Optimal. Leaf size=50 \[ \frac {(b \tan (e+f x))^{n+5} \, _2F_1\left (3,\frac {n+5}{2};\frac {n+7}{2};-\tan ^2(e+f x)\right )}{b^5 f (n+5)} \]
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Rubi [A] time = 0.06, antiderivative size = 50, 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 = {2591, 364} \[ \frac {(b \tan (e+f x))^{n+5} \, _2F_1\left (3,\frac {n+5}{2};\frac {n+7}{2};-\tan ^2(e+f x)\right )}{b^5 f (n+5)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 2591
Rubi steps
\begin {align*} \int \sin ^4(e+f x) (b \tan (e+f x))^n \, dx &=\frac {b \operatorname {Subst}\left (\int \frac {x^{4+n}}{\left (b^2+x^2\right )^3} \, dx,x,b \tan (e+f x)\right )}{f}\\ &=\frac {\, _2F_1\left (3,\frac {5+n}{2};\frac {7+n}{2};-\tan ^2(e+f x)\right ) (b \tan (e+f x))^{5+n}}{b^5 f (5+n)}\\ \end {align*}
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Mathematica [C] time = 4.71, size = 916, normalized size = 18.32 \[ \frac {64 (n+3) \left (F_1\left (\frac {n+1}{2};n,3;\frac {n+3}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 F_1\left (\frac {n+1}{2};n,4;\frac {n+3}{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 (\frac {n+1}{2};n,5;\frac {n+3}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \cos ^7\left (\frac {1}{2} (e+f x)\right ) \sin ^5\left (\frac {1}{2} (e+f x)\right ) (b \tan (e+f x))^n}{f (n+1) \left ((n+3) F_1\left (\frac {n+1}{2};n,3;\frac {n+3}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (\cos (e+f x)+1)+(n+3) F_1\left (\frac {n+1}{2};n,5;\frac {n+3}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (\cos (e+f x)+1)+2 \left (-2 n F_1\left (\frac {n+1}{2};n,4;\frac {n+3}{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 )-6 F_1\left (\frac {n+1}{2};n,4;\frac {n+3}{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 )-5 F_1\left (\frac {n+3}{2};n,6;\frac {n+5}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+n F_1\left (\frac {n+3}{2};n+1,3;\frac {n+5}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 n F_1\left (\frac {n+3}{2};n+1,4;\frac {n+5}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+n F_1\left (\frac {n+3}{2};n+1,5;\frac {n+5}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+3 F_1\left (\frac {n+3}{2};n,4;\frac {n+5}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (\cos (e+f x)-1)-8 F_1\left (\frac {n+3}{2};n,5;\frac {n+5}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (\cos (e+f x)-1)+5 F_1\left (\frac {n+3}{2};n,6;\frac {n+5}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos (e+f x)-n F_1\left (\frac {n+3}{2};n+1,3;\frac {n+5}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos (e+f x)+2 n F_1\left (\frac {n+3}{2};n+1,4;\frac {n+5}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos (e+f x)-n F_1\left (\frac {n+3}{2};n+1,5;\frac {n+5}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos (e+f x)\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \left (b \tan \left (f x + e\right )\right )^{n}, x\right ) \]
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 \[ \int \left (b \tan \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.99, size = 0, normalized size = 0.00 \[ \int \left (\sin ^{4}\left (f x +e \right )\right ) \left (b \tan \left (f x +e \right )\right )^{n}\, dx \]
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 \[ \int \left (b \tan \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{4}\,{d x} \]
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 \[ \int {\sin \left (e+f\,x\right )}^4\,{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^n \,d x \]
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 \[ \int \left (b \tan {\left (e + f x \right )}\right )^{n} \sin ^{4}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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