Optimal. Leaf size=48 \[ \frac {\, _2F_1\left (2,\frac {4+m}{2};\frac {6+m}{2};\sin ^2(e+f x)\right ) (a \sin (e+f x))^{4+m}}{a^4 f (4+m)} \]
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Rubi [A]
time = 0.04, antiderivative size = 48, 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 = {2672, 371}
\begin {gather*} \frac {(a \sin (e+f x))^{m+4} \, _2F_1\left (2,\frac {m+4}{2};\frac {m+6}{2};\sin ^2(e+f x)\right )}{a^4 f (m+4)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 2672
Rubi steps
\begin {align*} \int (a \sin (e+f x))^m \tan ^3(e+f x) \, dx &=\frac {\text {Subst}\left (\int \frac {x^{3+m}}{\left (a^2-x^2\right )^2} \, dx,x,a \sin (e+f x)\right )}{f}\\ &=\frac {\, _2F_1\left (2,\frac {4+m}{2};\frac {6+m}{2};\sin ^2(e+f x)\right ) (a \sin (e+f x))^{4+m}}{a^4 f (4+m)}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 53, normalized size = 1.10 \begin {gather*} \frac {\, _2F_1\left (2,\frac {4+m}{2};1+\frac {4+m}{2};\sin ^2(e+f x)\right ) \sin ^4(e+f x) (a \sin (e+f x))^m}{f (4+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \left (a \sin \left (f x +e \right )\right )^{m} \left (\tan ^{3}\left (f x +e \right )\right )\, dx\]
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 ^{3}{\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 )}^3\,{\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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