Optimal. Leaf size=115 \[ -\frac{b \sin ^{\frac{7}{2}}(e+f x)}{5 f (b \sec (e+f x))^{3/2}}-\frac{7 b \sin ^{\frac{3}{2}}(e+f x)}{30 f (b \sec (e+f x))^{3/2}}+\frac{7 \sqrt{\sin (e+f x)} E\left (\left .e+f x-\frac{\pi }{4}\right |2\right )}{20 f \sqrt{\sin (2 e+2 f x)} \sqrt{b \sec (e+f x)}} \]
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Rubi [A] time = 0.164466, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2583, 2585, 2572, 2639} \[ -\frac{b \sin ^{\frac{7}{2}}(e+f x)}{5 f (b \sec (e+f x))^{3/2}}-\frac{7 b \sin ^{\frac{3}{2}}(e+f x)}{30 f (b \sec (e+f x))^{3/2}}+\frac{7 \sqrt{\sin (e+f x)} E\left (\left .e+f x-\frac{\pi }{4}\right |2\right )}{20 f \sqrt{\sin (2 e+2 f x)} \sqrt{b \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2583
Rule 2585
Rule 2572
Rule 2639
Rubi steps
\begin{align*} \int \frac{\sin ^{\frac{9}{2}}(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx &=-\frac{b \sin ^{\frac{7}{2}}(e+f x)}{5 f (b \sec (e+f x))^{3/2}}+\frac{7}{10} \int \frac{\sin ^{\frac{5}{2}}(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx\\ &=-\frac{7 b \sin ^{\frac{3}{2}}(e+f x)}{30 f (b \sec (e+f x))^{3/2}}-\frac{b \sin ^{\frac{7}{2}}(e+f x)}{5 f (b \sec (e+f x))^{3/2}}+\frac{7}{20} \int \frac{\sqrt{\sin (e+f x)}}{\sqrt{b \sec (e+f x)}} \, dx\\ &=-\frac{7 b \sin ^{\frac{3}{2}}(e+f x)}{30 f (b \sec (e+f x))^{3/2}}-\frac{b \sin ^{\frac{7}{2}}(e+f x)}{5 f (b \sec (e+f x))^{3/2}}+\frac{7 \int \sqrt{b \cos (e+f x)} \sqrt{\sin (e+f x)} \, dx}{20 \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}\\ &=-\frac{7 b \sin ^{\frac{3}{2}}(e+f x)}{30 f (b \sec (e+f x))^{3/2}}-\frac{b \sin ^{\frac{7}{2}}(e+f x)}{5 f (b \sec (e+f x))^{3/2}}+\frac{\left (7 \sqrt{\sin (e+f x)}\right ) \int \sqrt{\sin (2 e+2 f x)} \, dx}{20 \sqrt{b \sec (e+f x)} \sqrt{\sin (2 e+2 f x)}}\\ &=-\frac{7 b \sin ^{\frac{3}{2}}(e+f x)}{30 f (b \sec (e+f x))^{3/2}}-\frac{b \sin ^{\frac{7}{2}}(e+f x)}{5 f (b \sec (e+f x))^{3/2}}+\frac{7 E\left (\left .e-\frac{\pi }{4}+f x\right |2\right ) \sqrt{\sin (e+f x)}}{20 f \sqrt{b \sec (e+f x)} \sqrt{\sin (2 e+2 f x)}}\\ \end{align*}
Mathematica [C] time = 0.536627, size = 86, normalized size = 0.75 \[ -\frac{b \left (42 \sqrt [4]{-\tan ^2(e+f x)} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{1}{2};\sec ^2(e+f x)\right )-26 \cos (2 (e+f x))+3 \cos (4 (e+f x))+23\right )}{120 f \sqrt{\sin (e+f x)} (b \sec (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.118, size = 532, normalized size = 4.6 \begin{align*} \text{result too large to display} \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{\sin \left (f x + e\right )^{\frac{9}{2}}}{\sqrt{b \sec \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt{b \sec \left (f x + e\right )} \sqrt{\sin \left (f x + e\right )}}{b \sec \left (f x + e\right )}, x\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{\sin \left (f x + e\right )^{\frac{9}{2}}}{\sqrt{b \sec \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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