Optimal. Leaf size=61 \[ -\frac{8 b}{21 f \sin ^{\frac{3}{2}}(e+f x) (b \sec (e+f x))^{3/2}}-\frac{2 b}{7 f \sin ^{\frac{7}{2}}(e+f x) (b \sec (e+f x))^{3/2}} \]
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Rubi [A] time = 0.0780422, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2584, 2578} \[ -\frac{8 b}{21 f \sin ^{\frac{3}{2}}(e+f x) (b \sec (e+f x))^{3/2}}-\frac{2 b}{7 f \sin ^{\frac{7}{2}}(e+f x) (b \sec (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2584
Rule 2578
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{b \sec (e+f x)} \sin ^{\frac{9}{2}}(e+f x)} \, dx &=-\frac{2 b}{7 f (b \sec (e+f x))^{3/2} \sin ^{\frac{7}{2}}(e+f x)}+\frac{4}{7} \int \frac{1}{\sqrt{b \sec (e+f x)} \sin ^{\frac{5}{2}}(e+f x)} \, dx\\ &=-\frac{2 b}{7 f (b \sec (e+f x))^{3/2} \sin ^{\frac{7}{2}}(e+f x)}-\frac{8 b}{21 f (b \sec (e+f x))^{3/2} \sin ^{\frac{3}{2}}(e+f x)}\\ \end{align*}
Mathematica [A] time = 0.138539, size = 42, normalized size = 0.69 \[ \frac{2 b (2 \cos (2 (e+f x))-5)}{21 f \sin ^{\frac{7}{2}}(e+f x) (b \sec (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.107, size = 82, normalized size = 1.3 \begin{align*}{\frac{32\,\cos \left ( fx+e \right ) \left ( 4\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-7 \right ) \left ( -1+\cos \left ( fx+e \right ) \right ) ^{4}}{21\,f \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{2}+ \left ( \cos \left ( fx+e \right ) \right ) ^{2}-2\,\cos \left ( fx+e \right ) +1 \right ) ^{4}} \left ( \sin \left ( fx+e \right ) \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt{{\frac{b}{\cos \left ( fx+e \right ) }}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sec \left (f x + e\right )} \sin \left (f x + e\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.47585, size = 181, normalized size = 2.97 \begin{align*} \frac{2 \,{\left (4 \, \cos \left (f x + e\right )^{4} - 7 \, \cos \left (f x + e\right )^{2}\right )} \sqrt{\frac{b}{\cos \left (f x + e\right )}} \sqrt{\sin \left (f x + e\right )}}{21 \,{\left (b f \cos \left (f x + e\right )^{4} - 2 \, b f \cos \left (f x + e\right )^{2} + b f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sec \left (f x + e\right )} \sin \left (f x + e\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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