Optimal. Leaf size=97 \[ -\frac{2 d^2 E\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{b \tan (e+f x)}}{b^2 f \sqrt{\sin (e+f x)} \sqrt{d \sec (e+f x)}}-\frac{2 d^2}{b f \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)}} \]
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Rubi [A] time = 0.122558, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2608, 2616, 2640, 2639} \[ -\frac{2 d^2 E\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{b \tan (e+f x)}}{b^2 f \sqrt{\sin (e+f x)} \sqrt{d \sec (e+f x)}}-\frac{2 d^2}{b f \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2608
Rule 2616
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{(d \sec (e+f x))^{3/2}}{(b \tan (e+f x))^{3/2}} \, dx &=-\frac{2 d^2}{b f \sqrt{d \sec (e+f x)} \sqrt{b \tan (e+f x)}}-\frac{d^2 \int \frac{\sqrt{b \tan (e+f x)}}{\sqrt{d \sec (e+f x)}} \, dx}{b^2}\\ &=-\frac{2 d^2}{b f \sqrt{d \sec (e+f x)} \sqrt{b \tan (e+f x)}}-\frac{\left (d^2 \sqrt{b \tan (e+f x)}\right ) \int \sqrt{b \sin (e+f x)} \, dx}{b^2 \sqrt{d \sec (e+f x)} \sqrt{b \sin (e+f x)}}\\ &=-\frac{2 d^2}{b f \sqrt{d \sec (e+f x)} \sqrt{b \tan (e+f x)}}-\frac{\left (d^2 \sqrt{b \tan (e+f x)}\right ) \int \sqrt{\sin (e+f x)} \, dx}{b^2 \sqrt{d \sec (e+f x)} \sqrt{\sin (e+f x)}}\\ &=-\frac{2 d^2}{b f \sqrt{d \sec (e+f x)} \sqrt{b \tan (e+f x)}}-\frac{2 d^2 E\left (\left .\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )\right |2\right ) \sqrt{b \tan (e+f x)}}{b^2 f \sqrt{d \sec (e+f x)} \sqrt{\sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.608908, size = 70, normalized size = 0.72 \[ \frac{2 d^2 \left (\sqrt [4]{-\tan ^2(e+f x)} \, _2F_1\left (-\frac{1}{4},\frac{1}{4};\frac{3}{4};\sec ^2(e+f x)\right )-1\right )}{b f \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.185, size = 538, normalized size = 5.6 \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{\left (d \sec \left (f x + e\right )\right )^{\frac{3}{2}}}{\left (b \tan \left (f x + e\right )\right )^{\frac{3}{2}}}\,{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{\sqrt{d \sec \left (f x + e\right )} \sqrt{b \tan \left (f x + e\right )} d \sec \left (f x + e\right )}{b^{2} \tan \left (f x + e\right )^{2}}, 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{\left (d \sec \left (f x + e\right )\right )^{\frac{3}{2}}}{\left (b \tan \left (f x + e\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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