Optimal. Leaf size=63 \[ -\frac {\cot ^5(e+f x) (b \csc (e+f x))^m \, _2F_1\left (\frac {5}{2},\frac {5+m}{2};\frac {7}{2};\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac {5+m}{2}}}{5 f} \]
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Rubi [A]
time = 0.03, antiderivative size = 63, 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 {\cot ^5(e+f x) \sin ^2(e+f x)^{\frac {m+5}{2}} (b \csc (e+f x))^m \, _2F_1\left (\frac {5}{2},\frac {m+5}{2};\frac {7}{2};\cos ^2(e+f x)\right )}{5 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2697
Rubi steps
\begin {align*} \int \cot ^4(e+f x) (b \csc (e+f x))^m \, dx &=-\frac {\cot ^5(e+f x) (b \csc (e+f x))^m \, _2F_1\left (\frac {5}{2},\frac {5+m}{2};\frac {7}{2};\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac {5+m}{2}}}{5 f}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 106, normalized size = 1.68 \begin {gather*} -\frac {\cot (e+f x) (b \csc (e+f x))^m \left (\, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3}{2};\cos ^2(e+f x)\right )-2 \, _2F_1\left (\frac {1}{2},\frac {3+m}{2};\frac {3}{2};\cos ^2(e+f x)\right )+\, _2F_1\left (\frac {1}{2},\frac {5+m}{2};\frac {3}{2};\cos ^2(e+f x)\right )\right ) \sin ^2(e+f x)^{\frac {1+m}{2}}}{f} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \left (\cot ^{4}\left (f x +e \right )\right ) \left (b \csc \left (f x +e \right )\right )^{m}\, dx\]
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 \csc {\left (e + f x \right )}\right )^{m} \cot ^{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 {\mathrm {cot}\left (e+f\,x\right )}^4\,{\left (\frac {b}{\sin \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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