Integrand size = 34, antiderivative size = 140 \[ \int F^{c (a+b x)} \sqrt {f \tan (d+e x)} \sqrt {g \tan (d+e x)} \, dx=-\frac {i f F^{c (a+b x)} \sqrt {g \tan (d+e x)}}{b c \log (F) \sqrt {f \tan (d+e x)}}+\frac {2 i f F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {i b c \log (F)}{2 e},1-\frac {i b c \log (F)}{2 e},-e^{2 i (d+e x)}\right ) \sqrt {g \tan (d+e x)}}{b c \log (F) \sqrt {f \tan (d+e x)}} \] Output:
-I*f*F^(c*(b*x+a))*(g*tan(e*x+d))^(1/2)/b/c/ln(F)/(f*tan(e*x+d))^(1/2)+2*I *f*F^(c*(b*x+a))*hypergeom([1, -1/2*I*b*c*ln(F)/e],[1-1/2*I*b*c*ln(F)/e],- exp(2*I*(e*x+d)))*(g*tan(e*x+d))^(1/2)/b/c/ln(F)/(f*tan(e*x+d))^(1/2)
Time = 0.80 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.68 \[ \int F^{c (a+b x)} \sqrt {f \tan (d+e x)} \sqrt {g \tan (d+e x)} \, dx=\frac {i f F^{c (a+b x)} \left (-1+2 \operatorname {Hypergeometric2F1}\left (1,-\frac {i b c \log (F)}{2 e},1-\frac {i b c \log (F)}{2 e},-e^{2 i (d+e x)}\right )\right ) \sqrt {g \tan (d+e x)}}{b c \log (F) \sqrt {f \tan (d+e x)}} \] Input:
Integrate[F^(c*(a + b*x))*Sqrt[f*Tan[d + e*x]]*Sqrt[g*Tan[d + e*x]],x]
Output:
(I*f*F^(c*(a + b*x))*(-1 + 2*Hypergeometric2F1[1, ((-1/2*I)*b*c*Log[F])/e, 1 - ((I/2)*b*c*Log[F])/e, -E^((2*I)*(d + e*x))])*Sqrt[g*Tan[d + e*x]])/(b *c*Log[F]*Sqrt[f*Tan[d + e*x]])
Time = 0.33 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.82, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {2031, 4942, 2009}
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.
\(\displaystyle \int F^{c (a+b x)} \sqrt {f \tan (d+e x)} \sqrt {g \tan (d+e x)} \, dx\) |
\(\Big \downarrow \) 2031 |
\(\displaystyle \frac {f \sqrt {g \tan (d+e x)} \int F^{c (a+b x)} \tan (d+e x)dx}{\sqrt {f \tan (d+e x)}}\) |
\(\Big \downarrow \) 4942 |
\(\displaystyle \frac {i f \sqrt {g \tan (d+e x)} \int \left (\frac {2 F^{c (a+b x)}}{1+e^{2 i (d+e x)}}-F^{c (a+b x)}\right )dx}{\sqrt {f \tan (d+e x)}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {i f \sqrt {g \tan (d+e x)} \left (-\frac {F^{c (a+b x)}}{b c \log (F)}+\frac {2 F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {i b c \log (F)}{2 e},1-\frac {i b c \log (F)}{2 e},-e^{2 i (d+e x)}\right )}{b c \log (F)}\right )}{\sqrt {f \tan (d+e x)}}\) |
Input:
Int[F^(c*(a + b*x))*Sqrt[f*Tan[d + e*x]]*Sqrt[g*Tan[d + e*x]],x]
Output:
(I*f*(-(F^(c*(a + b*x))/(b*c*Log[F])) + (2*F^(c*(a + b*x))*Hypergeometric2 F1[1, ((-1/2*I)*b*c*Log[F])/e, 1 - ((I/2)*b*c*Log[F])/e, -E^((2*I)*(d + e* x))])/(b*c*Log[F]))*Sqrt[g*Tan[d + e*x]])/Sqrt[f*Tan[d + e*x]]
Int[(Fx_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Simp[a^(m + 1/ 2)*b^(n - 1/2)*(Sqrt[b*v]/Sqrt[a*v]) Int[v^(m + n)*Fx, x], x] /; FreeQ[{a , b, m}, x] && !IntegerQ[m] && IGtQ[n + 1/2, 0] && IntegerQ[m + n]
Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Tan[(d_.) + (e_.)*(x_)]^(n_.), x_Symb ol] :> Simp[I^n Int[ExpandIntegrand[F^(c*(a + b*x))*((1 - E^(2*I*(d + e*x )))^n/(1 + E^(2*I*(d + e*x)))^n), x], x], x] /; FreeQ[{F, a, b, c, d, e}, x ] && IntegerQ[n]
\[\int F^{c \left (b x +a \right )} \sqrt {f \tan \left (e x +d \right )}\, \sqrt {g \tan \left (e x +d \right )}d x\]
Input:
int(F^(c*(b*x+a))*(f*tan(e*x+d))^(1/2)*(g*tan(e*x+d))^(1/2),x)
Output:
int(F^(c*(b*x+a))*(f*tan(e*x+d))^(1/2)*(g*tan(e*x+d))^(1/2),x)
\[ \int F^{c (a+b x)} \sqrt {f \tan (d+e x)} \sqrt {g \tan (d+e x)} \, dx=\int { \sqrt {f \tan \left (e x + d\right )} \sqrt {g \tan \left (e x + d\right )} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(f*tan(e*x+d))^(1/2)*(g*tan(e*x+d))^(1/2),x, algor ithm="fricas")
Output:
integral(sqrt(f*tan(e*x + d))*sqrt(g*tan(e*x + d))*F^(b*c*x + a*c), x)
\[ \int F^{c (a+b x)} \sqrt {f \tan (d+e x)} \sqrt {g \tan (d+e x)} \, dx=\int F^{c \left (a + b x\right )} \sqrt {f \tan {\left (d + e x \right )}} \sqrt {g \tan {\left (d + e x \right )}}\, dx \] Input:
integrate(F**(c*(b*x+a))*(f*tan(e*x+d))**(1/2)*(g*tan(e*x+d))**(1/2),x)
Output:
Integral(F**(c*(a + b*x))*sqrt(f*tan(d + e*x))*sqrt(g*tan(d + e*x)), x)
\[ \int F^{c (a+b x)} \sqrt {f \tan (d+e x)} \sqrt {g \tan (d+e x)} \, dx=\int { \sqrt {f \tan \left (e x + d\right )} \sqrt {g \tan \left (e x + d\right )} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(f*tan(e*x+d))^(1/2)*(g*tan(e*x+d))^(1/2),x, algor ithm="maxima")
Output:
integrate(sqrt(f*tan(e*x + d))*sqrt(g*tan(e*x + d))*F^((b*x + a)*c), x)
\[ \int F^{c (a+b x)} \sqrt {f \tan (d+e x)} \sqrt {g \tan (d+e x)} \, dx=\int { \sqrt {f \tan \left (e x + d\right )} \sqrt {g \tan \left (e x + d\right )} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(f*tan(e*x+d))^(1/2)*(g*tan(e*x+d))^(1/2),x, algor ithm="giac")
Output:
integrate(sqrt(f*tan(e*x + d))*sqrt(g*tan(e*x + d))*F^((b*x + a)*c), x)
Timed out. \[ \int F^{c (a+b x)} \sqrt {f \tan (d+e x)} \sqrt {g \tan (d+e x)} \, dx=\int F^{c\,\left (a+b\,x\right )}\,\sqrt {f\,\mathrm {tan}\left (d+e\,x\right )}\,\sqrt {g\,\mathrm {tan}\left (d+e\,x\right )} \,d x \] Input:
int(F^(c*(a + b*x))*(f*tan(d + e*x))^(1/2)*(g*tan(d + e*x))^(1/2),x)
Output:
int(F^(c*(a + b*x))*(f*tan(d + e*x))^(1/2)*(g*tan(d + e*x))^(1/2), x)
\[ \int F^{c (a+b x)} \sqrt {f \tan (d+e x)} \sqrt {g \tan (d+e x)} \, dx=\sqrt {g}\, f^{a c +\frac {1}{2}} \left (\int f^{b c x} \tan \left (e x +d \right )d x \right ) \] Input:
int(F^(c*(b*x+a))*(f*tan(e*x+d))^(1/2)*(g*tan(e*x+d))^(1/2),x)
Output:
sqrt(g)*f**((2*a*c + 1)/2)*int(f**(b*c*x)*tan(d + e*x),x)