Optimal. Leaf size=88 \[ -\frac {(a \cos (e+f x))^{1+m} \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+m-n),\frac {1}{2} (3+m-n),\cos ^2(e+f x)\right ) (b \sec (e+f x))^n \sin (e+f x)}{a f (1+m-n) \sqrt {\sin ^2(e+f x)}} \]
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Rubi [A]
time = 0.05, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2668, 2722}
\begin {gather*} -\frac {\sin (e+f x) (a \cos (e+f x))^{m+1} (b \sec (e+f x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (m-n+1);\frac {1}{2} (m-n+3);\cos ^2(e+f x)\right )}{a f (m-n+1) \sqrt {\sin ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2668
Rule 2722
Rubi steps
\begin {align*} \int (a \cos (e+f x))^m (b \sec (e+f x))^n \, dx &=\left ((a \cos (e+f x))^n (b \sec (e+f x))^n\right ) \int (a \cos (e+f x))^{m-n} \, dx\\ &=-\frac {(a \cos (e+f x))^{1+m} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (1+m-n);\frac {1}{2} (3+m-n);\cos ^2(e+f x)\right ) (b \sec (e+f x))^n \sin (e+f x)}{a f (1+m-n) \sqrt {\sin ^2(e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 6.76, size = 89, normalized size = 1.01 \begin {gather*} -\frac {\cos (e+f x) (a \cos (e+f x))^m \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+m-n),\frac {1}{2} (3+m-n),\cos ^2(e+f x)\right ) (b \sec (e+f x))^n \sin (e+f x)}{f (1+m-n) \sqrt {\sin ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \left (a \cos \left (f x +e \right )\right )^{m} \left (b \sec \left (f x +e \right )\right )^{n}\, dx\]
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 \cos {\left (e + f x \right )}\right )^{m} \left (b \sec {\left (e + f x \right )}\right )^{n}\, 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 {\left (a\,\cos \left (e+f\,x\right )\right )}^m\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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