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

Optimal. Leaf size=81 \[ -\frac{2 e^{i (d+e x)} F^{c (a+b x)} \text{Hypergeometric2F1}\left (1,\frac{e-i b c \log (F)}{2 e},\frac{1}{2} \left (3-\frac{i b c \log (F)}{e}\right ),e^{2 i (d+e x)}\right )}{e-i b c \log (F)} \]

[Out]

(-2*E^(I*(d + e*x))*F^(c*(a + b*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])

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Rubi [A]  time = 0.0216913, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {4453} \[ -\frac{2 e^{i (d+e x)} F^{c (a+b x)} \, _2F_1\left (1,\frac{e-i b c \log (F)}{2 e};\frac{1}{2} \left (3-\frac{i b c \log (F)}{e}\right );e^{2 i (d+e x)}\right )}{e-i b c \log (F)} \]

Antiderivative was successfully verified.

[In]

Int[F^(c*(a + b*x))*Csc[d + e*x],x]

[Out]

(-2*E^(I*(d + e*x))*F^(c*(a + b*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])

Rule 4453

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 (d+e x) \, dx &=-\frac{2 e^{i (d+e x)} F^{c (a+b x)} \, _2F_1\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)}\\ \end{align*}

Mathematica [A]  time = 1.77452, size = 114, normalized size = 1.41 \[ \frac{i F^{c (a+b x)} \left (\text{Hypergeometric2F1}\left (1,-\frac{i b c \log (F)}{e},1-\frac{i b c \log (F)}{e},-\cos (d+e x)-i \sin (d+e x)\right )-\text{Hypergeometric2F1}\left (1,-\frac{i b c \log (F)}{e},1-\frac{i b c \log (F)}{e},\cos (d+e x)+i \sin (d+e x)\right )\right )}{b c \log (F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{F}^{c \left ( bx+a \right ) }\csc \left ( ex+d \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (F^{b c x + a c} \csc \left (e x + d\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int F^{c \left (a + b x\right )} \csc{\left (d + e x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int F^{{\left (b x + a\right )} c} \csc \left (e x + d\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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