Integrand size = 34, antiderivative size = 107 \[ \int F^{c (a+b x)} \sqrt {f \cosh (d+e x)} \sqrt {g \tanh (d+e x)} \, dx=-\frac {2 F^{c (a+b x)} \sqrt {f \cosh (d+e x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4} \left (-1+\frac {2 b c \log (F)}{e}\right ),\frac {1}{4} \left (3+\frac {2 b c \log (F)}{e}\right ),e^{2 d+2 e x}\right ) \sqrt {g \tanh (d+e x)}}{\sqrt {1-e^{2 d+2 e x}} (e-2 b c \log (F))} \] Output:
-2*F^(c*(b*x+a))*(f*cosh(e*x+d))^(1/2)*hypergeom([-1/2, -1/4+1/2*b*c*ln(F) /e],[3/4+1/2*b*c*ln(F)/e],exp(2*e*x+2*d))*(g*tanh(e*x+d))^(1/2)/(1-exp(2*e *x+2*d))^(1/2)/(e-2*b*c*ln(F))
Time = 1.57 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.06 \[ \int F^{c (a+b x)} \sqrt {f \cosh (d+e x)} \sqrt {g \tanh (d+e x)} \, dx=\frac {4 F^{c (a+b x)} \sqrt {f \cosh (d+e x)} \operatorname {Hypergeometric2F1}\left (1,\frac {5}{4}+\frac {b c \log (F)}{2 e},\frac {3}{4}+\frac {b c \log (F)}{2 e},\cosh (2 (d+e x))+\sinh (2 (d+e x))\right ) \sinh (d+e x) (\cosh (d+e x)+\sinh (d+e x)) \sqrt {g \tanh (d+e x)}}{e-2 b c \log (F)} \] Input:
Integrate[F^(c*(a + b*x))*Sqrt[f*Cosh[d + e*x]]*Sqrt[g*Tanh[d + e*x]],x]
Output:
(4*F^(c*(a + b*x))*Sqrt[f*Cosh[d + e*x]]*Hypergeometric2F1[1, 5/4 + (b*c*L og[F])/(2*e), 3/4 + (b*c*Log[F])/(2*e), Cosh[2*(d + e*x)] + Sinh[2*(d + e* x)]]*Sinh[d + e*x]*(Cosh[d + e*x] + Sinh[d + e*x])*Sqrt[g*Tanh[d + e*x]])/ (e - 2*b*c*Log[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.
| \(\displaystyle \int F^{c (a+b x)} \sqrt {f \cosh (d+e x)} \sqrt {g \tanh (d+e x)} \, dx\) |
| \(\Big \downarrow \) 7271 |
| \(\displaystyle \frac {\sqrt {f \cosh (d+e x)} \int F^{c (a+b x)} \sqrt {\cosh (d+e x)} \sqrt {g \tanh (d+e x)}dx}{\sqrt {\cosh (d+e x)}}\) |
| \(\Big \downarrow \) 7271 |
| \(\displaystyle \frac {\sqrt {f \cosh (d+e x)} \sqrt {g \tanh (d+e x)} \int F^{c (a+b x)} \sqrt {\cosh (d+e x)} \sqrt {\tanh (d+e x)}dx}{\sqrt {\cosh (d+e x)} \sqrt {\tanh (d+e x)}}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {\sqrt {f \cosh (d+e x)} \sqrt {g \tanh (d+e x)} \int F^{a c+b x c} \sqrt {\cosh (d+e x)} \sqrt {\tanh (d+e x)}dx}{\sqrt {\cosh (d+e x)} \sqrt {\tanh (d+e x)}}\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \frac {\sqrt {f \cosh (d+e x)} \sqrt {g \tanh (d+e x)} \int F^{a c+b x c} \sqrt {\cosh (d+e x)} \sqrt {\tanh (d+e x)}dx}{\sqrt {\cosh (d+e x)} \sqrt {\tanh (d+e x)}}\) |
Input:
Int[F^(c*(a + b*x))*Sqrt[f*Cosh[d + e*x]]*Sqrt[g*Tanh[d + e*x]],x]
Output:
$Aborted
\[\int F^{c \left (b x +a \right )} \sqrt {f \cosh \left (e x +d \right )}\, \sqrt {g \tanh \left (e x +d \right )}d x\]
Input:
int(F^(c*(b*x+a))*(f*cosh(e*x+d))^(1/2)*(g*tanh(e*x+d))^(1/2),x)
Output:
int(F^(c*(b*x+a))*(f*cosh(e*x+d))^(1/2)*(g*tanh(e*x+d))^(1/2),x)
\[ \int F^{c (a+b x)} \sqrt {f \cosh (d+e x)} \sqrt {g \tanh (d+e x)} \, dx=\int { \sqrt {f \cosh \left (e x + d\right )} \sqrt {g \tanh \left (e x + d\right )} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(f*cosh(e*x+d))^(1/2)*(g*tanh(e*x+d))^(1/2),x, alg orithm="fricas")
Output:
integral(sqrt(f*cosh(e*x + d))*sqrt(g*tanh(e*x + d))*F^(b*c*x + a*c), x)
Timed out. \[ \int F^{c (a+b x)} \sqrt {f \cosh (d+e x)} \sqrt {g \tanh (d+e x)} \, dx=\text {Timed out} \] Input:
integrate(F**(c*(b*x+a))*(f*cosh(e*x+d))**(1/2)*(g*tanh(e*x+d))**(1/2),x)
Output:
Timed out
\[ \int F^{c (a+b x)} \sqrt {f \cosh (d+e x)} \sqrt {g \tanh (d+e x)} \, dx=\int { \sqrt {f \cosh \left (e x + d\right )} \sqrt {g \tanh \left (e x + d\right )} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(f*cosh(e*x+d))^(1/2)*(g*tanh(e*x+d))^(1/2),x, alg orithm="maxima")
Output:
integrate(sqrt(f*cosh(e*x + d))*sqrt(g*tanh(e*x + d))*F^((b*x + a)*c), x)
\[ \int F^{c (a+b x)} \sqrt {f \cosh (d+e x)} \sqrt {g \tanh (d+e x)} \, dx=\int { \sqrt {f \cosh \left (e x + d\right )} \sqrt {g \tanh \left (e x + d\right )} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(f*cosh(e*x+d))^(1/2)*(g*tanh(e*x+d))^(1/2),x, alg orithm="giac")
Output:
integrate(sqrt(f*cosh(e*x + d))*sqrt(g*tanh(e*x + d))*F^((b*x + a)*c), x)
Timed out. \[ \int F^{c (a+b x)} \sqrt {f \cosh (d+e x)} \sqrt {g \tanh (d+e x)} \, dx=\int F^{c\,\left (a+b\,x\right )}\,\sqrt {f\,\mathrm {cosh}\left (d+e\,x\right )}\,\sqrt {g\,\mathrm {tanh}\left (d+e\,x\right )} \,d x \] Input:
int(F^(c*(a + b*x))*(f*cosh(d + e*x))^(1/2)*(g*tanh(d + e*x))^(1/2),x)
Output:
int(F^(c*(a + b*x))*(f*cosh(d + e*x))^(1/2)*(g*tanh(d + e*x))^(1/2), x)
\[ \int F^{c (a+b x)} \sqrt {f \cosh (d+e x)} \sqrt {g \tanh (d+e x)} \, dx=\frac {\sqrt {g}\, f^{a c +\frac {1}{2}} \left (2 f^{b c x} \sqrt {\tanh \left (e x +d \right )}\, \sqrt {\cosh \left (e x +d \right )}-\left (\int \frac {f^{b c x} \sqrt {\tanh \left (e x +d \right )}\, \sqrt {\cosh \left (e x +d \right )}\, \sinh \left (e x +d \right )}{\cosh \left (e x +d \right )}d x \right ) e -\left (\int \frac {f^{b c x} \sqrt {\tanh \left (e x +d \right )}\, \sqrt {\cosh \left (e x +d \right )}}{\tanh \left (e x +d \right )}d x \right ) e +\left (\int f^{b c x} \sqrt {\tanh \left (e x +d \right )}\, \sqrt {\cosh \left (e x +d \right )}\, \tanh \left (e x +d \right )d x \right ) e \right )}{2 \,\mathrm {log}\left (f \right ) b c} \] Input:
int(F^(c*(b*x+a))*(f*cosh(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(cosh(d + e*x)) - int((f**(b*c*x)*sqrt(tanh(d + e*x))*sqrt(cosh(d + e*x))*sinh(d + e*x))/cosh(d + e*x),x)*e - int((f**(b*c*x)*sqrt(tanh(d + e*x))*sqrt(cosh( d + e*x)))/tanh(d + e*x),x)*e + int(f**(b*c*x)*sqrt(tanh(d + e*x))*sqrt(co sh(d + e*x))*tanh(d + e*x),x)*e))/(2*log(f)*b*c)