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