Optimal. Leaf size=79 \[ -\frac {(d \cot (e+f x))^{1+n} \csc ^3(e+f x) \, _2F_1\left (\frac {1+n}{2},\frac {4+n}{2};\frac {3+n}{2};\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac {4+n}{2}}}{d f (1+n)} \]
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Rubi [A]
time = 0.03, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2697}
\begin {gather*} -\frac {\csc ^3(e+f x) \sin ^2(e+f x)^{\frac {n+4}{2}} (d \cot (e+f x))^{n+1} \, _2F_1\left (\frac {n+1}{2},\frac {n+4}{2};\frac {n+3}{2};\cos ^2(e+f x)\right )}{d f (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2697
Rubi steps
\begin {align*} \int (d \cot (e+f x))^n \csc ^3(e+f x) \, dx &=-\frac {(d \cot (e+f x))^{1+n} \csc ^3(e+f x) \, _2F_1\left (\frac {1+n}{2},\frac {4+n}{2};\frac {3+n}{2};\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac {4+n}{2}}}{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 = 14.17, size = 784, normalized size = 9.92 \begin {gather*} -\frac {\cot ^2\left (\frac {1}{2} (e+f x)\right ) (d \cot (e+f x))^n \, _2F_1\left (-1-\frac {n}{2},-n;-\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^{-n}}{f (8+4 n)}+\frac {8 (-4+n) \cos ^6\left (\frac {1}{2} (e+f x)\right ) (d \cot (e+f x))^n \csc ^2(e+f x) \left (n F_1\left (1-\frac {n}{2};-n,1;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+n) \cot ^2\left (\frac {1}{2} (e+f x)\right ) \, _2F_1\left (-n,-\frac {n}{2};1-\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \sin ^4\left (\frac {1}{2} (e+f x)\right )}{f (-2+n) n \left (-8 n F_1\left (2-\frac {n}{2};1-n,1;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 ) \sin ^4\left (\frac {1}{2} (e+f x)\right )-8 F_1\left (2-\frac {n}{2};-n,2;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 ) \sin ^4\left (\frac {1}{2} (e+f x)\right )+(-4+n) \left (4 \cos ^4\left (\frac {1}{2} (e+f x)\right ) \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^n-F_1\left (1-\frac {n}{2};-n,1;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(e+f x)\right )\right )}+\frac {(d \cot (e+f x))^n \, _2F_1\left (1-\frac {n}{2},-n;2-\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^{-n} \tan ^2\left (\frac {1}{2} (e+f x)\right )}{f (8-4 n)}+\frac {(-4+n) F_1\left (1-\frac {n}{2};-n,1;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 ) (d \cot (e+f x))^n \sin ^2\left (\frac {1}{2} (e+f x)\right )}{f (4-2 n) \left ((-4+n) F_1\left (1-\frac {n}{2};-n,1;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 \left (n F_1\left (2-\frac {n}{2};1-n,1;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};-n,2;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 ) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.30, size = 0, normalized size = 0.00 \[\int \left (d \cot \left (f x +e \right )\right )^{n} \left (\csc ^{3}\left (f x +e \right )\right )\, dx\]
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} \csc ^{3}{\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 = -0.01 \begin {gather*} \int \frac {{\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^n}{{\sin \left (e+f\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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