Optimal. Leaf size=174 \[ \frac {\sec ^n(e+f x) \sin (e+f x)}{f (a+a \sec (e+f x))}+\frac {(1-n) \, _2F_1\left (\frac {1}{2},\frac {2-n}{2};\frac {4-n}{2};\cos ^2(e+f x)\right ) \sec ^{-2+n}(e+f x) \sin (e+f x)}{a f (2-n) \sqrt {\sin ^2(e+f x)}}-\frac {\, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(e+f x)\right ) \sec ^{-1+n}(e+f x) \sin (e+f x)}{a f \sqrt {\sin ^2(e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.12, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3905, 3872,
3857, 2722} \begin {gather*} \frac {(1-n) \sin (e+f x) \sec ^{n-2}(e+f x) \, _2F_1\left (\frac {1}{2},\frac {2-n}{2};\frac {4-n}{2};\cos ^2(e+f x)\right )}{a f (2-n) \sqrt {\sin ^2(e+f x)}}-\frac {\sin (e+f x) \sec ^{n-1}(e+f x) \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(e+f x)\right )}{a f \sqrt {\sin ^2(e+f x)}}+\frac {\sin (e+f x) \sec ^n(e+f x)}{f (a \sec (e+f x)+a)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2722
Rule 3857
Rule 3872
Rule 3905
Rubi steps
\begin {align*} \int \frac {\sec ^n(e+f x)}{a+a \sec (e+f x)} \, dx &=\frac {\sec ^n(e+f x) \sin (e+f x)}{f (a+a \sec (e+f x))}-\frac {(1-n) \int \sec ^{-1+n}(e+f x) (a-a \sec (e+f x)) \, dx}{a^2}\\ &=\frac {\sec ^n(e+f x) \sin (e+f x)}{f (a+a \sec (e+f x))}-\frac {(1-n) \int \sec ^{-1+n}(e+f x) \, dx}{a}+\frac {(1-n) \int \sec ^n(e+f x) \, dx}{a}\\ &=\frac {\sec ^n(e+f x) \sin (e+f x)}{f (a+a \sec (e+f x))}-\frac {\left ((1-n) \cos ^n(e+f x) \sec ^n(e+f x)\right ) \int \cos ^{1-n}(e+f x) \, dx}{a}+\frac {\left ((1-n) \cos ^n(e+f x) \sec ^n(e+f x)\right ) \int \cos ^{-n}(e+f x) \, dx}{a}\\ &=\frac {\sec ^n(e+f x) \sin (e+f x)}{f (a+a \sec (e+f x))}+\frac {(1-n) \, _2F_1\left (\frac {1}{2},\frac {2-n}{2};\frac {4-n}{2};\cos ^2(e+f x)\right ) \sec ^{-2+n}(e+f x) \sin (e+f x)}{a f (2-n) \sqrt {\sin ^2(e+f x)}}-\frac {\, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(e+f x)\right ) \sec ^{-1+n}(e+f x) \sin (e+f x)}{a f \sqrt {\sin ^2(e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F]
time = 0.90, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sec ^n(e+f x)}{a+a \sec (e+f x)} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\sec ^{n}\left (f x +e \right )}{a +a \sec \left (f x +e \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\sec ^{n}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (\frac {1}{\cos \left (e+f\,x\right )}\right )}^n}{a+\frac {a}{\cos \left (e+f\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________