Optimal. Leaf size=63 \[ \frac {\cos ^2(e+f x)^{\frac {3+m}{2}} \, _2F_1\left (\frac {3}{2},\frac {3+m}{2};\frac {5}{2};\sin ^2(e+f x)\right ) (b \sec (e+f x))^m \tan ^3(e+f x)}{3 f} \]
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Rubi [A]
time = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2697}
\begin {gather*} \frac {\tan ^3(e+f x) \cos ^2(e+f x)^{\frac {m+3}{2}} (b \sec (e+f x))^m \, _2F_1\left (\frac {3}{2},\frac {m+3}{2};\frac {5}{2};\sin ^2(e+f x)\right )}{3 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2697
Rubi steps
\begin {align*} \int (b \sec (e+f x))^m \tan ^2(e+f x) \, dx &=\frac {\cos ^2(e+f x)^{\frac {3+m}{2}} \, _2F_1\left (\frac {3}{2},\frac {3+m}{2};\frac {5}{2};\sin ^2(e+f x)\right ) (b \sec (e+f x))^m \tan ^3(e+f x)}{3 f}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 25.32, size = 6726, normalized size = 106.76 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \left (b \sec \left (f x +e \right )\right )^{m} \left (\tan ^{2}\left (f x +e \right )\right )\, dx\]
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 (b \sec {\left (e + f x \right )}\right )^{m} \tan ^{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 = -0.02 \begin {gather*} \int {\mathrm {tan}\left (e+f\,x\right )}^2\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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