Optimal. Leaf size=78 \[ -\frac{\cos (e+f x) \sin ^2(e+f x)^{-n/2} (b \tan (e+f x))^n \, _2F_1\left (\frac{1-n}{2},\frac{6-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right )}{f (1-n)} \]
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Rubi [A] time = 0.082808, antiderivative size = 78, normalized size of antiderivative = 1., 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 = {2601, 2576} \[ -\frac{\cos (e+f x) \sin ^2(e+f x)^{-n/2} (b \tan (e+f x))^n \, _2F_1\left (\frac{1-n}{2},\frac{6-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right )}{f (1-n)} \]
Antiderivative was successfully verified.
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Rule 2601
Rule 2576
Rubi steps
\begin{align*} \int \csc ^5(e+f x) (b \tan (e+f x))^n \, dx &=\left (\cos ^n(e+f x) \sin ^{-n}(e+f x) (b \tan (e+f x))^n\right ) \int \cos ^{-n}(e+f x) \sin ^{-5+n}(e+f x) \, dx\\ &=-\frac{\cos (e+f x) \, _2F_1\left (\frac{1-n}{2},\frac{6-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{-n/2} (b \tan (e+f x))^n}{f (1-n)}\\ \end{align*}
Mathematica [B] time = 7.63341, size = 254, normalized size = 3.26 \[ \frac{(b \tan (e+f x))^n \left (\cos (e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right )\right )^n \left (\frac{\tan ^4\left (\frac{1}{2} (e+f x)\right ) \, _2F_1\left (\frac{n}{2}+2,n;\frac{n}{2}+3;\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{n+4}+\frac{4 \tan ^2\left (\frac{1}{2} (e+f x)\right ) \, _2F_1\left (\frac{n}{2}+1,n;\frac{n}{2}+2;\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{n+2}+\frac{6 \, _2F_1\left (\frac{n}{2},n;\frac{n}{2}+1;\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{n}+\frac{\cot ^4\left (\frac{1}{2} (e+f x)\right ) \, _2F_1\left (\frac{n}{2}-2,n;\frac{n}{2}-1;\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{n-4}+\frac{4 \cot ^2\left (\frac{1}{2} (e+f x)\right ) \, _2F_1\left (\frac{n}{2}-1,n;\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{n-2}\right )}{16 f} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.337, size = 0, normalized size = 0. \begin{align*} \int \left ( \csc \left ( fx+e \right ) \right ) ^{5} \left ( b\tan \left ( fx+e \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \tan \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (b \tan \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{5}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \tan \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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