Optimal. Leaf size=51 \[ -\frac {(d \cot (e+f x))^{1+n} \, _2F_1\left (3,\frac {1+n}{2};\frac {3+n}{2};-\cot ^2(e+f x)\right )}{d f (1+n)} \]
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Rubi [A]
time = 0.04, antiderivative size = 51, 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 = {2687, 371}
\begin {gather*} -\frac {(d \cot (e+f x))^{n+1} \, _2F_1\left (3,\frac {n+1}{2};\frac {n+3}{2};-\cot ^2(e+f x)\right )}{d f (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 2687
Rubi steps
\begin {align*} \int (d \cot (e+f x))^n \sin ^4(e+f x) \, dx &=\frac {\text {Subst}\left (\int \frac {(-d x)^n}{\left (1+x^2\right )^3} \, dx,x,-\cot (e+f x)\right )}{f}\\ &=-\frac {(d \cot (e+f x))^{1+n} \, _2F_1\left (3,\frac {1+n}{2};\frac {3+n}{2};-\cot ^2(e+f x)\right )}{d f (1+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 = 8.49, size = 1099, normalized size = 21.55 \begin {gather*} \frac {2 (-3+n) \left (F_1\left (\frac {1}{2}-\frac {n}{2};-n,3;\frac {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 )-2 F_1\left (\frac {1}{2}-\frac {n}{2};-n,4;\frac {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 (\frac {1}{2}-\frac {n}{2};-n,5;\frac {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 )\right ) \cos ^3\left (\frac {1}{2} (e+f x)\right ) (d \cot (e+f x))^n \sin \left (\frac {1}{2} (e+f x)\right ) \sin ^4(e+f x)}{f (-1+n) \left (-3 F_1\left (\frac {3}{2}-\frac {n}{2};-n,4;\frac {5}{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 )+8 F_1\left (\frac {3}{2}-\frac {n}{2};-n,5;\frac {5}{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 )-5 F_1\left (\frac {3}{2}-\frac {n}{2};-n,6;\frac {5}{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 )+3 F_1\left (\frac {1}{2}-\frac {n}{2};-n,3;\frac {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 ) \cos ^2\left (\frac {1}{2} (e+f x)\right )-n F_1\left (\frac {1}{2}-\frac {n}{2};-n,3;\frac {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 ) \cos ^2\left (\frac {1}{2} (e+f x)\right )-6 F_1\left (\frac {1}{2}-\frac {n}{2};-n,4;\frac {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 ) \cos ^2\left (\frac {1}{2} (e+f x)\right )+2 n F_1\left (\frac {1}{2}-\frac {n}{2};-n,4;\frac {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 ) \cos ^2\left (\frac {1}{2} (e+f x)\right )+3 F_1\left (\frac {1}{2}-\frac {n}{2};-n,5;\frac {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 ) \cos ^2\left (\frac {1}{2} (e+f x)\right )-n F_1\left (\frac {1}{2}-\frac {n}{2};-n,5;\frac {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 ) \cos ^2\left (\frac {1}{2} (e+f x)\right )+n F_1\left (\frac {3}{2}-\frac {n}{2};1-n,3;\frac {5}{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))+n F_1\left (\frac {3}{2}-\frac {n}{2};1-n,5;\frac {5}{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))+3 F_1\left (\frac {3}{2}-\frac {n}{2};-n,4;\frac {5}{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 (e+f x)-8 F_1\left (\frac {3}{2}-\frac {n}{2};-n,5;\frac {5}{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 (e+f x)+5 F_1\left (\frac {3}{2}-\frac {n}{2};-n,6;\frac {5}{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 (e+f x)+4 n F_1\left (\frac {3}{2}-\frac {n}{2};1-n,4;\frac {5}{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 ) \sin ^2\left (\frac {1}{2} (e+f x)\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 1.14, size = 0, normalized size = 0.00 \[\int \left (d \cot \left (f x +e \right )\right )^{n} \left (\sin ^{4}\left (f x +e \right )\right )\, dx\]
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 (d \cot {\left (e + f x \right )}\right )^{n} \sin ^{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 = -0.02 \begin {gather*} \int {\sin \left (e+f\,x\right )}^4\,{\left (d\,\mathrm {cot}\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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