\(\int F^{c (a+b x)} \csc ^2(d+e x) \, dx\) [6]

Optimal result

Integrand size = 18, antiderivative size = 78 \[ \int F^{c (a+b x)} \csc ^2(d+e x) \, dx=-\frac {4 e^{2 i (d+e x)} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {i b c \log (F)}{2 e},2-\frac {i b c \log (F)}{2 e},e^{2 i (d+e x)}\right )}{2 i e+b c \log (F)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {4538} \[ \int F^{c (a+b x)} \csc ^2(d+e x) \, dx=-\frac {4 e^{2 i (d+e x)} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {i b c \log (F)}{2 e},2-\frac {i b c \log (F)}{2 e},e^{2 i (d+e x)}\right )}{b c \log (F)+2 i e} \]

[In]

Int[F^(c*(a + b*x))*Csc[d + e*x]^2,x]

[Out]

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

Rule 4538

Int[Csc[(d_.) + (e_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[(-2*I)^n*E^(I*n*(d + e*
x))*(F^(c*(a + b*x))/(I*e*n + b*c*Log[F]))*Hypergeometric2F1[n, n/2 - I*b*c*(Log[F]/(2*e)), 1 + n/2 - I*b*c*(L
og[F]/(2*e)), E^(2*I*(d + e*x))], x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = -\frac {4 e^{2 i (d+e x)} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {i b c \log (F)}{2 e},2-\frac {i b c \log (F)}{2 e},e^{2 i (d+e x)}\right )}{2 i e+b c \log (F)} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.31 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.29 \[ \int F^{c (a+b x)} \csc ^2(d+e x) \, dx=-\frac {2 i F^{c (a+b x)} \left (\left (-1+e^{2 i d}\right ) \operatorname {Hypergeometric2F1}\left (1,-\frac {i b c \log (F)}{2 e},1-\frac {i b c \log (F)}{2 e},e^{2 i (d+e x)}\right )+\csc (d+e x) \sin (d) (\cos (e x)-i \sin (e x))\right )}{e \left (-1+e^{2 i d}\right )} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

4*(24*F^(b*c*x)*F^(a*c)*b*c*e^2*log(F) + 2*(F^(a*c)*b^3*c^3*log(F)^3 + 16*F^(a*c)*b*c*e^2*log(F))*F^(b*c*x)*co
s(2*e*x + 2*d)^2 + 2*(F^(a*c)*b^3*c^3*log(F)^3 + 16*F^(a*c)*b*c*e^2*log(F))*F^(b*c*x)*sin(2*e*x + 2*d)^2 - (F^
(a*c)*b^3*c^3*log(F)^3 + 64*F^(a*c)*b*c*e^2*log(F))*F^(b*c*x)*cos(2*e*x + 2*d) + 2*(5*F^(a*c)*b^2*c^2*e*log(F)
^2 - 16*F^(a*c)*e^3)*F^(b*c*x)*sin(2*e*x + 2*d) + (24*F^(b*c*x)*F^(a*c)*b*c*e^2*log(F) - (F^(a*c)*b^3*c^3*log(
F)^3 + 16*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*log(F)^2 + 16*F^(a*c)*e^3)
*F^(b*c*x)*sin(2*e*x + 2*d))*cos(4*e*x + 4*d) + 4*(F^(a*c)*b^5*c^5*e*log(F)^5 + 20*F^(a*c)*b^3*c^3*e^3*log(F)^
3 + 64*F^(a*c)*b*c*e^5*log(F) + (F^(a*c)*b^5*c^5*e*log(F)^5 + 20*F^(a*c)*b^3*c^3*e^3*log(F)^3 + 64*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 + 20*F^(a*c)*b^3*c^3*e^3*log(F)^3 + 64*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 + 20*F^(a*c)*b^3*c^3*e^3*log(F)^3 + 64*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 + 20*F^(a*c)*b^3*c^3*e^3*log(F)^3 + 64*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*e*log(F)^5 + 20*F^(a*c)*b^3*c^3*
e^3*log(F)^3 + 64*F^(a*c)*b*c*e^5*log(F))*sin(2*e*x + 2*d)^2 + 2*(F^(a*c)*b^5*c^5*e*log(F)^5 + 20*F^(a*c)*b^3*
c^3*e^3*log(F)^3 + 64*F^(a*c)*b*c*e^5*log(F) - 2*(F^(a*c)*b^5*c^5*e*log(F)^5 + 20*F^(a*c)*b^3*c^3*e^3*log(F)^3
+ 64*F^(a*c)*b*c*e^5*log(F))*cos(2*e*x + 2*d))*cos(4*e*x + 4*d) - 4*(F^(a*c)*b^5*c^5*e*log(F)^5 + 20*F^(a*c)*
b^3*c^3*e^3*log(F)^3 + 64*F^(a*c)*b*c*e^5*log(F))*cos(2*e*x + 2*d))*integrate((6*F^(b*c*x)*b*c*e*cos(6*e*x + 6
*d)*log(F) - 18*F^(b*c*x)*b*c*e*cos(4*e*x + 4*d)*log(F) + 18*F^(b*c*x)*b*c*e*cos(2*e*x + 2*d)*log(F) - 6*F^(b*
c*x)*b*c*e*log(F) - (b^2*c^2*log(F)^2 - 8*e^2)*F^(b*c*x)*sin(6*e*x + 6*d) + 3*(b^2*c^2*log(F)^2 - 8*e^2)*F^(b*
c*x)*sin(4*e*x + 4*d) - 3*(b^2*c^2*log(F)^2 - 8*e^2)*F^(b*c*x)*sin(2*e*x + 2*d))/(b^4*c^4*log(F)^4 + 20*b^2*c^
2*e^2*log(F)^2 + 64*e^4 + (b^4*c^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + 64*e^4)*cos(6*e*x + 6*d)^2 + 9*(b^4*c^
4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + 64*e^4)*cos(4*e*x + 4*d)^2 + 9*(b^4*c^4*log(F)^4 + 20*b^2*c^2*e^2*log(F
)^2 + 64*e^4)*cos(2*e*x + 2*d)^2 + (b^4*c^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + 64*e^4)*sin(6*e*x + 6*d)^2 +
9*(b^4*c^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + 64*e^4)*sin(4*e*x + 4*d)^2 - 18*(b^4*c^4*log(F)^4 + 20*b^2*c^2
*e^2*log(F)^2 + 64*e^4)*sin(4*e*x + 4*d)*sin(2*e*x + 2*d) + 9*(b^4*c^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + 64
*e^4)*sin(2*e*x + 2*d)^2 - 2*(b^4*c^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + 64*e^4 + 3*(b^4*c^4*log(F)^4 + 20*b
^2*c^2*e^2*log(F)^2 + 64*e^4)*cos(4*e*x + 4*d) - 3*(b^4*c^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + 64*e^4)*cos(2
*e*x + 2*d))*cos(6*e*x + 6*d) + 6*(b^4*c^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + 64*e^4 - 3*(b^4*c^4*log(F)^4 +
20*b^2*c^2*e^2*log(F)^2 + 64*e^4)*cos(2*e*x + 2*d))*cos(4*e*x + 4*d) - 6*(b^4*c^4*log(F)^4 + 20*b^2*c^2*e^2*l
og(F)^2 + 64*e^4)*cos(2*e*x + 2*d) - 6*((b^4*c^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + 64*e^4)*sin(4*e*x + 4*d)
- (b^4*c^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + 64*e^4)*sin(2*e*x + 2*d))*sin(6*e*x + 6*d)), x) - (2*(F^(a*c)
*b^2*c^2*e*log(F)^2 + 16*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 + 16*F^(a*c)*b*c*
e^2*log(F))*F^(b*c*x)*sin(2*e*x + 2*d) + 4*(F^(a*c)*b^2*c^2*e*log(F)^2 - 8*F^(a*c)*e^3)*F^(b*c*x))*sin(4*e*x +
4*d))/(b^4*c^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + 64*e^4 + (b^4*c^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + 64
*e^4)*cos(4*e*x + 4*d)^2 + 4*(b^4*c^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + 64*e^4)*cos(2*e*x + 2*d)^2 + (b^4*c
^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + 64*e^4)*sin(4*e*x + 4*d)^2 - 4*(b^4*c^4*log(F)^4 + 20*b^2*c^2*e^2*log(
F)^2 + 64*e^4)*sin(4*e*x + 4*d)*sin(2*e*x + 2*d) + 4*(b^4*c^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + 64*e^4)*sin
(2*e*x + 2*d)^2 + 2*(b^4*c^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + 64*e^4 - 2*(b^4*c^4*log(F)^4 + 20*b^2*c^2*e^
2*log(F)^2 + 64*e^4)*cos(2*e*x + 2*d))*cos(4*e*x + 4*d) - 4*(b^4*c^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + 64*e
^4)*cos(2*e*x + 2*d))

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(F^(c*(a + b*x))/sin(d + e*x)^2,x)

[Out]

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