Optimal. Leaf size=39 \[ \frac {\, _2F_1\left (2,\frac {m}{2};\frac {2+m}{2};\sec ^2(e+f x)\right ) (b \sec (e+f x))^m}{f m} \]
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Rubi [A]
time = 0.03, antiderivative size = 39, 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 = {2686, 371}
\begin {gather*} \frac {(b \sec (e+f x))^m \, _2F_1\left (2,\frac {m}{2};\frac {m+2}{2};\sec ^2(e+f x)\right )}{f m} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 2686
Rubi steps
\begin {align*} \int \cot ^3(e+f x) (b \sec (e+f x))^m \, dx &=\frac {b \text {Subst}\left (\int \frac {(b x)^{-1+m}}{\left (-1+x^2\right )^2} \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac {\, _2F_1\left (2,\frac {m}{2};\frac {2+m}{2};\sec ^2(e+f x)\right ) (b \sec (e+f x))^m}{f m}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 12.54, size = 815, normalized size = 20.90 \begin {gather*} \frac {\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) \left (-((1+\cos (e+f x)) \, _2F_1(1,1-m;2-m;\cos (e+f x)))+2^m \, _2F_1\left (1-m,1-m;2-m;\frac {1}{2} \cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right )^{-m}\right ) (b \sec (e+f x))^m}{4 f (-1+m)}-\frac {2 \cot \left (\frac {1}{2} (e+f x)\right ) \cot (e+f x) \csc ^2(e+f x) \left (F_1\left (1;m,-m;2;\cot ^2\left (\frac {1}{2} (e+f x)\right ),-\cot ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cot ^4\left (\frac {1}{2} (e+f x)\right ) \left (-\cos (e+f x) \csc ^2\left (\frac {1}{2} (e+f x)\right )\right )^m \sec ^2\left (\frac {1}{2} (e+f x)\right )^m+F_1\left (1;m,-m;2;\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \csc ^2\left (\frac {1}{2} (e+f x)\right )^m \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^m\right ) (b \sec (e+f x))^m}{f \left (2 F_1\left (1;m,-m;2;\cot ^2\left (\frac {1}{2} (e+f x)\right ),-\cot ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cot ^6\left (\frac {1}{2} (e+f x)\right ) \left (-\cos (e+f x) \csc ^2\left (\frac {1}{2} (e+f x)\right )\right )^m \sec ^2\left (\frac {1}{2} (e+f x)\right )^{1+m}+m F_1\left (2;m,1-m;3;\cot ^2\left (\frac {1}{2} (e+f x)\right ),-\cot ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cot ^8\left (\frac {1}{2} (e+f x)\right ) \left (-\cos (e+f x) \csc ^2\left (\frac {1}{2} (e+f x)\right )\right )^m \sec ^2\left (\frac {1}{2} (e+f x)\right )^{1+m}+m F_1\left (2;1+m,-m;3;\cot ^2\left (\frac {1}{2} (e+f x)\right ),-\cot ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cot ^8\left (\frac {1}{2} (e+f x)\right ) \left (-\cos (e+f x) \csc ^2\left (\frac {1}{2} (e+f x)\right )\right )^m \sec ^2\left (\frac {1}{2} (e+f x)\right )^{1+m}-2 F_1\left (1;m,-m;2;\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \csc ^2\left (\frac {1}{2} (e+f x)\right )^{1+m} \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^m-m F_1\left (2;m,1-m;3;\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \csc ^2\left (\frac {1}{2} (e+f x)\right )^m \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^{1+m} \sec (e+f x)-m F_1\left (2;1+m,-m;3;\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \csc ^2\left (\frac {1}{2} (e+f x)\right )^m \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^{1+m} \sec (e+f x)\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \left (\cot ^{3}\left (f x +e \right )\right ) \left (b \sec \left (f x +e \right )\right )^{m}\, dx\]
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 (b \sec {\left (e + f x \right )}\right )^{m} \cot ^{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 = -0.03 \begin {gather*} \int {\mathrm {cot}\left (e+f\,x\right )}^3\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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