Optimal. Leaf size=71 \[ -\frac {8 b}{5 a^3 f \sqrt {a \sin (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {2 b}{5 a f (a \sin (e+f x))^{5/2} \sqrt {b \sec (e+f x)}} \]
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Rubi [A] time = 0.11, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2584, 2578} \[ -\frac {8 b}{5 a^3 f \sqrt {a \sin (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {2 b}{5 a f (a \sin (e+f x))^{5/2} \sqrt {b \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2578
Rule 2584
Rubi steps
\begin {align*} \int \frac {\sqrt {b \sec (e+f x)}}{(a \sin (e+f x))^{7/2}} \, dx &=-\frac {2 b}{5 a f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{5/2}}+\frac {4 \int \frac {\sqrt {b \sec (e+f x)}}{(a \sin (e+f x))^{3/2}} \, dx}{5 a^2}\\ &=-\frac {2 b}{5 a f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{5/2}}-\frac {8 b}{5 a^3 f \sqrt {b \sec (e+f x)} \sqrt {a \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 52, normalized size = 0.73 \[ \frac {2 (2 \cos (2 (e+f x))-3) \cot (e+f x) \sqrt {b \sec (e+f x)}}{5 a^2 f (a \sin (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.13, size = 73, normalized size = 1.03 \[ -\frac {2 \, {\left (4 \, \cos \left (f x + e\right )^{3} - 5 \, \cos \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{5 \, {\left (a^{4} f \cos \left (f x + e\right )^{2} - a^{4} f\right )} \sin \left (f x + e\right )} \]
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 \[ \int \frac {\sqrt {b \sec \left (f x + e\right )}}{\left (a \sin \left (f x + e\right )\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 52, normalized size = 0.73 \[ \frac {2 \left (4 \left (\cos ^{2}\left (f x +e \right )\right )-5\right ) \cos \left (f x +e \right ) \sqrt {\frac {b}{\cos \left (f x +e \right )}}\, \sin \left (f x +e \right )}{5 f \left (a \sin \left (f x +e \right )\right )^{\frac {7}{2}}} \]
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 \[ \int \frac {\sqrt {b \sec \left (f x + e\right )}}{\left (a \sin \left (f x + e\right )\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.81, size = 83, normalized size = 1.17 \[ -\frac {4\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}\,\left (3\,\cos \left (e+f\,x\right )-4\,\cos \left (3\,e+3\,f\,x\right )+\cos \left (5\,e+5\,f\,x\right )\right )}{5\,a^3\,f\,\sqrt {a\,\sin \left (e+f\,x\right )}\,\left (\cos \left (4\,e+4\,f\,x\right )-4\,\cos \left (2\,e+2\,f\,x\right )+3\right )} \]
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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