Optimal. Leaf size=132 \[ \frac {3 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{20 b^2 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {\csc ^5(e+f x)}{5 b f (b \sec (e+f x))^{3/2}}+\frac {\csc ^3(e+f x)}{10 b f (b \sec (e+f x))^{3/2}}+\frac {3 \csc (e+f x)}{20 b f (b \sec (e+f x))^{3/2}} \]
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Rubi [A] time = 0.15, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2623, 2625, 3771, 2639} \[ \frac {3 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{20 b^2 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {\csc ^5(e+f x)}{5 b f (b \sec (e+f x))^{3/2}}+\frac {\csc ^3(e+f x)}{10 b f (b \sec (e+f x))^{3/2}}+\frac {3 \csc (e+f x)}{20 b f (b \sec (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2623
Rule 2625
Rule 2639
Rule 3771
Rubi steps
\begin {align*} \int \frac {\csc ^6(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx &=-\frac {\csc ^5(e+f x)}{5 b f (b \sec (e+f x))^{3/2}}-\frac {3 \int \frac {\csc ^4(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx}{10 b^2}\\ &=\frac {\csc ^3(e+f x)}{10 b f (b \sec (e+f x))^{3/2}}-\frac {\csc ^5(e+f x)}{5 b f (b \sec (e+f x))^{3/2}}-\frac {3 \int \frac {\csc ^2(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx}{20 b^2}\\ &=\frac {3 \csc (e+f x)}{20 b f (b \sec (e+f x))^{3/2}}+\frac {\csc ^3(e+f x)}{10 b f (b \sec (e+f x))^{3/2}}-\frac {\csc ^5(e+f x)}{5 b f (b \sec (e+f x))^{3/2}}+\frac {3 \int \frac {1}{\sqrt {b \sec (e+f x)}} \, dx}{40 b^2}\\ &=\frac {3 \csc (e+f x)}{20 b f (b \sec (e+f x))^{3/2}}+\frac {\csc ^3(e+f x)}{10 b f (b \sec (e+f x))^{3/2}}-\frac {\csc ^5(e+f x)}{5 b f (b \sec (e+f x))^{3/2}}+\frac {3 \int \sqrt {\cos (e+f x)} \, dx}{40 b^2 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}\\ &=\frac {3 \csc (e+f x)}{20 b f (b \sec (e+f x))^{3/2}}+\frac {\csc ^3(e+f x)}{10 b f (b \sec (e+f x))^{3/2}}-\frac {\csc ^5(e+f x)}{5 b f (b \sec (e+f x))^{3/2}}+\frac {3 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{20 b^2 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 87, normalized size = 0.66 \[ \frac {\sin (e+f x) \sqrt {b \sec (e+f x)} \left (-4 \csc ^6(e+f x)+6 \csc ^4(e+f x)+\csc ^2(e+f x)+3 \sqrt {\cos (e+f x)} \csc (e+f x) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )-3\right )}{20 b^3 f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sec \left (f x + e\right )} \csc \left (f x + e\right )^{6}}{b^{3} \sec \left (f x + e\right )^{3}}, x\right ) \]
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 \[ \int \frac {\csc \left (f x + e\right )^{6}}{\left (b \sec \left (f x + e\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.25, size = 923, normalized size = 6.99 \[ \text {result too large to display} \]
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 \[ \int \frac {\csc \left (f x + e\right )^{6}}{\left (b \sec \left (f x + e\right )\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\sin \left (e+f\,x\right )}^6\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \]
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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