Optimal. Leaf size=50 \[ \frac{(a+b) \tan ^5(e+f x)}{5 f}+\frac{(2 a+b) \tan ^3(e+f x)}{3 f}+\frac{a \tan (e+f x)}{f} \]
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Rubi [A] time = 0.0444607, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3191, 373} \[ \frac{(a+b) \tan ^5(e+f x)}{5 f}+\frac{(2 a+b) \tan ^3(e+f x)}{3 f}+\frac{a \tan (e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 3191
Rule 373
Rubi steps
\begin{align*} \int \sec ^6(e+f x) \left (a+b \sin ^2(e+f x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \left (1+x^2\right ) \left (a+(a+b) x^2\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \left (a+(2 a+b) x^2+(a+b) x^4\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{a \tan (e+f x)}{f}+\frac{(2 a+b) \tan ^3(e+f x)}{3 f}+\frac{(a+b) \tan ^5(e+f x)}{5 f}\\ \end{align*}
Mathematica [A] time = 0.178007, size = 64, normalized size = 1.28 \[ \frac{\tan (e+f x) \left (3 a \tan ^4(e+f x)+10 a \tan ^2(e+f x)+15 a+3 b \sec ^4(e+f x)-b \sec ^2(e+f x)-2 b\right )}{15 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 76, normalized size = 1.5 \begin{align*}{\frac{1}{f} \left ( -a \left ( -{\frac{8}{15}}-{\frac{ \left ( \sec \left ( fx+e \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \sec \left ( fx+e \right ) \right ) ^{2}}{15}} \right ) \tan \left ( fx+e \right ) +b \left ({\frac{ \left ( \sin \left ( fx+e \right ) \right ) ^{3}}{5\, \left ( \cos \left ( fx+e \right ) \right ) ^{5}}}+{\frac{2\, \left ( \sin \left ( fx+e \right ) \right ) ^{3}}{15\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.992635, size = 58, normalized size = 1.16 \begin{align*} \frac{3 \,{\left (a + b\right )} \tan \left (f x + e\right )^{5} + 5 \,{\left (2 \, a + b\right )} \tan \left (f x + e\right )^{3} + 15 \, a \tan \left (f x + e\right )}{15 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94451, size = 143, normalized size = 2.86 \begin{align*} \frac{{\left (2 \,{\left (4 \, a - b\right )} \cos \left (f x + e\right )^{4} +{\left (4 \, a - b\right )} \cos \left (f x + e\right )^{2} + 3 \, a + 3 \, b\right )} \sin \left (f x + e\right )}{15 \, f \cos \left (f x + e\right )^{5}} \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 [A] time = 1.13642, size = 86, normalized size = 1.72 \begin{align*} \frac{3 \, a \tan \left (f x + e\right )^{5} + 3 \, b \tan \left (f x + e\right )^{5} + 10 \, a \tan \left (f x + e\right )^{3} + 5 \, b \tan \left (f x + e\right )^{3} + 15 \, a \tan \left (f x + e\right )}{15 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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