Optimal. Leaf size=84 \[ \frac {2 e^{i (d+e x)} F^{c (a+b x)} \, _2F_1\left (1,\frac {e-i b c \log (F)}{2 e};\frac {1}{2} \left (3-\frac {i b c \log (F)}{e}\right );-e^{2 i (d+e x)}\right )}{b c \log (F)+i e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 84, normalized size of antiderivative = 1.00, 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 = {4451} \[ \frac {2 e^{i (d+e x)} F^{c (a+b x)} \, _2F_1\left (1,\frac {e-i b c \log (F)}{2 e};\frac {1}{2} \left (3-\frac {i b c \log (F)}{e}\right );-e^{2 i (d+e x)}\right )}{b c \log (F)+i e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4451
Rubi steps
\begin {align*} \int F^{c (a+b x)} \sec (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 )}{i e+b c \log (F)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 84, normalized size = 1.00 \[ \frac {2 e^{i (d+e x)} F^{c (a+b x)} \, _2F_1\left (1,\frac {1}{2}-\frac {i b c \log (F)}{2 e};\frac {3}{2}-\frac {i b c \log (F)}{2 e};-e^{2 i (d+e x)}\right )}{b c \log (F)+i e} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (F^{b c x + a c} \sec \left (e x + d\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{{\left (b x + a\right )} c} \sec \left (e x + d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int F^{c \left (b x +a \right )} \sec \left (e x +d \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {F^{c\,\left (a+b\,x\right )}}{\cos \left (d+e\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{c \left (a + b x\right )} \sec {\left (d + e x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________