x},\frac {1}{2} \left (1-e^{2 d+2 e x}\right )\right ) (f \text {csch}(d+e x))^p (g \tanh (d+e x))^q}{e (1-p+q)} \] Output:
-2^(-1-q)*(1-exp(2*e*x+2*d))*(1+exp(2*e*x+2*d))^q*F^(c*(b*x+a))*AppellF1(1 -p+q,1-1/2*p-1/2*b*c*ln(F)/e,q,2-p+q,1-exp(2*e*x+2*d),1/2-1/2*exp(2*e*x+2* d))*(f*csch(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-p+q)
\[ \int F^{c (a+b x)} (f \text {csch}(d+e x))^p (g \tanh (d+e x))^q \, dx=\int F^{c (a+b x)} (f \text {csch}(d+e x))^p (g \tanh (d+e x))^q \, dx \] Input:
Integrate[F^(c*(a + b*x))*(f*Csch[d + e*x])^p*(g*Tanh[d + e*x])^q,x]
Output:
Integrate[F^(c*(a + b*x))*(f*Csch[d + e*x])^p*(g*Tanh[d + e*x])^q, x]
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.
\(\displaystyle \int F^{c (a+b x)} (f \text {csch}(d+e x))^p (g \tanh (d+e x))^q \, dx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \text {csch}^{-p}(d+e x) (f \text {csch}(d+e x))^p \int F^{c (a+b x)} \text {csch}^p(d+e x) (g \tanh (d+e x))^qdx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \text {csch}^{-p}(d+e x) \tanh ^{-q}(d+e x) (f \text {csch}(d+e x))^p (g \tanh (d+e x))^q \int F^{c (a+b x)} \text {csch}^p(d+e x) \tanh ^q(d+e x)dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \text {csch}^{-p}(d+e x) \tanh ^{-q}(d+e x) (f \text {csch}(d+e x))^p (g \tanh (d+e x))^q \int F^{a c+b x c} \text {csch}^p(d+e x) \tanh ^q(d+e x)dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \text {csch}^{-p}(d+e x) \tanh ^{-q}(d+e x) (f \text {csch}(d+e x))^p (g \tanh (d+e x))^q \int F^{a c+b x c} \text {csch}^p(d+e x) \tanh ^q(d+e x)dx\) |
Input:
Int[F^(c*(a + b*x))*(f*Csch[d + e*x])^p*(g*Tanh[d + e*x])^q,x]
Output:
$Aborted
\[\int F^{c \left (b x +a \right )} \left (f \,\operatorname {csch}\left (e x +d \right )\right )^{p} \left (g \tanh \left (e x +d \right )\right )^{q}d x\]
Input:
int(F^(c*(b*x+a))*(f*csch(e*x+d))^p*(g*tanh(e*x+d))^q,x)
Output:
int(F^(c*(b*x+a))*(f*csch(e*x+d))^p*(g*tanh(e*x+d))^q,x)
\[ \int F^{c (a+b x)} (f \text {csch}(d+e x))^p (g \tanh (d+e x))^q \, dx=\int { \left (f \operatorname {csch}\left (e x + d\right )\right )^{p} \left (g \tanh \left (e x + d\right )\right )^{q} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(f*csch(e*x+d))^p*(g*tanh(e*x+d))^q,x, algorithm=" fricas")
Output:
integral((f*csch(e*x + d))^p*(g*tanh(e*x + d))^q*F^(b*c*x + a*c), x)
\[ \int F^{c (a+b x)} (f \text {csch}(d+e x))^p (g \tanh (d+e x))^q \, dx=\int F^{c \left (a + b x\right )} \left (f \operatorname {csch}{\left (d + e x \right )}\right )^{p} \left (g \tanh {\left (d + e x \right )}\right )^{q}\, dx \] Input:
integrate(F**(c*(b*x+a))*(f*csch(e*x+d))**p*(g*tanh(e*x+d))**q,x)
Output:
Integral(F**(c*(a + b*x))*(f*csch(d + e*x))**p*(g*tanh(d + e*x))**q, x)
\[ \int F^{c (a+b x)} (f \text {csch}(d+e x))^p (g \tanh (d+e x))^q \, dx=\int { \left (f \operatorname {csch}\left (e x + d\right )\right )^{p} \left (g \tanh \left (e x + d\right )\right )^{q} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(f*csch(e*x+d))^p*(g*tanh(e*x+d))^q,x, algorithm=" maxima")
Output:
integrate((f*csch(e*x + d))^p*(g*tanh(e*x + d))^q*F^((b*x + a)*c), x)
\[ \int F^{c (a+b x)} (f \text {csch}(d+e x))^p (g \tanh (d+e x))^q \, dx=\int { \left (f \operatorname {csch}\left (e x + d\right )\right )^{p} \left (g \tanh \left (e x + d\right )\right )^{q} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(f*csch(e*x+d))^p*(g*tanh(e*x+d))^q,x, algorithm=" giac")
Output:
integrate((f*csch(e*x + d))^p*(g*tanh(e*x + d))^q*F^((b*x + a)*c), x)
Timed out. \[ \int F^{c (a+b x)} (f \text {csch}(d+e x))^p (g \tanh (d+e x))^q \, dx=\int F^{c\,\left (a+b\,x\right )}\,{\left (g\,\mathrm {tanh}\left (d+e\,x\right )\right )}^q\,{\left (\frac {f}{\mathrm {sinh}\left (d+e\,x\right )}\right )}^p \,d x \] Input:
int(F^(c*(a + b*x))*(g*tanh(d + e*x))^q*(f/sinh(d + e*x))^p,x)
Output:
int(F^(c*(a + b*x))*(g*tanh(d + e*x))^q*(f/sinh(d + e*x))^p, x)
\[ \int F^{c (a+b x)} (f \text {csch}(d+e x))^p (g \tanh (d+e x))^q \, dx=g^{q} f^{a c +p} \left (\int f^{b c x} \tanh \left (e x +d \right )^{q} \mathrm {csch}\left (e x +d \right )^{p}d x \right ) \] Input:
int(F^(c*(b*x+a))*(f*csch(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*csch(d + e*x)**p,x)