Optimal. Leaf size=41 \[ \frac {2 b^3}{3 f (b \sec (e+f x))^{3/2}}+\frac {2 b \sqrt {b \sec (e+f x)}}{f} \]
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Rubi [A] time = 0.05, antiderivative size = 41, 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, 14} \[ \frac {2 b^3}{3 f (b \sec (e+f x))^{3/2}}+\frac {2 b \sqrt {b \sec (e+f x)}}{f} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2622
Rubi steps
\begin {align*} \int (b \sec (e+f x))^{3/2} \sin ^3(e+f x) \, dx &=\frac {b^3 \operatorname {Subst}\left (\int \frac {-1+\frac {x^2}{b^2}}{x^{5/2}} \, dx,x,b \sec (e+f x)\right )}{f}\\ &=\frac {b^3 \operatorname {Subst}\left (\int \left (-\frac {1}{x^{5/2}}+\frac {1}{b^2 \sqrt {x}}\right ) \, dx,x,b \sec (e+f x)\right )}{f}\\ &=\frac {2 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 = 0.07, size = 30, normalized size = 0.73 \[ \frac {b (\cos (2 (e+f x))+7) \sqrt {b \sec (e+f x)}}{3 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 31, normalized size = 0.76 \[ \frac {2 \, {\left (b \cos \left (f x + e\right )^{2} + 3 \, b\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{3 \, f} \]
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 \[ \int \left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}} \sin \left (f x + e\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.19, size = 949, normalized size = 23.15 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 37, normalized size = 0.90 \[ \frac {2 \, b {\left (\frac {b^{2}}{\left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {3}{2}}} + 3 \, \sqrt {\frac {b}{\cos \left (f x + e\right )}}\right )}}{3 \, f} \]
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 \[ \int {\sin \left (e+f\,x\right )}^3\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2} \,d x \]
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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