Optimal. Leaf size=88 \[ \frac{b \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)}}{f}-\frac{b^2 \sqrt{\sin (e+f x)} F\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{d \sec (e+f x)}}{f \sqrt{b \tan (e+f x)}} \]
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Rubi [A] time = 0.115462, antiderivative size = 88, 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 = {2611, 2616, 2642, 2641} \[ \frac{b \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)}}{f}-\frac{b^2 \sqrt{\sin (e+f x)} F\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{d \sec (e+f x)}}{f \sqrt{b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2611
Rule 2616
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \sqrt{d \sec (e+f x)} (b \tan (e+f x))^{3/2} \, dx &=\frac{b \sqrt{d \sec (e+f x)} \sqrt{b \tan (e+f x)}}{f}-\frac{1}{2} b^2 \int \frac{\sqrt{d \sec (e+f x)}}{\sqrt{b \tan (e+f x)}} \, dx\\ &=\frac{b \sqrt{d \sec (e+f x)} \sqrt{b \tan (e+f x)}}{f}-\frac{\left (b^2 \sqrt{d \sec (e+f x)} \sqrt{b \sin (e+f x)}\right ) \int \frac{1}{\sqrt{b \sin (e+f x)}} \, dx}{2 \sqrt{b \tan (e+f x)}}\\ &=\frac{b \sqrt{d \sec (e+f x)} \sqrt{b \tan (e+f x)}}{f}-\frac{\left (b^2 \sqrt{d \sec (e+f x)} \sqrt{\sin (e+f x)}\right ) \int \frac{1}{\sqrt{\sin (e+f x)}} \, dx}{2 \sqrt{b \tan (e+f x)}}\\ &=-\frac{b^2 F\left (\left .\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )\right |2\right ) \sqrt{d \sec (e+f x)} \sqrt{\sin (e+f x)}}{f \sqrt{b \tan (e+f x)}}+\frac{b \sqrt{d \sec (e+f x)} \sqrt{b \tan (e+f x)}}{f}\\ \end{align*}
Mathematica [C] time = 0.73783, size = 105, normalized size = 1.19 \[ \frac{b \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)} \left (\sec ^{\frac{3}{2}}(e+f x)-\frac{\sec ^2\left (\frac{1}{2} (e+f x)\right ) \sqrt{\sec (e+f x)+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{\sqrt{2}}\right )}{f \sec ^{\frac{3}{2}}(e+f x)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.226, size = 211, normalized size = 2.4 \begin{align*}{\frac{\cos \left ( fx+e \right ) \sqrt{2}}{2\,f \left ( \cos \left ( fx+e \right ) -1 \right ) \sin \left ( fx+e \right ) } \left ({\frac{b\sin \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }} \right ) ^{{\frac{3}{2}}}\sqrt{{\frac{d}{\cos \left ( fx+e \right ) }}} \left ( i\cos \left ( fx+e \right ) \sin \left ( fx+e \right ) \sqrt{{\frac{i\cos \left ( fx+e \right ) -i+\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{-{\frac{i\cos \left ( fx+e \right ) -i-\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( fx+e \right ) -i+\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{{\frac{-i \left ( \cos \left ( fx+e \right ) -1 \right ) }{\sin \left ( fx+e \right ) }}}+\cos \left ( fx+e \right ) \sqrt{2}-\sqrt{2} \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 \sqrt{d \sec \left (f x + e\right )} \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 (\sqrt{d \sec \left (f x + e\right )} \sqrt{b \tan \left (f x + e\right )} b \tan \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 \sqrt{d \sec \left (f x + e\right )} \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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