\(\int F^{c (a+b x)} \cot ^2(d+e x) \csc (d+e x) \, dx\) [100]

Optimal result

Integrand size = 24, antiderivative size = 193 \[ \int F^{c (a+b x)} \cot ^2(d+e x) \csc (d+e x) \, dx=\frac {2 e^{i (d+e x)} F^{c (a+b x)}}{e \left (1-e^{2 i (d+e x)}\right )^2}-\frac {e^{i (d+e x)} F^{c (a+b x)} (e-i b c \log (F))}{e^2 \left (1-e^{2 i (d+e x)}\right )}+\frac {e^{i (d+e x)} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {e-i b c \log (F)}{2 e},\frac {1}{2} \left (3-\frac {i b c \log (F)}{e}\right ),e^{2 i (d+e x)}\right ) (e-b c \log (F)) (e+b c \log (F))}{e^2 (e-i b c \log (F))} \] Output:

2*exp(I*(e*x+d))*F^(c*(b*x+a))/e/(1-exp(2*I*(e*x+d)))^2-exp(I*(e*x+d))*F^( 
c*(b*x+a))*(e-I*b*c*ln(F))/e^2/(1-exp(2*I*(e*x+d)))+exp(I*(e*x+d))*F^(c*(b 
*x+a))*hypergeom([1, 1/2*(e-I*b*c*ln(F))/e],[3/2-1/2*I*b*c*ln(F)/e],exp(2* 
I*(e*x+d)))*(e-b*c*ln(F))*(e+b*c*ln(F))/e^2/(e-I*b*c*ln(F))
 

Mathematica [A] (verified)

Time = 2.59 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.85 \[ \int F^{c (a+b x)} \cot ^2(d+e x) \csc (d+e x) \, dx=\frac {F^{c \left (a-\frac {b d}{e}\right )} \left (-4 F^{\frac {b c (d+e x)}{e}} \csc (d+e x) (e-i b c \log (F)) (e \cot (d+e x)+b c \log (F))+8 e^{\frac {(d+e x) (i e+b c \log (F))}{e}} \operatorname {Hypergeometric2F1}\left (1,\frac {e-i b c \log (F)}{2 e},\frac {3}{2}-\frac {i b c \log (F)}{2 e},e^{2 i (d+e x)}\right ) \left (e^2-b^2 c^2 \log ^2(F)\right )\right )}{8 e^2 (e-i b c \log (F))} \] Input:

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

Output:

(F^(c*(a - (b*d)/e))*(-4*F^((b*c*(d + e*x))/e)*Csc[d + e*x]*(e - I*b*c*Log 
[F])*(e*Cot[d + e*x] + b*c*Log[F]) + 8*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))]*(e^2 - b^2*c^2*Log[F]^2)))/(8*e^2*(e - I*b*c*Log[F 
]))
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.33, 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.

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

\(\Big \downarrow \) 4974

\(\displaystyle \int \left (-\frac {2 i e^{i d+i e x} F^{a c+b c x}}{-1+e^{2 i (d+e x)}}-\frac {8 i e^{i d+i e x} F^{a c+b c x}}{\left (-1+e^{2 i (d+e x)}\right )^2}-\frac {8 i e^{i d+i e x} F^{a c+b c x}}{\left (-1+e^{2 i (d+e x)}\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 e^{i d+i e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (1,\frac {e-i b c \log (F)}{2 e},\frac {1}{2} \left (3-\frac {i b c \log (F)}{e}\right ),e^{2 i (d+e x)}\right )}{e-i b c \log (F)}-\frac {8 e^{i d+i e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (2,\frac {e-i b c \log (F)}{2 e},\frac {1}{2} \left (3-\frac {i b c \log (F)}{e}\right ),e^{2 i (d+e x)}\right )}{e-i b c \log (F)}+\frac {8 e^{i d+i e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (3,\frac {e-i b c \log (F)}{2 e},\frac {1}{2} \left (3-\frac {i b c \log (F)}{e}\right ),e^{2 i (d+e x)}\right )}{e-i b c \log (F)}\)

Input:

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

Output:

(2*E^(I*d + I*e*x)*F^(a*c + b*c*x)*Hypergeometric2F1[1, (e - I*b*c*Log[F]) 
/(2*e), (3 - (I*b*c*Log[F])/e)/2, E^((2*I)*(d + e*x))])/(e - I*b*c*Log[F]) 
 - (8*E^(I*d + I*e*x)*F^(a*c + b*c*x)*Hypergeometric2F1[2, (e - I*b*c*Log[ 
F])/(2*e), (3 - (I*b*c*Log[F])/e)/2, E^((2*I)*(d + e*x))])/(e - I*b*c*Log[ 
F]) + (8*E^(I*d + I*e*x)*F^(a*c + b*c*x)*Hypergeometric2F1[3, (e - I*b*c*L 
og[F])/(2*e), (3 - (I*b*c*Log[F])/e)/2, E^((2*I)*(d + e*x))])/(e - I*b*c*L 
og[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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-2*((23*F^(a*c)*b^4*c^4*e*log(F)^4 - 418*F^(a*c)*b^2*c^2*e^3*log(F)^2 + 13 
5*F^(a*c)*e^5)*F^(b*c*x)*cos(e*x + d) + (F^(a*c)*b^5*c^5*log(F)^5 - 158*F^ 
(a*c)*b^3*c^3*e^2*log(F)^3 + 417*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x)*sin(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 + 2 
25*F^(a*c)*e^5)*F^(b*c*x)*cos(5*e*x + 5*d) + 12*(F^(a*c)*b^4*c^4*e*log(F)^ 
4 + 24*F^(a*c)*b^2*c^2*e^3*log(F)^2 - 25*F^(a*c)*e^5)*F^(b*c*x)*cos(3*e*x 
+ 3*d) + (23*F^(a*c)*b^4*c^4*e*log(F)^4 - 418*F^(a*c)*b^2*c^2*e^3*log(F)^2 
 + 135*F^(a*c)*e^5)*F^(b*c*x)*cos(e*x + d) + (F^(a*c)*b^5*c^5*log(F)^5 + 3 
4*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)*sin 
(5*e*x + 5*d) + 2*(F^(a*c)*b^5*c^5*log(F)^5 + 14*F^(a*c)*b^3*c^3*e^2*log(F 
)^3 - 275*F^(a*c)*b*c*e^4*log(F))*F^(b*c*x)*sin(3*e*x + 3*d) + (F^(a*c)*b^ 
5*c^5*log(F)^5 - 158*F^(a*c)*b^3*c^3*e^2*log(F)^3 + 417*F^(a*c)*b*c*e^4*lo 
g(F))*F^(b*c*x)*sin(e*x + d))*cos(6*e*x + 6*d) + (3*(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)*cos(4 
*e*x + 4*d) - 3*(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)*cos(2*e*x + 2*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)*sin(4*e*x + 4*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)*sin(2*e*x + 2*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*...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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