Optimal. Leaf size=48 \[ \frac {(b \csc (e+f x))^{5+n} \, _2F_1\left (3,\frac {5+n}{2};\frac {7+n}{2};\csc ^2(e+f x)\right )}{b^5 f (5+n)} \]
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Rubi [A]
time = 0.04, antiderivative size = 48, 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 = {2701, 371}
\begin {gather*} \frac {(b \csc (e+f x))^{n+5} \, _2F_1\left (3,\frac {n+5}{2};\frac {n+7}{2};\csc ^2(e+f x)\right )}{b^5 f (n+5)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 2701
Rubi steps
\begin {align*} \int (b \csc (e+f x))^n \sec ^5(e+f x) \, dx &=-\frac {\text {Subst}\left (\int \frac {x^{4+n}}{\left (-1+\frac {x^2}{b^2}\right )^3} \, dx,x,b \csc (e+f x)\right )}{b^5 f}\\ &=\frac {(b \csc (e+f x))^{5+n} \, _2F_1\left (3,\frac {5+n}{2};\frac {7+n}{2};\csc ^2(e+f x)\right )}{b^5 f (5+n)}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 51, normalized size = 1.06 \begin {gather*} -\frac {b (b \csc (e+f x))^{-1+n} \, _2F_1\left (3,\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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Maple [F]
time = 0.30, size = 0, normalized size = 0.00 \[\int \left (b \csc \left (f x +e \right )\right )^{n} \left (\sec ^{5}\left (f x +e \right )\right )\, dx\]
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} \sec ^{5}{\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 \frac {{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^n}{{\cos \left (e+f\,x\right )}^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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