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 = 0.0572467, antiderivative size = 50, normalized size of antiderivative = 1., 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 2591
Rule 364
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*}
Mathematica [C] time = 4.74523, 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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Maple [F] time = 0.826, size = 0, normalized size = 0. \begin{align*} \int \left ( \sin \left ( fx+e \right ) \right ) ^{4} \left ( b\tan \left ( fx+e \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \tan \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \tan \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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