\(\int F^{c (a+b x)} \cot (d+e x) \csc (d+e x) \, dx\) [92]

Optimal result

Integrand size = 22, antiderivative size = 133 \[ \int F^{c (a+b x)} \cot (d+e x) \csc (d+e x) \, dx=\frac {2 i e^{i (d+e x)} F^{c (a+b x)}}{e \left (1-e^{2 i (d+e x)}\right )}-\frac {2 b c e^{i (d+e x)} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {e-i b c \log (F)}{2 e},\frac {1}{2} \left (3-\frac {i b c \log (F)}{e}\right ),e^{2 i (d+e x)}\right ) \log (F)}{e (e-i b c \log (F))} \] Output:

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

Mathematica [A] (verified)

Time = 3.95 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.09 \[ \int F^{c (a+b x)} \cot (d+e x) \csc (d+e x) \, dx=\frac {i F^{c (a+b x)} \csc \left (\frac {1}{2} (d+e x)\right ) \sec \left (\frac {1}{2} (d+e x)\right ) \left (i+\operatorname {Hypergeometric2F1}\left (1,-\frac {i b c \log (F)}{e},1-\frac {i b c \log (F)}{e},-\cos (d+e x)-i \sin (d+e x)\right ) \sin (d+e x)-\operatorname {Hypergeometric2F1}\left (1,-\frac {i b c \log (F)}{e},1-\frac {i b c \log (F)}{e},\cos (d+e x)+i \sin (d+e x)\right ) \sin (d+e x)\right )}{2 e} \] Input:

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

Output:

((I/2)*F^(c*(a + b*x))*Csc[(d + e*x)/2]*Sec[(d + e*x)/2]*(I + Hypergeometr 
ic2F1[1, ((-I)*b*c*Log[F])/e, 1 - (I*b*c*Log[F])/e, -Cos[d + e*x] - I*Sin[ 
d + e*x]]*Sin[d + e*x] - Hypergeometric2F1[1, ((-I)*b*c*Log[F])/e, 1 - (I* 
b*c*Log[F])/e, Cos[d + e*x] + I*Sin[d + e*x]]*Sin[d + e*x]))/e
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.30, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.

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

\(\Big \downarrow \) 4974

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 e^{i d+i e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (1,\frac {e-i b c \log (F)}{2 e},\frac {1}{2} \left (3-\frac {i b c \log (F)}{e}\right ),e^{2 i (d+e x)}\right )}{b c \log (F)+i e}-\frac {4 e^{i d+i e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (2,\frac {e-i b c \log (F)}{2 e},\frac {1}{2} \left (3-\frac {i b c \log (F)}{e}\right ),e^{2 i (d+e x)}\right )}{b c \log (F)+i e}\)

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-2*((F^(a*c)*b^3*c^3*log(F)^3 - 15*F^(a*c)*b*c*e^2*log(F))*F^(b*c*x)*cos(e 
*x + d) - (7*F^(a*c)*b^2*c^2*e*log(F)^2 - 9*F^(a*c)*e^3)*F^(b*c*x)*sin(e*x 
 + d) + ((F^(a*c)*b^3*c^3*log(F)^3 + 9*F^(a*c)*b*c*e^2*log(F))*F^(b*c*x)*c 
os(3*e*x + 3*d) + (F^(a*c)*b^3*c^3*log(F)^3 - 15*F^(a*c)*b*c*e^2*log(F))*F 
^(b*c*x)*cos(e*x + d) - (F^(a*c)*b^2*c^2*e*log(F)^2 + 9*F^(a*c)*e^3)*F^(b* 
c*x)*sin(3*e*x + 3*d) - (7*F^(a*c)*b^2*c^2*e*log(F)^2 - 9*F^(a*c)*e^3)*F^( 
b*c*x)*sin(e*x + d))*cos(4*e*x + 4*d) - (2*(F^(a*c)*b^3*c^3*log(F)^3 + 9*F 
^(a*c)*b*c*e^2*log(F))*F^(b*c*x)*cos(2*e*x + 2*d) + 2*(F^(a*c)*b^2*c^2*e*l 
og(F)^2 + 9*F^(a*c)*e^3)*F^(b*c*x)*sin(2*e*x + 2*d) - (F^(a*c)*b^3*c^3*log 
(F)^3 + 9*F^(a*c)*b*c*e^2*log(F))*F^(b*c*x))*cos(3*e*x + 3*d) - 2*((F^(a*c 
)*b^3*c^3*log(F)^3 - 15*F^(a*c)*b*c*e^2*log(F))*F^(b*c*x)*cos(e*x + d) - ( 
7*F^(a*c)*b^2*c^2*e*log(F)^2 - 9*F^(a*c)*e^3)*F^(b*c*x)*sin(e*x + d))*cos( 
2*e*x + 2*d) - 8*(F^(a*c)*b^5*c^5*e*log(F)^5 + 10*F^(a*c)*b^3*c^3*e^3*log( 
F)^3 + 9*F^(a*c)*b*c*e^5*log(F) + (F^(a*c)*b^5*c^5*e*log(F)^5 + 10*F^(a*c) 
*b^3*c^3*e^3*log(F)^3 + 9*F^(a*c)*b*c*e^5*log(F))*cos(4*e*x + 4*d)^2 + 4*( 
F^(a*c)*b^5*c^5*e*log(F)^5 + 10*F^(a*c)*b^3*c^3*e^3*log(F)^3 + 9*F^(a*c)*b 
*c*e^5*log(F))*cos(2*e*x + 2*d)^2 + (F^(a*c)*b^5*c^5*e*log(F)^5 + 10*F^(a* 
c)*b^3*c^3*e^3*log(F)^3 + 9*F^(a*c)*b*c*e^5*log(F))*sin(4*e*x + 4*d)^2 - 4 
*(F^(a*c)*b^5*c^5*e*log(F)^5 + 10*F^(a*c)*b^3*c^3*e^3*log(F)^3 + 9*F^(a*c) 
*b*c*e^5*log(F))*sin(4*e*x + 4*d)*sin(2*e*x + 2*d) + 4*(F^(a*c)*b^5*c^5...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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