Optimal. Leaf size=88 \[ \frac {b \cos ^{-1+m}(e+f x) \cos ^2(e+f x)^{\frac {1-m}{2}} (b \csc (e+f x))^{-1+n} \, _2F_1\left (\frac {1-m}{2},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(e+f x)\right )}{f (1-n)} \]
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Rubi [A]
time = 0.06, antiderivative size = 88, 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 = {2667, 2657}
\begin {gather*} \frac {b \cos ^{m-1}(e+f x) \cos ^2(e+f x)^{\frac {1-m}{2}} (b \csc (e+f x))^{n-1} \, _2F_1\left (\frac {1-m}{2},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(e+f x)\right )}{f (1-n)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2657
Rule 2667
Rubi steps
\begin {align*} \int \cos ^m(e+f x) (b \csc (e+f x))^n \, dx &=\left (b^2 (b \csc (e+f x))^{-1+n} (b \sin (e+f x))^{-1+n}\right ) \int \cos ^m(e+f x) (b \sin (e+f x))^{-n} \, dx\\ &=\frac {b \cos ^{-1+m}(e+f x) \cos ^2(e+f x)^{\frac {1-m}{2}} (b \csc (e+f x))^{-1+n} \, _2F_1\left (\frac {1-m}{2},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(e+f x)\right )}{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 = 0.17, size = 314, normalized size = 3.57 \begin {gather*} -\frac {2 (-3+n) F_1\left (\frac {1}{2}-\frac {n}{2};-m,1+m-n;\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 ^3\left (\frac {1}{2} (e+f x)\right ) \cos ^m(e+f x) (b \csc (e+f x))^n \sin \left (\frac {1}{2} (e+f x)\right )}{f (-1+n) \left ((-3+n) F_1\left (\frac {1}{2}-\frac {n}{2};-m,1+m-n;\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 (m F_1\left (\frac {3}{2}-\frac {n}{2};1-m,1+m-n;\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 )+(1+m-n) F_1\left (\frac {3}{2}-\frac {n}{2};-m,2+m-n;\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 ) \sin ^2\left (\frac {1}{2} (e+f x)\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \left (\cos ^{m}\left (f x +e \right )\right ) \left (b \csc \left (f x +e \right )\right )^{n}\, dx\]
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 )^{n} \cos ^{m}{\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 {\cos \left (e+f\,x\right )}^m\,{\left (\frac {b}{\sin \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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