Optimal. Leaf size=141 \[ -\frac{e^{i (d+e x)} F^{c (a+b x)} (-b c \log (F)+i e) \text{Hypergeometric2F1}\left (1,\frac{e-i b c \log (F)}{2 e},\frac{1}{2} \left (3-\frac{i b c \log (F)}{e}\right ),-e^{2 i (d+e x)}\right )}{e^2}-\frac{b c \log (F) \sec (d+e x) F^{c (a+b x)}}{2 e^2}+\frac{\tan (d+e x) \sec (d+e x) F^{c (a+b x)}}{2 e} \]
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Rubi [A] time = 0.048772, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {4448, 4451} \[ -\frac{e^{i (d+e x)} F^{c (a+b x)} (-b c \log (F)+i e) \, _2F_1\left (1,\frac{e-i b c \log (F)}{2 e};\frac{1}{2} \left (3-\frac{i b c \log (F)}{e}\right );-e^{2 i (d+e x)}\right )}{e^2}-\frac{b c \log (F) \sec (d+e x) F^{c (a+b x)}}{2 e^2}+\frac{\tan (d+e x) \sec (d+e x) F^{c (a+b x)}}{2 e} \]
Antiderivative was successfully verified.
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Rule 4448
Rule 4451
Rubi steps
\begin{align*} \int F^{c (a+b x)} \sec ^3(d+e x) \, dx &=-\frac{b c F^{c (a+b x)} \log (F) \sec (d+e x)}{2 e^2}+\frac{F^{c (a+b x)} \sec (d+e x) \tan (d+e x)}{2 e}+\frac{1}{2} \left (1+\frac{b^2 c^2 \log ^2(F)}{e^2}\right ) \int F^{c (a+b x)} \sec (d+e x) \, dx\\ &=-\frac{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 ) (i e-b c \log (F))}{e^2}-\frac{b c F^{c (a+b x)} \log (F) \sec (d+e x)}{2 e^2}+\frac{F^{c (a+b x)} \sec (d+e x) \tan (d+e x)}{2 e}\\ \end{align*}
Mathematica [A] time = 0.26007, size = 112, normalized size = 0.79 \[ \frac{F^{c (a+b x)} \left (\sec (d+e x) (e \tan (d+e x)-b c \log (F))+2 e^{i (d+e x)} (b c \log (F)-i e) \text{Hypergeometric2F1}\left (1,\frac{e-i b c \log (F)}{2 e},\frac{3}{2}-\frac{i b c \log (F)}{2 e},-e^{2 i (d+e x)}\right )\right )}{2 e^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.205, size = 0, normalized size = 0. \begin{align*} \int{F}^{c \left ( bx+a \right ) } \left ( \sec \left ( ex+d \right ) \right ) ^{3}\, 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} \sec \left (e x + d\right )^{3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int F^{{\left (b x + a\right )} c} \sec \left (e x + d\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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