Optimal. Leaf size=83 \[ -\frac {(b \cot (e+f x))^{1+n} (a \csc (e+f x))^m \, _2F_1\left (\frac {1+n}{2},\frac {1}{2} (1+m+n);\frac {3+n}{2};\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac {1}{2} (1+m+n)}}{b f (1+n)} \]
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Rubi [A]
time = 0.03, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2697}
\begin {gather*} -\frac {(a \csc (e+f x))^m (b \cot (e+f x))^{n+1} \sin ^2(e+f x)^{\frac {1}{2} (m+n+1)} \, _2F_1\left (\frac {n+1}{2},\frac {1}{2} (m+n+1);\frac {n+3}{2};\cos ^2(e+f x)\right )}{b f (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2697
Rubi steps
\begin {align*} \int (b \cot (e+f x))^n (a \csc (e+f x))^m \, dx &=-\frac {(b \cot (e+f x))^{1+n} (a \csc (e+f x))^m \, _2F_1\left (\frac {1+n}{2},\frac {1}{2} (1+m+n);\frac {3+n}{2};\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac {1}{2} (1+m+n)}}{b 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 = 2.06, size = 306, normalized size = 3.69 \begin {gather*} -\frac {a (-3+m+n) F_1\left (\frac {1}{2} (1-m-n);-n,1-m;\frac {1}{2} (3-m-n);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (b \cot (e+f x))^n (a \csc (e+f x))^{-1+m}}{f (-1+m+n) \left ((-3+m+n) F_1\left (\frac {1}{2} (1-m-n);-n,1-m;\frac {1}{2} (3-m-n);\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 (\frac {1}{2} (3-m-n);1-n,1-m;\frac {1}{2} (5-m-n);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-(-1+m) F_1\left (\frac {1}{2} (3-m-n);-n,2-m;\frac {1}{2} (5-m-n);\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 = 0.40, size = 0, normalized size = 0.00 \[\int \left (b \cot \left (f x +e \right )\right )^{n} \left (a \csc \left (f x +e \right )\right )^{m}\, dx\]
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 (a \csc {\left (e + f x \right )}\right )^{m} \left (b \cot {\left (e + f x \right )}\right )^{n}\, 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 {\left (b\,\mathrm {cot}\left (e+f\,x\right )\right )}^n\,{\left (\frac {a}{\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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