Optimal. Leaf size=81 \[ -\frac{2 e^{i (d+e x)} F^{c (a+b x)} \text{Hypergeometric2F1}\left (1,\frac{e-i b c \log (F)}{2 e},\frac{1}{2} \left (3-\frac{i b c \log (F)}{e}\right ),e^{2 i (d+e x)}\right )}{e-i b c \log (F)} \]
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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.
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Rule 4453
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.
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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.
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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.
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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.
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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.
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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.
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