Integrand size = 30, antiderivative size = 118 \[ \int F^{c (a+b x)} (f \cosh (d+e x))^p (g \text {sech}(d+e x))^q \, dx=-\frac {\left (1+e^{2 d+2 e x}\right )^{-p+q} F^{c (a+b x)} (f \cosh (d+e x))^p \operatorname {Hypergeometric2F1}\left (-p+q,\frac {1}{2} \left (-p+q+\frac {b c \log (F)}{e}\right ),\frac {1}{2} \left (2-p+q+\frac {b c \log (F)}{e}\right ),-e^{2 d+2 e x}\right ) (g \text {sech}(d+e x))^q}{e p-e q-b c \log (F)} \] Output:
-(1+exp(2*e*x+2*d))^(-p+q)*F^(c*(b*x+a))*(f*cosh(e*x+d))^p*hypergeom([-p+q , -1/2*p+1/2*q+1/2*b*c*ln(F)/e],[1-1/2*p+1/2*q+1/2*b*c*ln(F)/e],-exp(2*e*x +2*d))*(g*sech(e*x+d))^q/(e*p-e*q-b*c*ln(F))
Timed out. \[ \int F^{c (a+b x)} (f \cosh (d+e x))^p (g \text {sech}(d+e x))^q \, dx=\text {\$Aborted} \] Input:
Integrate[F^(c*(a + b*x))*(f*Cosh[d + e*x])^p*(g*Sech[d + e*x])^q,x]
Output:
$Aborted
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.
\(\displaystyle \int F^{c (a+b x)} (f \cosh (d+e x))^p (g \text {sech}(d+e x))^q \, dx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \cosh ^{-p}(d+e x) (f \cosh (d+e x))^p \int F^{c (a+b x)} \cosh ^p(d+e x) (g \text {sech}(d+e x))^qdx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \cosh ^{-p}(d+e x) \text {sech}^{-q}(d+e x) (f \cosh (d+e x))^p (g \text {sech}(d+e x))^q \int F^{c (a+b x)} \cosh ^p(d+e x) \text {sech}^q(d+e x)dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \cosh ^{-p}(d+e x) \text {sech}^{-q}(d+e x) (f \cosh (d+e x))^p (g \text {sech}(d+e x))^q \int F^{a c+b x c} \cosh ^p(d+e x) \text {sech}^q(d+e x)dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \cosh ^{-p}(d+e x) \text {sech}^{-q}(d+e x) (f \cosh (d+e x))^p (g \text {sech}(d+e x))^q \int F^{a c+b x c} \cosh ^p(d+e x) \text {sech}^q(d+e x)dx\) |
Input:
Int[F^(c*(a + b*x))*(f*Cosh[d + e*x])^p*(g*Sech[d + e*x])^q,x]
Output:
$Aborted
\[\int F^{c \left (b x +a \right )} \left (f \cosh \left (e x +d \right )\right )^{p} \left (g \,\operatorname {sech}\left (e x +d \right )\right )^{q}d x\]
Input:
int(F^(c*(b*x+a))*(f*cosh(e*x+d))^p*(g*sech(e*x+d))^q,x)
Output:
int(F^(c*(b*x+a))*(f*cosh(e*x+d))^p*(g*sech(e*x+d))^q,x)
\[ \int F^{c (a+b x)} (f \cosh (d+e x))^p (g \text {sech}(d+e x))^q \, dx=\int { \left (f \cosh \left (e x + d\right )\right )^{p} \left (g \operatorname {sech}\left (e x + d\right )\right )^{q} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(f*cosh(e*x+d))^p*(g*sech(e*x+d))^q,x, algorithm=" fricas")
Output:
integral((f*cosh(e*x + d))^p*(g*sech(e*x + d))^q*F^(b*c*x + a*c), x)
Timed out. \[ \int F^{c (a+b x)} (f \cosh (d+e x))^p (g \text {sech}(d+e x))^q \, dx=\text {Timed out} \] Input:
integrate(F**(c*(b*x+a))*(f*cosh(e*x+d))**p*(g*sech(e*x+d))**q,x)
Output:
Timed out
\[ \int F^{c (a+b x)} (f \cosh (d+e x))^p (g \text {sech}(d+e x))^q \, dx=\int { \left (f \cosh \left (e x + d\right )\right )^{p} \left (g \operatorname {sech}\left (e x + d\right )\right )^{q} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(f*cosh(e*x+d))^p*(g*sech(e*x+d))^q,x, algorithm=" maxima")
Output:
integrate((f*cosh(e*x + d))^p*(g*sech(e*x + d))^q*F^((b*x + a)*c), x)
\[ \int F^{c (a+b x)} (f \cosh (d+e x))^p (g \text {sech}(d+e x))^q \, dx=\int { \left (f \cosh \left (e x + d\right )\right )^{p} \left (g \operatorname {sech}\left (e x + d\right )\right )^{q} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(f*cosh(e*x+d))^p*(g*sech(e*x+d))^q,x, algorithm=" giac")
Output:
integrate((f*cosh(e*x + d))^p*(g*sech(e*x + d))^q*F^((b*x + a)*c), x)
Timed out. \[ \int F^{c (a+b x)} (f \cosh (d+e x))^p (g \text {sech}(d+e x))^q \, dx=\int F^{c\,\left (a+b\,x\right )}\,{\left (f\,\mathrm {cosh}\left (d+e\,x\right )\right )}^p\,{\left (\frac {g}{\mathrm {cosh}\left (d+e\,x\right )}\right )}^q \,d x \] Input:
int(F^(c*(a + b*x))*(f*cosh(d + e*x))^p*(g/cosh(d + e*x))^q,x)
Output:
int(F^(c*(a + b*x))*(f*cosh(d + e*x))^p*(g/cosh(d + e*x))^q, x)
\[ \int F^{c (a+b x)} (f \cosh (d+e x))^p (g \text {sech}(d+e x))^q \, dx=g^{q} f^{a c +p} \left (\int f^{b c x} \mathrm {sech}\left (e x +d \right )^{q} \cosh \left (e x +d \right )^{p}d x \right ) \] Input:
int(F^(c*(b*x+a))*(f*cosh(e*x+d))^p*(g*sech(e*x+d))^q,x)
Output:
g**q*f**(a*c + p)*int(f**(b*c*x)*sech(d + e*x)**q*cosh(d + e*x)**p,x)