Optimal. Leaf size=68 \[ \frac {\sqrt {\cos ^2(e+f x)} \, _2F_1\left (\frac {5}{2},\frac {5+m}{2};\frac {7+m}{2};\sin ^2(e+f x)\right ) \sec (e+f x) (a \sin (e+f x))^{5+m}}{a^5 f (5+m)} \]
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Rubi [A]
time = 0.06, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2680, 2657}
\begin {gather*} \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) (a \sin (e+f x))^{m+5} \, _2F_1\left (\frac {5}{2},\frac {m+5}{2};\frac {m+7}{2};\sin ^2(e+f x)\right )}{a^5 f (m+5)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2657
Rule 2680
Rubi steps
\begin {align*} \int (a \sin (e+f x))^m \tan ^4(e+f x) \, dx &=\frac {\int \sec ^4(e+f x) (a \sin (e+f x))^{4+m} \, dx}{a^4}\\ &=\frac {\sqrt {\cos ^2(e+f x)} \, _2F_1\left (\frac {5}{2},\frac {5+m}{2};\frac {7+m}{2};\sin ^2(e+f x)\right ) \sec (e+f x) (a \sin (e+f x))^{5+m}}{a^5 f (5+m)}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 71, normalized size = 1.04 \begin {gather*} \frac {\sqrt {\cos ^2(e+f x)} \, _2F_1\left (\frac {5}{2},\frac {5+m}{2};\frac {7+m}{2};\sin ^2(e+f x)\right ) \sin ^4(e+f x) (a \sin (e+f x))^m \tan (e+f x)}{f (5+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \left (a \sin \left (f x +e \right )\right )^{m} \left (\tan ^{4}\left (f x +e \right )\right )\, dx\]
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 (a \sin {\left (e + f x \right )}\right )^{m} \tan ^{4}{\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 {\mathrm {tan}\left (e+f\,x\right )}^4\,{\left (a\,\sin \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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