Optimal. Leaf size=89 \[ \frac {\cos ^2(e+f x)^{\frac {n+1}{2}} (a \csc (e+f x))^m (b \tan (e+f x))^{n+1} \, _2F_1\left (\frac {n+1}{2},\frac {1}{2} (-m+n+1);\frac {1}{2} (-m+n+3);\sin ^2(e+f x)\right )}{b f (-m+n+1)} \]
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Rubi [A] time = 0.15, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2618, 2602, 2577} \[ \frac {\cos ^2(e+f x)^{\frac {n+1}{2}} (a \csc (e+f x))^m (b \tan (e+f x))^{n+1} \, _2F_1\left (\frac {n+1}{2},\frac {1}{2} (-m+n+1);\frac {1}{2} (-m+n+3);\sin ^2(e+f x)\right )}{b f (-m+n+1)} \]
Antiderivative was successfully verified.
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Rule 2577
Rule 2602
Rule 2618
Rubi steps
\begin {align*} \int (a \csc (e+f x))^m (b \tan (e+f x))^n \, dx &=\left ((a \csc (e+f x))^m \left (\frac {\sin (e+f x)}{a}\right )^m\right ) \int \left (\frac {\sin (e+f x)}{a}\right )^{-m} (b \tan (e+f x))^n \, dx\\ &=\frac {\left (\cos ^{1+n}(e+f x) (a \csc (e+f x))^{1+m} \left (\frac {\sin (e+f x)}{a}\right )^{m-n} (b \tan (e+f x))^{1+n}\right ) \int \cos ^{-n}(e+f x) \left (\frac {\sin (e+f x)}{a}\right )^{-m+n} \, dx}{a b}\\ &=\frac {\cos ^2(e+f x)^{\frac {1+n}{2}} (a \csc (e+f x))^{1+m} \, _2F_1\left (\frac {1+n}{2},\frac {1}{2} (1-m+n);\frac {1}{2} (3-m+n);\sin ^2(e+f x)\right ) \sin (e+f x) (b \tan (e+f x))^{1+n}}{a b f (1-m+n)}\\ \end {align*}
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Mathematica [C] time = 2.04, size = 287, normalized size = 3.22 \[ -\frac {a (m-n-3) (a \csc (e+f x))^{m-1} (b \tan (e+f x))^n F_1\left (\frac {1}{2} (-m+n+1);n,1-m;\frac {1}{2} (-m+n+3);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{f (m-n-1) \left ((m-n-3) F_1\left (\frac {1}{2} (-m+n+1);n,1-m;\frac {1}{2} (-m+n+3);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 \tan ^2\left (\frac {1}{2} (e+f x)\right ) \left ((m-1) F_1\left (\frac {1}{2} (-m+n+3);n,2-m;\frac {1}{2} (-m+n+5);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+n F_1\left (\frac {1}{2} (-m+n+3);n+1,1-m;\frac {1}{2} (-m+n+5);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (a \csc \left (f x + e\right )\right )^{m} \left (b \tan \left (f x + e\right )\right )^{n}, x\right ) \]
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 \[ \int \left (a \csc \left (f x + e\right )\right )^{m} \left (b \tan \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.53, size = 0, normalized size = 0.00 \[ \int \left (a \csc \left (f x +e \right )\right )^{m} \left (b \tan \left (f x +e \right )\right )^{n}\, dx \]
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 \[ \int \left (a \csc \left (f x + e\right )\right )^{m} \left (b \tan \left (f x + e\right )\right )^{n}\,{d x} \]
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 \[ \int {\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\,{\left (\frac {a}{\sin \left (e+f\,x\right )}\right )}^m \,d x \]
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 \[ \int \left (a \csc {\left (e + f x \right )}\right )^{m} \left (b \tan {\left (e + f x \right )}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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