Optimal. Leaf size=72 \[ \frac {b \sqrt {\cos ^2(e+f x)} \sec (e+f x) (b \csc (e+f x))^{n-1} \, _2F_1\left (\frac {7}{2},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(e+f x)\right )}{f (1-n)} \]
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Rubi [A] time = 0.08, antiderivative size = 72, 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 = {2631, 2577} \[ \frac {b \sqrt {\cos ^2(e+f x)} \sec (e+f x) (b \csc (e+f x))^{n-1} \, _2F_1\left (\frac {7}{2},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(e+f x)\right )}{f (1-n)} \]
Antiderivative was successfully verified.
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Rule 2577
Rule 2631
Rubi steps
\begin {align*} \int (b \csc (e+f x))^n \sec ^6(e+f x) \, dx &=\left (b^2 (b \csc (e+f x))^{-1+n} (b \sin (e+f x))^{-1+n}\right ) \int \sec ^6(e+f x) (b \sin (e+f x))^{-n} \, dx\\ &=\frac {b \sqrt {\cos ^2(e+f x)} (b \csc (e+f x))^{-1+n} \, _2F_1\left (\frac {7}{2},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(e+f x)\right ) \sec (e+f x)}{f (1-n)}\\ \end {align*}
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Mathematica [A] time = 0.61, size = 77, normalized size = 1.07 \[ \frac {\tan (e+f x) \sec ^2(e+f x)^{-n/2} (b \csc (e+f x))^n \, _2F_1\left (-\frac {n}{2}-2,\frac {1}{2}-\frac {n}{2};\frac {3}{2}-\frac {n}{2};-\tan ^2(e+f x)\right )}{f (1-n)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.06, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \csc \left (f x + e\right )\right )^{n} \sec \left (f x + e\right )^{6}, x\right ) \]
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 \[ \int \left (b \csc \left (f x + e\right )\right )^{n} \sec \left (f x + e\right )^{6}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.28, size = 0, normalized size = 0.00 \[ \int \left (b \csc \left (f x +e \right )\right )^{n} \left (\sec ^{6}\left (f x +e \right )\right )\, dx \]
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 \[ \int \left (b \csc \left (f x + e\right )\right )^{n} \sec \left (f x + e\right )^{6}\,{d x} \]
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 \[ \int \frac {{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^n}{{\cos \left (e+f\,x\right )}^6} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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