Optimal. Leaf size=43 \[ \frac{(3 a+2 b) \tan (e+f x)}{3 f}+\frac{b \tan (e+f x) \sec ^2(e+f x)}{3 f} \]
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Rubi [A] time = 0.0383772, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4046, 3767, 8} \[ \frac{(3 a+2 b) \tan (e+f x)}{3 f}+\frac{b \tan (e+f x) \sec ^2(e+f x)}{3 f} \]
Antiderivative was successfully verified.
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Rule 4046
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \sec ^2(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx &=\frac{b \sec ^2(e+f x) \tan (e+f x)}{3 f}+\frac{1}{3} (3 a+2 b) \int \sec ^2(e+f x) \, dx\\ &=\frac{b \sec ^2(e+f x) \tan (e+f x)}{3 f}-\frac{(3 a+2 b) \operatorname{Subst}(\int 1 \, dx,x,-\tan (e+f x))}{3 f}\\ &=\frac{(3 a+2 b) \tan (e+f x)}{3 f}+\frac{b \sec ^2(e+f x) \tan (e+f x)}{3 f}\\ \end{align*}
Mathematica [A] time = 0.0878204, size = 36, normalized size = 0.84 \[ \frac{a \tan (e+f x)}{f}+\frac{b \left (\frac{1}{3} \tan ^3(e+f x)+\tan (e+f x)\right )}{f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 35, normalized size = 0.8 \begin{align*}{\frac{1}{f} \left ( a\tan \left ( fx+e \right ) -b \left ( -{\frac{2}{3}}-{\frac{ \left ( \sec \left ( fx+e \right ) \right ) ^{2}}{3}} \right ) \tan \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02257, size = 46, normalized size = 1.07 \begin{align*} \frac{{\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} b + 3 \, a \tan \left (f x + e\right )}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.458157, size = 95, normalized size = 2.21 \begin{align*} \frac{{\left ({\left (3 \, a + 2 \, b\right )} \cos \left (f x + e\right )^{2} + b\right )} \sin \left (f x + e\right )}{3 \, f \cos \left (f x + e\right )^{3}} \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 \left (a + b \sec ^{2}{\left (e + f x \right )}\right ) \sec ^{2}{\left (e + f x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26565, size = 50, normalized size = 1.16 \begin{align*} \frac{b \tan \left (f x + e\right )^{3} + 3 \, a \tan \left (f x + e\right ) + 3 \, b \tan \left (f x + e\right )}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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