Optimal. Leaf size=107 \[ \frac {(a+a \sec (e+f x))^m \tan (e+f x)}{f (1+m)}+\frac {2^{\frac {1}{2}+m} m \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {1}{2} (1-\sec (e+f x))\right ) (1+\sec (e+f x))^{-\frac {1}{2}-m} (a+a \sec (e+f x))^m \tan (e+f x)}{f (1+m)} \]
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Rubi [A]
time = 0.08, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3883, 3913,
3912, 71} \begin {gather*} \frac {2^{m+\frac {1}{2}} m \tan (e+f x) (\sec (e+f x)+1)^{-m-\frac {1}{2}} (a \sec (e+f x)+a)^m \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {1}{2} (1-\sec (e+f x))\right )}{f (m+1)}+\frac {\tan (e+f x) (a \sec (e+f x)+a)^m}{f (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 3883
Rule 3912
Rule 3913
Rubi steps
\begin {align*} \int \sec ^2(e+f x) (a+a \sec (e+f x))^m \, dx &=\frac {(a+a \sec (e+f x))^m \tan (e+f x)}{f (1+m)}+\frac {m \int \sec (e+f x) (a+a \sec (e+f x))^m \, dx}{1+m}\\ &=\frac {(a+a \sec (e+f x))^m \tan (e+f x)}{f (1+m)}+\frac {\left (m (1+\sec (e+f x))^{-m} (a+a \sec (e+f x))^m\right ) \int \sec (e+f x) (1+\sec (e+f x))^m \, dx}{1+m}\\ &=\frac {(a+a \sec (e+f x))^m \tan (e+f x)}{f (1+m)}-\frac {\left (m (1+\sec (e+f x))^{-\frac {1}{2}-m} (a+a \sec (e+f x))^m \tan (e+f x)\right ) \text {Subst}\left (\int \frac {(1+x)^{-\frac {1}{2}+m}}{\sqrt {1-x}} \, dx,x,\sec (e+f x)\right )}{f (1+m) \sqrt {1-\sec (e+f x)}}\\ &=\frac {(a+a \sec (e+f x))^m \tan (e+f x)}{f (1+m)}+\frac {2^{\frac {1}{2}+m} m \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {1}{2} (1-\sec (e+f x))\right ) (1+\sec (e+f x))^{-\frac {1}{2}-m} (a+a \sec (e+f x))^m \tan (e+f x)}{f (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 95, normalized size = 0.89 \begin {gather*} \frac {(1+\sec (e+f x))^{-\frac {1}{2}-m} (a (1+\sec (e+f x)))^m \left (2^{\frac {3}{2}+m} m \, _2F_1\left (\frac {1}{2},-\frac {1}{2}-m;\frac {3}{2};\frac {1}{2} (1-\sec (e+f x))\right )+(1+\sec (e+f x))^{\frac {1}{2}+m}\right ) \tan (e+f x)}{f+2 f m} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \left (\sec ^{2}\left (f x +e \right )\right ) \left (a +a \sec \left (f x +e \right )\right )^{m}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{m} \sec ^{2}{\left (e + f x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^m}{{\cos \left (e+f\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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