3.1.6 \(\int F^{c (a+b x)} \csc ^2(d+e x) \, dx\) [6]
Optimal. Leaf size=78 \[ -\frac {4 e^{2 i (d+e x)} F^{c (a+b x)} \, _2F_1\left (2,1-\frac {i b c \log (F)}{2 e};2-\frac {i b c \log (F)}{2 e};e^{2 i (d+e x)}\right )}{2 i e+b c \log (F)} \]
[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))
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Rubi [A]
time = 0.02, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {4538}
\begin {gather*} -\frac {4 e^{2 i (d+e x)} F^{c (a+b x)} \, _2F_1\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} \end {gather*}
Antiderivative was successfully verified.
[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*} \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)} \, _2F_1\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*}
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Mathematica [A]
time = 1.31, size = 101, normalized size = 1.29 \begin {gather*} -\frac {2 i F^{c (a+b x)} \left (\left (-1+e^{2 i d}\right ) \, _2F_1\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 )} \end {gather*}
Antiderivative was successfully verified.
[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)))
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int F^{c \left (b x +a \right )} \left (\csc ^{2}\left (e x +d \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[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)
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(c*(b*x+a))*csc(e*x+d)^2,x, algorithm="maxima")
[Out]
4*(24*F^(a*c)*b*c*e^(b*c*x*log(F) + 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)*cos(2*x*e + 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*x*e + 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*x*e + 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*x*e + 2*d) + (24*F^(a*c)*b*c*e^(b*c*x*log(F) + 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*x*e + 2*d) + 2*(F^(a*c)*b^2*c^2*e*log(F)^2 + 1
6*F^(a*c)*e^3)*F^(b*c*x)*sin(2*x*e + 2*d))*cos(4*x*e + 4*d) - 4*(F^(a*c)*b^5*c^5*log(F)^5 + 20*F^(a*c)*b^3*c^3
*e^2*log(F)^3 + 64*F^(a*c)*b*c*e^4*log(F) + (F^(a*c)*b^5*c^5*log(F)^5 + 20*F^(a*c)*b^3*c^3*e^2*log(F)^3 + 64*F
^(a*c)*b*c*e^4*log(F))*cos(4*x*e + 4*d)^2 + 4*(F^(a*c)*b^5*c^5*log(F)^5 + 20*F^(a*c)*b^3*c^3*e^2*log(F)^3 + 64
*F^(a*c)*b*c*e^4*log(F))*cos(2*x*e + 2*d)^2 + (F^(a*c)*b^5*c^5*log(F)^5 + 20*F^(a*c)*b^3*c^3*e^2*log(F)^3 + 64
*F^(a*c)*b*c*e^4*log(F))*sin(4*x*e + 4*d)^2 - 4*(F^(a*c)*b^5*c^5*log(F)^5 + 20*F^(a*c)*b^3*c^3*e^2*log(F)^3 +
64*F^(a*c)*b*c*e^4*log(F))*sin(4*x*e + 4*d)*sin(2*x*e + 2*d) + 4*(F^(a*c)*b^5*c^5*log(F)^5 + 20*F^(a*c)*b^3*c^
3*e^2*log(F)^3 + 64*F^(a*c)*b*c*e^4*log(F))*sin(2*x*e + 2*d)^2 + 2*(F^(a*c)*b^5*c^5*log(F)^5 + 20*F^(a*c)*b^3*
c^3*e^2*log(F)^3 + 64*F^(a*c)*b*c*e^4*log(F) - 2*(F^(a*c)*b^5*c^5*log(F)^5 + 20*F^(a*c)*b^3*c^3*e^2*log(F)^3 +
64*F^(a*c)*b*c*e^4*log(F))*cos(2*x*e + 2*d))*cos(4*x*e + 4*d) - 4*(F^(a*c)*b^5*c^5*log(F)^5 + 20*F^(a*c)*b^3*
c^3*e^2*log(F)^3 + 64*F^(a*c)*b*c*e^4*log(F))*cos(2*x*e + 2*d))*integrate(-(6*b*c*cos(6*x*e + 6*d)*e^(b*c*x*lo
g(F) + 2)*log(F) - 18*b*c*cos(4*x*e + 4*d)*e^(b*c*x*log(F) + 2)*log(F) + 18*b*c*cos(2*x*e + 2*d)*e^(b*c*x*log(
F) + 2)*log(F) - 6*b*c*e^(b*c*x*log(F) + 2)*log(F) - (b^2*c^2*e*log(F)^2 - 8*e^3)*F^(b*c*x)*sin(6*x*e + 6*d) +
3*(b^2*c^2*e*log(F)^2 - 8*e^3)*F^(b*c*x)*sin(4*x*e + 4*d) - 3*(b^2*c^2*e*log(F)^2 - 8*e^3)*F^(b*c*x)*sin(2*x*
e + 2*d))/(b^4*c^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + (b^4*c^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + 64*e^4)*
cos(6*x*e + 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*x*e + 4*d)^2 + 9*(b^4*c^4*l
og(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + 64*e^4)*cos(2*x*e + 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*x*e + 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*x*e + 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*x*e + 4*d)*sin(2*x*e + 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*x*e + 2*d)^2 - 2*(b^4*c^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + 3*(b^
4*c^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + 64*e^4)*cos(4*x*e + 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*x*e + 2*d) + 64*e^4)*cos(6*x*e + 6*d) + 6*(b^4*c^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 -
3*(b^4*c^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + 64*e^4)*cos(2*x*e + 2*d) + 64*e^4)*cos(4*x*e + 4*d) - 6*(b^4*c
^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + 64*e^4)*cos(2*x*e + 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*x*e + 4*d) - (b^4*c^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + 64*e^4)*sin(2*x*e + 2*d))*sin(6
*x*e + 6*d) + 64*e^4), 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*x*e + 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*x*e + 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*x*e + 4*d))/(b^4*c^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + (b^4*c^4*log(F)^4
+ 20*b^2*c^2*e^2*log(F)^2 + 64*e^4)*cos(4*x*e + 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*x*e + 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*x*e + 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*x*e + 4*d)*sin(2*x*e + 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*x*e + 2*d)^2 + 2*(b^4*c^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 - 2*(b^4*c^4*log
(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + 64*e^4)*cos(2*x*e + 2*d) + 64*e^4)*cos(4*x*e + 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*x*e + 2*d) + 64*e^4)
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(c*(b*x+a))*csc(e*x+d)^2,x, algorithm="fricas")
[Out]
integral(F^(b*c*x + a*c)*csc(x*e + d)^2, x)
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int F^{c \left (a + b x\right )} \csc ^{2}{\left (d + e x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {F^{c\,\left (a+b\,x\right )}}{{\sin \left (d+e\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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