Optimal. Leaf size=51 \[ -\frac {(d \cot (e+f x))^{1+n} \, _2F_1\left (2,\frac {1+n}{2};\frac {3+n}{2};-\cot ^2(e+f x)\right )}{d f (1+n)} \]
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Rubi [A]
time = 0.03, 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 (2,\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 ^2(e+f x) \, dx &=\frac {\text {Subst}\left (\int \frac {(-d x)^n}{\left (1+x^2\right )^2} \, dx,x,-\cot (e+f x)\right )}{f}\\ &=-\frac {(d \cot (e+f x))^{1+n} \, _2F_1\left (2,\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 = 3.42, size = 509, normalized size = 9.98 \begin {gather*} -\frac {4 (-3+n) \left (F_1\left (\frac {1}{2}-\frac {n}{2};-n,2;\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,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 )\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 ^2(e+f x)}{f (-1+n) \left (2 (-3+n) F_1\left (\frac {1}{2}-\frac {n}{2};-n,2;\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 (-3+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 )-2 \left (n F_1\left (\frac {3}{2}-\frac {n}{2};1-n,2;\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 )-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 )+2 F_1\left (\frac {3}{2}-\frac {n}{2};-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 )-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 )\right ) (-1+\cos (e+f x))\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.77, size = 0, normalized size = 0.00 \[\int \left (d \cot \left (f x +e \right )\right )^{n} \left (\sin ^{2}\left (f x +e \right )\right )\, dx\]
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 ^{2}{\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 )}^2\,{\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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