\(\int F^{c (a+b x)} \csc ^3(d+e x) \sec ^3(d+e x) \, dx\) [128]

Optimal result

Integrand size = 26, antiderivative size = 83 \[ \int F^{c (a+b x)} \csc ^3(d+e x) \sec ^3(d+e x) \, dx=\frac {64 e^{6 i (d+e x)} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (3,\frac {1}{4} \left (6-\frac {i b c \log (F)}{e}\right ),\frac {1}{4} \left (10-\frac {i b c \log (F)}{e}\right ),e^{4 i (d+e x)}\right )}{6 e-i b c \log (F)} \] Output:

64*exp(6*I*(e*x+d))*F^(c*(b*x+a))*hypergeom([3, 3/2-1/4*I*b*c*ln(F)/e],[5/ 
2-1/4*I*b*c*ln(F)/e],exp(4*I*(e*x+d)))/(6*e-I*b*c*ln(F))
 

Mathematica [A] (verified)

Time = 2.81 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.87 \[ \int F^{c (a+b x)} \csc ^3(d+e x) \sec ^3(d+e x) \, dx=-\frac {F^{c \left (a-\frac {b d}{e}\right )} \left (4 e^{\frac {(d+e x) (2 i e+b c \log (F))}{e}} \cot (d+e x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}-\frac {i b c \log (F)}{4 e},\frac {3}{2}-\frac {i b c \log (F)}{4 e},e^{4 i (d+e x)}\right ) (2 e+i b c \log (F))+F^{\frac {b c (d+e x)}{e}} \csc ^2(d+e x) (2 e \cot (2 (d+e x))+b c \log (F))\right ) \tan (d+e x)}{2 e^2} \] Input:

Integrate[F^(c*(a + b*x))*Csc[d + e*x]^3*Sec[d + e*x]^3,x]
 

Output:

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

Rubi [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(732\) vs. \(2(83)=166\).

Time = 1.44 (sec) , antiderivative size = 732, normalized size of antiderivative = 8.82, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, 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.

\(\displaystyle \int \csc ^3(d+e x) \sec ^3(d+e x) F^{c (a+b x)} \, dx\)

\(\Big \downarrow \) 4974

\(\displaystyle \int \left (-\frac {21 i e^{6 i d+6 i e x} F^{a c+b c x}}{2 \left (-1+e^{i d+i e x}\right )}+\frac {21 i e^{6 i d+6 i e x} F^{a c+b c x}}{2 \left (1+e^{i d+i e x}\right )}+\frac {12 i e^{6 i d+6 i e x} F^{a c+b c x}}{1+e^{2 i d+2 i e x}}+\frac {9 i e^{6 i d+6 i e x} F^{a c+b c x}}{2 \left (-1+e^{i d+i e x}\right )^2}+\frac {9 i e^{6 i d+6 i e x} F^{a c+b c x}}{2 \left (1+e^{i d+i e x}\right )^2}+\frac {12 i e^{6 i d+6 i e x} F^{a c+b c x}}{\left (1+e^{2 i d+2 i e x}\right )^2}-\frac {i e^{6 i d+6 i e x} F^{a c+b c x}}{\left (-1+e^{i d+i e x}\right )^3}+\frac {i e^{6 i d+6 i e x} F^{a c+b c x}}{\left (1+e^{i d+i e x}\right )^3}+\frac {8 i e^{6 i d+6 i e x} F^{a c+b c x}}{\left (1+e^{2 i d+2 i e x}\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {8 i F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (3,\frac {i b c \log (F)}{2 e},\frac {i b c \log (F)}{2 e}+1,-e^{-2 i (d+e x)}\right )}{b c \log (F)}+\frac {12 e^{6 i d+6 i e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (6-\frac {i b c \log (F)}{e}\right ),\frac {1}{2} \left (8-\frac {i b c \log (F)}{e}\right ),-e^{2 i (d+e x)}\right )}{6 e-i b c \log (F)}+\frac {21 e^{6 i d+6 i e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (1,6-\frac {i b c \log (F)}{e},7-\frac {i b c \log (F)}{e},-e^{i (d+e x)}\right )}{2 (6 e-i b c \log (F))}+\frac {21 e^{6 i d+6 i e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (1,6-\frac {i b c \log (F)}{e},7-\frac {i b c \log (F)}{e},e^{i (d+e x)}\right )}{2 (6 e-i b c \log (F))}+\frac {12 e^{6 i d+6 i e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (2,\frac {1}{2} \left (6-\frac {i b c \log (F)}{e}\right ),\frac {1}{2} \left (8-\frac {i b c \log (F)}{e}\right ),-e^{2 i (d+e x)}\right )}{6 e-i b c \log (F)}+\frac {9 e^{6 i d+6 i e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (2,6-\frac {i b c \log (F)}{e},7-\frac {i b c \log (F)}{e},-e^{i (d+e x)}\right )}{2 (6 e-i b c \log (F))}+\frac {9 e^{6 i d+6 i e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (2,6-\frac {i b c \log (F)}{e},7-\frac {i b c \log (F)}{e},e^{i (d+e x)}\right )}{2 (6 e-i b c \log (F))}+\frac {e^{6 i d+6 i e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (3,6-\frac {i b c \log (F)}{e},7-\frac {i b c \log (F)}{e},-e^{i (d+e x)}\right )}{6 e-i b c \log (F)}+\frac {e^{6 i d+6 i e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (3,6-\frac {i b c \log (F)}{e},7-\frac {i b c \log (F)}{e},e^{i (d+e x)}\right )}{6 e-i b c \log (F)}\)

Input:

Int[F^(c*(a + b*x))*Csc[d + e*x]^3*Sec[d + e*x]^3,x]
 

Output:

((8*I)*F^(a*c + b*c*x)*Hypergeometric2F1[3, ((I/2)*b*c*Log[F])/e, 1 + ((I/ 
2)*b*c*Log[F])/e, -E^((-2*I)*(d + e*x))])/(b*c*Log[F]) + (12*E^((6*I)*d + 
(6*I)*e*x)*F^(a*c + b*c*x)*Hypergeometric2F1[1, (6 - (I*b*c*Log[F])/e)/2, 
(8 - (I*b*c*Log[F])/e)/2, -E^((2*I)*(d + e*x))])/(6*e - I*b*c*Log[F]) + (2 
1*E^((6*I)*d + (6*I)*e*x)*F^(a*c + b*c*x)*Hypergeometric2F1[1, 6 - (I*b*c* 
Log[F])/e, 7 - (I*b*c*Log[F])/e, -E^(I*(d + e*x))])/(2*(6*e - I*b*c*Log[F] 
)) + (21*E^((6*I)*d + (6*I)*e*x)*F^(a*c + b*c*x)*Hypergeometric2F1[1, 6 - 
(I*b*c*Log[F])/e, 7 - (I*b*c*Log[F])/e, E^(I*(d + e*x))])/(2*(6*e - I*b*c* 
Log[F])) + (12*E^((6*I)*d + (6*I)*e*x)*F^(a*c + b*c*x)*Hypergeometric2F1[2 
, (6 - (I*b*c*Log[F])/e)/2, (8 - (I*b*c*Log[F])/e)/2, -E^((2*I)*(d + e*x)) 
])/(6*e - I*b*c*Log[F]) + (9*E^((6*I)*d + (6*I)*e*x)*F^(a*c + b*c*x)*Hyper 
geometric2F1[2, 6 - (I*b*c*Log[F])/e, 7 - (I*b*c*Log[F])/e, -E^(I*(d + e*x 
))])/(2*(6*e - I*b*c*Log[F])) + (9*E^((6*I)*d + (6*I)*e*x)*F^(a*c + b*c*x) 
*Hypergeometric2F1[2, 6 - (I*b*c*Log[F])/e, 7 - (I*b*c*Log[F])/e, E^(I*(d 
+ e*x))])/(2*(6*e - I*b*c*Log[F])) + (E^((6*I)*d + (6*I)*e*x)*F^(a*c + b*c 
*x)*Hypergeometric2F1[3, 6 - (I*b*c*Log[F])/e, 7 - (I*b*c*Log[F])/e, -E^(I 
*(d + e*x))])/(6*e - I*b*c*Log[F]) + (E^((6*I)*d + (6*I)*e*x)*F^(a*c + b*c 
*x)*Hypergeometric2F1[3, 6 - (I*b*c*Log[F])/e, 7 - (I*b*c*Log[F])/e, E^(I* 
(d + e*x))])/(6*e - I*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]

\[\int F^{c \left (b x +a \right )} \csc \left (e x +d \right )^{3} \sec \left (e x +d \right )^{3}d x\]

Input:

int(F^(c*(b*x+a))*csc(e*x+d)^3*sec(e*x+d)^3,x)
 

Output:

int(F^(c*(b*x+a))*csc(e*x+d)^3*sec(e*x+d)^3,x)
 

Fricas [F]

\[ \int F^{c (a+b x)} \csc ^3(d+e x) \sec ^3(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \csc \left (e x + d\right )^{3} \sec \left (e x + d\right )^{3} \,d x } \] Input:

integrate(F^(c*(b*x+a))*csc(e*x+d)^3*sec(e*x+d)^3,x, algorithm="fricas")
 

Output:

integral(F^(b*c*x + a*c)*csc(e*x + d)^3*sec(e*x + d)^3, x)
 

Sympy [F(-1)]

Timed out. \[ \int F^{c (a+b x)} \csc ^3(d+e x) \sec ^3(d+e x) \, dx=\text {Timed out} \] Input:

integrate(F**(c*(b*x+a))*csc(e*x+d)**3*sec(e*x+d)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int F^{c (a+b x)} \csc ^3(d+e x) \sec ^3(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \csc \left (e x + d\right )^{3} \sec \left (e x + d\right )^{3} \,d x } \] Input:

integrate(F^(c*(b*x+a))*csc(e*x+d)^3*sec(e*x+d)^3,x, algorithm="maxima")
 

Output:

64*(192*F^(b*c*x)*F^(a*c)*b*c*e^2*log(F)*sin(2*e*x + 2*d) - 12*(F^(a*c)*b^ 
2*c^2*e*log(F)^2 - 60*F^(a*c)*e^3)*F^(b*c*x)*cos(2*e*x + 2*d) - (192*F^(b* 
c*x)*F^(a*c)*b*c*e^2*log(F)*sin(2*e*x + 2*d) - 6*(F^(a*c)*b^2*c^2*e*log(F) 
^2 + 100*F^(a*c)*e^3)*F^(b*c*x)*cos(6*e*x + 6*d) - 12*(F^(a*c)*b^2*c^2*e*l 
og(F)^2 - 60*F^(a*c)*e^3)*F^(b*c*x)*cos(2*e*x + 2*d) - (F^(a*c)*b^3*c^3*lo 
g(F)^3 + 100*F^(a*c)*b*c*e^2*log(F))*F^(b*c*x)*sin(6*e*x + 6*d))*cos(12*e* 
x + 12*d) + 3*(192*F^(b*c*x)*F^(a*c)*b*c*e^2*log(F)*sin(2*e*x + 2*d) - 6*( 
F^(a*c)*b^2*c^2*e*log(F)^2 + 100*F^(a*c)*e^3)*F^(b*c*x)*cos(6*e*x + 6*d) - 
 12*(F^(a*c)*b^2*c^2*e*log(F)^2 - 60*F^(a*c)*e^3)*F^(b*c*x)*cos(2*e*x + 2* 
d) - (F^(a*c)*b^3*c^3*log(F)^3 + 100*F^(a*c)*b*c*e^2*log(F))*F^(b*c*x)*sin 
(6*e*x + 6*d))*cos(8*e*x + 8*d) + 3*(6*(F^(a*c)*b^2*c^2*e*log(F)^2 + 100*F 
^(a*c)*e^3)*F^(b*c*x)*cos(4*e*x + 4*d) - (F^(a*c)*b^3*c^3*log(F)^3 + 100*F 
^(a*c)*b*c*e^2*log(F))*F^(b*c*x)*sin(4*e*x + 4*d) - 2*(F^(a*c)*b^2*c^2*e*l 
og(F)^2 + 100*F^(a*c)*e^3)*F^(b*c*x))*cos(6*e*x + 6*d) - 36*(16*F^(b*c*x)* 
F^(a*c)*b*c*e^2*log(F)*sin(2*e*x + 2*d) - (F^(a*c)*b^2*c^2*e*log(F)^2 - 60 
*F^(a*c)*e^3)*F^(b*c*x)*cos(2*e*x + 2*d))*cos(4*e*x + 4*d) - 12*(F^(a*c)*b 
^5*c^5*e*log(F)^5*sin(2*d) + 2*F^(a*c)*b^4*c^4*e^2*cos(2*d)*log(F)^4 + 136 
*F^(a*c)*b^3*c^3*e^3*log(F)^3*sin(2*d) + 272*F^(a*c)*b^2*c^2*e^4*cos(2*d)* 
log(F)^2 + 3600*F^(a*c)*b*c*e^5*log(F)*sin(2*d) + 7200*F^(a*c)*e^6*cos(2*d 
) + (F^(a*c)*b^5*c^5*e*log(F)^5*sin(2*d) + 2*F^(a*c)*b^4*c^4*e^2*cos(2*...
 

Giac [F]

\[ \int F^{c (a+b x)} \csc ^3(d+e x) \sec ^3(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \csc \left (e x + d\right )^{3} \sec \left (e x + d\right )^{3} \,d x } \] Input:

integrate(F^(c*(b*x+a))*csc(e*x+d)^3*sec(e*x+d)^3,x, algorithm="giac")
 

Output:

integrate(F^((b*x + a)*c)*csc(e*x + d)^3*sec(e*x + d)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int F^{c (a+b x)} \csc ^3(d+e x) \sec ^3(d+e x) \, dx=\int \frac {F^{c\,\left (a+b\,x\right )}}{{\cos \left (d+e\,x\right )}^3\,{\sin \left (d+e\,x\right )}^3} \,d x \] Input:

int(F^(c*(a + b*x))/(cos(d + e*x)^3*sin(d + e*x)^3),x)
 

Output:

int(F^(c*(a + b*x))/(cos(d + e*x)^3*sin(d + e*x)^3), x)
 

Reduce [F]

\[ \int F^{c (a+b x)} \csc ^3(d+e x) \sec ^3(d+e x) \, dx=f^{a c} \left (\int f^{b c x} \csc \left (e x +d \right )^{3} \sec \left (e x +d \right )^{3}d x \right ) \] Input:

int(F^(c*(b*x+a))*csc(e*x+d)^3*sec(e*x+d)^3,x)
 

Output:

f**(a*c)*int(f**(b*c*x)*csc(d + e*x)**3*sec(d + e*x)**3,x)