\(\int \frac {F^{a+b x} \cos (c+d x)}{e-e \sin (c+d x)} \, dx\) [66]

Optimal result

Integrand size = 27, antiderivative size = 82 \[ \int \frac {F^{a+b x} \cos (c+d x)}{e-e \sin (c+d x)} \, dx=-\frac {i F^{a+b x}}{b e \log (F)}+\frac {2 i F^{a+b x} \operatorname {Hypergeometric2F1}\left (1,-\frac {i b \log (F)}{d},1-\frac {i b \log (F)}{d},-i e^{i (c+d x)}\right )}{b e \log (F)} \] Output:

-I*F^(b*x+a)/b/e/ln(F)+2*I*F^(b*x+a)*hypergeom([1, -I*b*ln(F)/d],[1-I*b*ln 
(F)/d],-I*exp(I*(d*x+c)))/b/e/ln(F)
 

Mathematica [A] (verified)

Time = 3.19 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.78 \[ \int \frac {F^{a+b x} \cos (c+d x)}{e-e \sin (c+d x)} \, dx=\frac {i F^{a+b x} \left (-1+2 \operatorname {Hypergeometric2F1}\left (1,-\frac {i b \log (F)}{d},1-\frac {i b \log (F)}{d},-i e^{i (c+d x)}\right )\right )}{b e \log (F)} \] Input:

Integrate[(F^(a + b*x)*Cos[c + d*x])/(e - e*Sin[c + d*x]),x]
 

Output:

(I*F^(a + b*x)*(-1 + 2*Hypergeometric2F1[1, ((-I)*b*Log[F])/d, 1 - (I*b*Lo 
g[F])/d, (-I)*E^(I*(c + d*x))]))/(b*e*Log[F])
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {4962, 25, 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.

\(\displaystyle \int \frac {F^{a+b x} \cos (c+d x)}{e-e \sin (c+d x)} \, dx\)

\(\Big \downarrow \) 4962

\(\displaystyle -\frac {\int -F^{a+b x} \tan \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{e}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int F^{a+b x} \tan \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{e}\)

\(\Big \downarrow \) 4942

\(\displaystyle \frac {i \int \left (\frac {2 F^{a+b x}}{1+e^{\frac {1}{2} i (2 c+2 d x+\pi )}}-F^{a+b x}\right )dx}{e}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {i \left (-\frac {F^{a+b x}}{b \log (F)}+\frac {2 F^{a+b x} \operatorname {Hypergeometric2F1}\left (1,-\frac {i b \log (F)}{d},1-\frac {i b \log (F)}{d},-i e^{i (c+d x)}\right )}{b \log (F)}\right )}{e}\)

Input:

Int[(F^(a + b*x)*Cos[c + d*x])/(e - e*Sin[c + d*x]),x]
 

Output:

(I*(-(F^(a + b*x)/(b*Log[F])) + (2*F^(a + b*x)*Hypergeometric2F1[1, ((-I)* 
b*Log[F])/d, 1 - (I*b*Log[F])/d, (-I)*E^(I*(c + d*x))])/(b*Log[F])))/e
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 4942

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]
 

rule 4962

Int[Cos[(d_.) + (e_.)*(x_)]^(m_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*((f_) + 
 (g_.)*Sin[(d_.) + (e_.)*(x_)])^(n_.), x_Symbol] :> Simp[g^n   Int[F^(c*(a 
+ b*x))*Tan[f*(Pi/(4*g)) - d/2 - e*(x/2)]^m, x], x] /; FreeQ[{F, a, b, c, d 
, e, f, g}, x] && EqQ[f^2 - g^2, 0] && IntegersQ[m, n] && EqQ[m + n, 0]
 

Maple [F]

\[\int \frac {F^{b x +a} \cos \left (d x +c \right )}{e -e \sin \left (d x +c \right )}d x\]

Input:

int(F^(b*x+a)*cos(d*x+c)/(e-e*sin(d*x+c)),x)
 

Output:

int(F^(b*x+a)*cos(d*x+c)/(e-e*sin(d*x+c)),x)
 

Fricas [F]

\[ \int \frac {F^{a+b x} \cos (c+d x)}{e-e \sin (c+d x)} \, dx=\int { -\frac {F^{b x + a} \cos \left (d x + c\right )}{e \sin \left (d x + c\right ) - e} \,d x } \] Input:

integrate(F^(b*x+a)*cos(d*x+c)/(e-e*sin(d*x+c)),x, algorithm="fricas")
 

Output:

integral(-F^(b*x + a)*cos(d*x + c)/(e*sin(d*x + c) - e), x)
 

Sympy [F]

\[ \int \frac {F^{a+b x} \cos (c+d x)}{e-e \sin (c+d x)} \, dx=- \frac {\int \frac {F^{a + b x} \cos {\left (c + d x \right )}}{\sin {\left (c + d x \right )} - 1}\, dx}{e} \] Input:

integrate(F**(b*x+a)*cos(d*x+c)/(e-e*sin(d*x+c)),x)
 

Output:

-Integral(F**(a + b*x)*cos(c + d*x)/(sin(c + d*x) - 1), x)/e
 

Maxima [F]

\[ \int \frac {F^{a+b x} \cos (c+d x)}{e-e \sin (c+d x)} \, dx=\int { -\frac {F^{b x + a} \cos \left (d x + c\right )}{e \sin \left (d x + c\right ) - e} \,d x } \] Input:

integrate(F^(b*x+a)*cos(d*x+c)/(e-e*sin(d*x+c)),x, algorithm="maxima")
 

Output:

-(2*F^(b*x)*F^a*b*d*cos(d*x + c)*log(F) + 2*F^(b*x)*F^a*d^2*sin(d*x + c) - 
 (F^a*b^2*log(F)^2 + F^a*d^2)*F^(b*x)*cos(d*x + c)^2 - (F^a*b^2*log(F)^2 + 
 F^a*d^2)*F^(b*x)*sin(d*x + c)^2 + (F^a*b^2*log(F)^2 - F^a*d^2)*F^(b*x) + 
2*((F^a*b^3*d*log(F)^3 + F^a*b*d^3*log(F))*e*cos(d*x + c)^2 + (F^a*b^3*d*l 
og(F)^3 + F^a*b*d^3*log(F))*e*sin(d*x + c)^2 - 2*(F^a*b^3*d*log(F)^3 + F^a 
*b*d^3*log(F))*e*sin(d*x + c) + (F^a*b^3*d*log(F)^3 + F^a*b*d^3*log(F))*e) 
*integrate(-(2*F^(b*x)*b*cos(d*x + c)*log(F) - F^(b*x)*b*log(F)*sin(2*d*x 
+ 2*c) + F^(b*x)*d*cos(2*d*x + 2*c) + 2*F^(b*x)*d*sin(d*x + c) - F^(b*x)*d 
)/((b^2*log(F)^2 + d^2)*e*cos(2*d*x + 2*c)^2 + 4*(b^2*log(F)^2 + d^2)*e*co 
s(d*x + c)^2 - 4*(b^2*log(F)^2 + d^2)*e*cos(d*x + c)*sin(2*d*x + 2*c) + (b 
^2*log(F)^2 + d^2)*e*sin(2*d*x + 2*c)^2 + 4*(b^2*log(F)^2 + d^2)*e*sin(d*x 
 + c)^2 - 4*(b^2*log(F)^2 + d^2)*e*sin(d*x + c) + (b^2*log(F)^2 + d^2)*e + 
 2*(2*(b^2*log(F)^2 + d^2)*e*sin(d*x + c) - (b^2*log(F)^2 + d^2)*e)*cos(2* 
d*x + 2*c)), x))/((b^3*log(F)^3 + b*d^2*log(F))*e*cos(d*x + c)^2 + (b^3*lo 
g(F)^3 + b*d^2*log(F))*e*sin(d*x + c)^2 - 2*(b^3*log(F)^3 + b*d^2*log(F))* 
e*sin(d*x + c) + (b^3*log(F)^3 + b*d^2*log(F))*e)
 

Giac [F]

\[ \int \frac {F^{a+b x} \cos (c+d x)}{e-e \sin (c+d x)} \, dx=\int { -\frac {F^{b x + a} \cos \left (d x + c\right )}{e \sin \left (d x + c\right ) - e} \,d x } \] Input:

integrate(F^(b*x+a)*cos(d*x+c)/(e-e*sin(d*x+c)),x, algorithm="giac")
 

Output:

integrate(-F^(b*x + a)*cos(d*x + c)/(e*sin(d*x + c) - e), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {F^{a+b x} \cos (c+d x)}{e-e \sin (c+d x)} \, dx=\int \frac {F^{a+b\,x}\,\cos \left (c+d\,x\right )}{e-e\,\sin \left (c+d\,x\right )} \,d x \] Input:

int((F^(a + b*x)*cos(c + d*x))/(e - e*sin(c + d*x)),x)
 

Output:

int((F^(a + b*x)*cos(c + d*x))/(e - e*sin(c + d*x)), x)
 

Reduce [F]

\[ \int \frac {F^{a+b x} \cos (c+d x)}{e-e \sin (c+d x)} \, dx=-\frac {f^{a} \left (\int \frac {f^{b x} \cos \left (d x +c \right )}{\sin \left (d x +c \right )-1}d x \right )}{e} \] Input:

int(F^(b*x+a)*cos(d*x+c)/(e-e*sin(d*x+c)),x)
 

Output:

( - f**a*int((f**(b*x)*cos(c + d*x))/(sin(c + d*x) - 1),x))/e