Optimal. Leaf size=78 \[ \frac {\cos ^2(e+f x)^{\frac {1+n}{2}} \, _2F_1\left (\frac {1+n}{2},\frac {4+n}{2};\frac {6+n}{2};\sin ^2(e+f x)\right ) \sin ^3(e+f x) (b \tan (e+f x))^{1+n}}{b f (4+n)} \]
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Rubi [A]
time = 0.06, antiderivative size = 78, 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 = {2682, 2657}
\begin {gather*} \frac {\sin ^3(e+f x) \cos ^2(e+f x)^{\frac {n+1}{2}} (b \tan (e+f x))^{n+1} \, _2F_1\left (\frac {n+1}{2},\frac {n+4}{2};\frac {n+6}{2};\sin ^2(e+f x)\right )}{b f (n+4)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2657
Rule 2682
Rubi steps
\begin {align*} \int \sin ^3(e+f x) (b \tan (e+f x))^n \, dx &=\frac {\left (\cos ^{1+n}(e+f x) \sin ^{-1-n}(e+f x) (b \tan (e+f x))^{1+n}\right ) \int \cos ^{-n}(e+f x) \sin ^{3+n}(e+f x) \, dx}{b}\\ &=\frac {\cos ^2(e+f x)^{\frac {1+n}{2}} \, _2F_1\left (\frac {1+n}{2},\frac {4+n}{2};\frac {6+n}{2};\sin ^2(e+f x)\right ) \sin ^3(e+f x) (b \tan (e+f x))^{1+n}}{b f (4+n)}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 3.08, size = 456, normalized size = 5.85 \begin {gather*} \frac {4 (4+n) \left (F_1\left (1+\frac {n}{2};n,3;2+\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-F_1\left (1+\frac {n}{2};n,4;2+\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \cos ^3\left (\frac {1}{2} (e+f x)\right ) \sin \left (\frac {1}{2} (e+f x)\right ) \sin ^3(e+f x) (b \tan (e+f x))^n}{f (2+n) \left (-2 (4+n) F_1\left (1+\frac {n}{2};n,4;2+\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^2\left (\frac {1}{2} (e+f x)\right )+2 \left (3 F_1\left (2+\frac {n}{2};n,4;3+\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-4 F_1\left (2+\frac {n}{2};n,5;3+\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+n \left (-F_1\left (2+\frac {n}{2};1+n,3;3+\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+F_1\left (2+\frac {n}{2};1+n,4;3+\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )\right ) (-1+\cos (e+f x))+(4+n) F_1\left (1+\frac {n}{2};n,3;2+\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (1+\cos (e+f x))\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.50, size = 0, normalized size = 0.00 \[\int \left (\sin ^{3}\left (f x +e \right )\right ) \left (b \tan \left (f x +e \right )\right )^{n}\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {\sin \left (e+f\,x\right )}^3\,{\left (b\,\mathrm {tan}\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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