Optimal. Leaf size=58 \[ \frac{2 a \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{d \sec (e+f x)}}{f}+\frac{2 b \sqrt{d \sec (e+f x)}}{f} \]
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Rubi [A] time = 0.0474153, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3486, 3771, 2641} \[ \frac{2 a \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{d \sec (e+f x)}}{f}+\frac{2 b \sqrt{d \sec (e+f x)}}{f} \]
Antiderivative was successfully verified.
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Rule 3486
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \sqrt{d \sec (e+f x)} (a+b \tan (e+f x)) \, dx &=\frac{2 b \sqrt{d \sec (e+f x)}}{f}+a \int \sqrt{d \sec (e+f x)} \, dx\\ &=\frac{2 b \sqrt{d \sec (e+f x)}}{f}+\left (a \sqrt{\cos (e+f x)} \sqrt{d \sec (e+f x)}\right ) \int \frac{1}{\sqrt{\cos (e+f x)}} \, dx\\ &=\frac{2 b \sqrt{d \sec (e+f x)}}{f}+\frac{2 a \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{d \sec (e+f x)}}{f}\\ \end{align*}
Mathematica [A] time = 0.192757, size = 42, normalized size = 0.72 \[ \frac{2 \sqrt{d \sec (e+f x)} \left (a \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right )+b\right )}{f} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.25, size = 168, normalized size = 2.9 \begin{align*} 2\,{\frac{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2} \left ( \cos \left ( fx+e \right ) -1 \right ) ^{2}}{f \left ( \sin \left ( fx+e \right ) \right ) ^{4}}\sqrt{{\frac{d}{\cos \left ( fx+e \right ) }}} \left ( i\cos \left ( fx+e \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( \cos \left ( fx+e \right ) -1 \right ) }{\sin \left ( fx+e \right ) }},i \right ) a+i\sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( \cos \left ( fx+e \right ) -1 \right ) }{\sin \left ( fx+e \right ) }},i \right ) a+b \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 ) + a\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 (\sqrt{d \sec \left (f x + e\right )}{\left (b \tan \left (f x + e\right ) + a\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{d \sec{\left (e + f x \right )}} \left (a + b \tan{\left (e + f x \right )}\right )\, dx \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 ) + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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