Optimal. Leaf size=63 \[ -\frac {2 b^5}{7 f (b \sec (e+f x))^{7/2}}+\frac {4 b^3}{3 f (b \sec (e+f x))^{3/2}}+\frac {2 b \sqrt {b \sec (e+f x)}}{f} \]
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Rubi [A] time = 0.06, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2622, 270} \[ -\frac {2 b^5}{7 f (b \sec (e+f x))^{7/2}}+\frac {4 b^3}{3 f (b \sec (e+f x))^{3/2}}+\frac {2 b \sqrt {b \sec (e+f x)}}{f} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2622
Rubi steps
\begin {align*} \int (b \sec (e+f x))^{3/2} \sin ^5(e+f x) \, dx &=\frac {b^5 \operatorname {Subst}\left (\int \frac {\left (-1+\frac {x^2}{b^2}\right )^2}{x^{9/2}} \, dx,x,b \sec (e+f x)\right )}{f}\\ &=\frac {b^5 \operatorname {Subst}\left (\int \left (\frac {1}{x^{9/2}}-\frac {2}{b^2 x^{5/2}}+\frac {1}{b^4 \sqrt {x}}\right ) \, dx,x,b \sec (e+f x)\right )}{f}\\ &=-\frac {2 b^5}{7 f (b \sec (e+f x))^{7/2}}+\frac {4 b^3}{3 f (b \sec (e+f x))^{3/2}}+\frac {2 b \sqrt {b \sec (e+f x)}}{f}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 42, normalized size = 0.67 \[ \frac {b (44 \cos (2 (e+f x))-3 \cos (4 (e+f x))+215) \sqrt {b \sec (e+f x)}}{84 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 43, normalized size = 0.68 \[ -\frac {2 \, {\left (3 \, b \cos \left (f x + e\right )^{4} - 14 \, b \cos \left (f x + e\right )^{2} - 21 \, b\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{21 \, f} \]
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 \[ \int \left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}} \sin \left (f x + e\right )^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.23, size = 959, normalized size = 15.22 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 55, normalized size = 0.87 \[ -\frac {2 \, b {\left (\frac {3 \, b^{4}}{\left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {7}{2}}} - \frac {14 \, b^{2}}{\left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {3}{2}}} - 21 \, \sqrt {\frac {b}{\cos \left (f x + e\right )}}\right )}}{21 \, f} \]
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 \[ \int {\sin \left (e+f\,x\right )}^5\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2} \,d x \]
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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