\(\int F^{c (a+b x)} \cos (d+e x) \cot ^2(d+e x) \, dx\) [106]

Optimal result

Integrand size = 24, antiderivative size = 210 \[ \int F^{c (a+b x)} \cos (d+e x) \cot ^2(d+e x) \, dx=\frac {2 i e^{-i (d+e x)} F^{c (a+b x)}}{e \left (1-e^{2 i (d+e x)}\right )}+\frac {5 e^{-i (d+e x)} F^{c (a+b x)}}{2 (i e-b c \log (F))}+\frac {2 b c e^{-i (d+e x)} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {-e-i b c \log (F)}{2 e},\frac {e-i b c \log (F)}{2 e},e^{2 i (d+e x)}\right ) \log (F)}{e (e+i b c \log (F))}-\frac {e^{i (d+e x)} F^{c (a+b x)}}{2 (i e+b c \log (F))} \] Output:

2*I*F^(c*(b*x+a))/e/exp(I*(e*x+d))/(1-exp(2*I*(e*x+d)))+5/2*F^(c*(b*x+a))/ 
exp(I*(e*x+d))/(I*e-b*c*ln(F))+2*b*c*F^(c*(b*x+a))*hypergeom([1, 1/2*(-e-I 
*b*c*ln(F))/e],[1/2*(e-I*b*c*ln(F))/e],exp(2*I*(e*x+d)))*ln(F)/e/exp(I*(e* 
x+d))/(e+I*b*c*ln(F))-exp(I*(e*x+d))*F^(c*(b*x+a))/(2*I*e+2*b*c*ln(F))
 

Mathematica [A] (verified)

Time = 0.98 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.91 \[ \int F^{c (a+b x)} \cos (d+e x) \cot ^2(d+e x) \, dx=\frac {F^{c \left (a-\frac {b d}{e}\right )} \csc (d+e x) \left (-4 b c e^{\frac {(d+e x) (i e+b c \log (F))}{e}} \operatorname {Hypergeometric2F1}\left (1,\frac {e-i b c \log (F)}{2 e},\frac {3}{2}-\frac {i b c \log (F)}{2 e},e^{2 i (d+e x)}\right ) \log (F) (e+i b c \log (F)) \sin (d+e x)+F^{\frac {b c (d+e x)}{e}} \left (-3 e^2+e^2 \cos (2 (d+e x))-2 b^2 c^2 \log ^2(F)-b c e \log (F) \sin (2 (d+e x))\right )\right )}{2 \left (e^3+b^2 c^2 e \log ^2(F)\right )} \] Input:

Integrate[F^(c*(a + b*x))*Cos[d + e*x]*Cot[d + e*x]^2,x]
 

Output:

(F^(c*(a - (b*d)/e))*Csc[d + e*x]*(-4*b*c*E^(((d + e*x)*(I*e + b*c*Log[F]) 
)/e)*Hypergeometric2F1[1, (e - I*b*c*Log[F])/(2*e), 3/2 - ((I/2)*b*c*Log[F 
])/e, E^((2*I)*(d + e*x))]*Log[F]*(e + I*b*c*Log[F])*Sin[d + e*x] + F^((b* 
c*(d + e*x))/e)*(-3*e^2 + e^2*Cos[2*(d + e*x)] - 2*b^2*c^2*Log[F]^2 - b*c* 
e*Log[F]*Sin[2*(d + e*x)])))/(2*(e^3 + b^2*c^2*e*Log[F]^2))
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.26, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4974, 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.

\(\displaystyle \int \cos (d+e x) \cot ^2(d+e x) F^{c (a+b x)} \, dx\)

\(\Big \downarrow \) 4974

\(\displaystyle \int \left (-\frac {5}{2} e^{-i d-i e x} F^{a c+b c x}-\frac {1}{2} e^{2 i (d+e x)-i d-i e x} F^{a c+b c x}-\frac {6 e^{-i d-i e x} F^{a c+b c x}}{-1+e^{2 i (d+e x)}}-\frac {4 e^{-i d-i e x} F^{a c+b c x}}{\left (-1+e^{2 i (d+e x)}\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {6 e^{-i d-i e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (1,-\frac {e+i b c \log (F)}{2 e},\frac {e-i b c \log (F)}{2 e},e^{2 i (d+e x)}\right )}{-b c \log (F)+i e}+\frac {4 e^{-i d-i e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (2,-\frac {e+i b c \log (F)}{2 e},\frac {e-i b c \log (F)}{2 e},e^{2 i (d+e x)}\right )}{-b c \log (F)+i e}+\frac {5 F^{a c} e^{-x (-b c \log (F)+i e)-i d}}{2 (-b c \log (F)+i e)}-\frac {F^{a c} e^{x (b c \log (F)+i e)+i d}}{2 (b c \log (F)+i e)}\)

Input:

Int[F^(c*(a + b*x))*Cos[d + e*x]*Cot[d + e*x]^2,x]
 

Output:

(5*E^((-I)*d - x*(I*e - b*c*Log[F]))*F^(a*c))/(2*(I*e - b*c*Log[F])) - (6* 
E^((-I)*d - I*e*x)*F^(a*c + b*c*x)*Hypergeometric2F1[1, -1/2*(e + I*b*c*Lo 
g[F])/e, (e - I*b*c*Log[F])/(2*e), E^((2*I)*(d + e*x))])/(I*e - b*c*Log[F] 
) + (4*E^((-I)*d - I*e*x)*F^(a*c + b*c*x)*Hypergeometric2F1[2, -1/2*(e + I 
*b*c*Log[F])/e, (e - I*b*c*Log[F])/(2*e), E^((2*I)*(d + e*x))])/(I*e - b*c 
*Log[F]) - (E^(I*d + x*(I*e + b*c*Log[F]))*F^(a*c))/(2*(I*e + b*c*Log[F]))
 

Defintions of rubi rules used

rule 2009

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

rule 4974

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*(G_)[(d_.) + (e_.)*(x_)]^(m_.)*(H_)[( 
d_.) + (e_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigToExp[F^(c*(a + b*x)), 
 G[d + e*x]^m*H[d + e*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e}, x] && IGtQ[ 
m, 0] && IGtQ[n, 0] && TrigQ[G] && TrigQ[H]
 

Maple [F]

\[\int F^{c \left (b x +a \right )} \cos \left (e x +d \right ) \cot \left (e x +d \right )^{2}d x\]

Input:

int(F^(c*(b*x+a))*cos(e*x+d)*cot(e*x+d)^2,x)
 

Output:

int(F^(c*(b*x+a))*cos(e*x+d)*cot(e*x+d)^2,x)
 

Fricas [F]

\[ \int F^{c (a+b x)} \cos (d+e x) \cot ^2(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \cos \left (e x + d\right ) \cot \left (e x + d\right )^{2} \,d x } \] Input:

integrate(F^(c*(b*x+a))*cos(e*x+d)*cot(e*x+d)^2,x, algorithm="fricas")
 

Output:

integral(F^(b*c*x + a*c)*cos(e*x + d)*cot(e*x + d)^2, x)
 

Sympy [F]

\[ \int F^{c (a+b x)} \cos (d+e x) \cot ^2(d+e x) \, dx=\int F^{c \left (a + b x\right )} \cos {\left (d + e x \right )} \cot ^{2}{\left (d + e x \right )}\, dx \] Input:

integrate(F**(c*(b*x+a))*cos(e*x+d)*cot(e*x+d)**2,x)
 

Output:

Integral(F**(c*(a + b*x))*cos(d + e*x)*cot(d + e*x)**2, x)
 

Maxima [F]

\[ \int F^{c (a+b x)} \cos (d+e x) \cot ^2(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \cos \left (e x + d\right ) \cot \left (e x + d\right )^{2} \,d x } \] Input:

integrate(F^(c*(b*x+a))*cos(e*x+d)*cot(e*x+d)^2,x, algorithm="maxima")
 

Output:

-1/2*((F^(a*c)*b^5*c^5*log(F)^5 - 254*F^(a*c)*b^3*c^3*e^2*log(F)^3 + 705*F 
^(a*c)*b*c*e^4*log(F))*F^(b*c*x)*cos(e*x + d) + 3*(11*F^(a*c)*b^4*c^4*e*lo 
g(F)^4 - 234*F^(a*c)*b^2*c^2*e^3*log(F)^2 + 75*F^(a*c)*e^5)*F^(b*c*x)*sin( 
e*x + d) + ((F^(a*c)*b^5*c^5*log(F)^5 + 34*F^(a*c)*b^3*c^3*e^2*log(F)^3 + 
225*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x)*cos(5*e*x + 5*d) - 2*(F^(a*c)*b^5*c^ 
5*log(F)^5 + 34*F^(a*c)*b^3*c^3*e^2*log(F)^3 + 225*F^(a*c)*b*c*e^4*log(F)) 
*F^(b*c*x)*cos(3*e*x + 3*d) + (F^(a*c)*b^5*c^5*log(F)^5 + 34*F^(a*c)*b^3*c 
^3*e^2*log(F)^3 + 225*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x)*cos(e*x + d) - (F^ 
(a*c)*b^4*c^4*e*log(F)^4 + 34*F^(a*c)*b^2*c^2*e^3*log(F)^2 + 225*F^(a*c)*e 
^5)*F^(b*c*x)*sin(5*e*x + 5*d) + 2*(F^(a*c)*b^4*c^4*e*log(F)^4 + 34*F^(a*c 
)*b^2*c^2*e^3*log(F)^2 + 225*F^(a*c)*e^5)*F^(b*c*x)*sin(3*e*x + 3*d) - (F^ 
(a*c)*b^4*c^4*e*log(F)^4 + 34*F^(a*c)*b^2*c^2*e^3*log(F)^2 + 225*F^(a*c)*e 
^5)*F^(b*c*x)*sin(e*x + d))*cos(6*e*x + 6*d) + (3*(F^(a*c)*b^5*c^5*log(F)^ 
5 + 34*F^(a*c)*b^3*c^3*e^2*log(F)^3 + 225*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x 
)*cos(4*e*x + 4*d) + 3*(F^(a*c)*b^5*c^5*log(F)^5 + 2*F^(a*c)*b^3*c^3*e^2*l 
og(F)^3 - 575*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x)*cos(2*e*x + 2*d) - 7*(F^(a 
*c)*b^4*c^4*e*log(F)^4 + 34*F^(a*c)*b^2*c^2*e^3*log(F)^2 + 225*F^(a*c)*e^5 
)*F^(b*c*x)*sin(4*e*x + 4*d) - (25*F^(a*c)*b^4*c^4*e*log(F)^4 + 562*F^(a*c 
)*b^2*c^2*e^3*log(F)^2 - 1575*F^(a*c)*e^5)*F^(b*c*x)*sin(2*e*x + 2*d) + (F 
^(a*c)*b^5*c^5*log(F)^5 - 254*F^(a*c)*b^3*c^3*e^2*log(F)^3 + 705*F^(a*c...
 

Giac [F]

\[ \int F^{c (a+b x)} \cos (d+e x) \cot ^2(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \cos \left (e x + d\right ) \cot \left (e x + d\right )^{2} \,d x } \] Input:

integrate(F^(c*(b*x+a))*cos(e*x+d)*cot(e*x+d)^2,x, algorithm="giac")
 

Output:

integrate(F^((b*x + a)*c)*cos(e*x + d)*cot(e*x + d)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int F^{c (a+b x)} \cos (d+e x) \cot ^2(d+e x) \, dx=\int F^{c\,\left (a+b\,x\right )}\,\cos \left (d+e\,x\right )\,{\mathrm {cot}\left (d+e\,x\right )}^2 \,d x \] Input:

int(F^(c*(a + b*x))*cos(d + e*x)*cot(d + e*x)^2,x)
 

Output:

int(F^(c*(a + b*x))*cos(d + e*x)*cot(d + e*x)^2, x)
 

Reduce [F]

\[ \int F^{c (a+b x)} \cos (d+e x) \cot ^2(d+e x) \, dx=f^{a c} \left (\int f^{b c x} \cos \left (e x +d \right ) \cot \left (e x +d \right )^{2}d x \right ) \] Input:

int(F^(c*(b*x+a))*cos(e*x+d)*cot(e*x+d)^2,x)
 

Output:

f**(a*c)*int(f**(b*c*x)*cos(d + e*x)*cot(d + e*x)**2,x)