Optimal. Leaf size=31 \[ \frac{\tan ^7(a+b x)}{7 b}+\frac{\tan ^5(a+b x)}{5 b} \]
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Rubi [A] time = 0.0320576, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2607, 14} \[ \frac{\tan ^7(a+b x)}{7 b}+\frac{\tan ^5(a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 2607
Rule 14
Rubi steps
\begin{align*} \int \sec ^4(a+b x) \tan ^4(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac{\tan ^5(a+b x)}{5 b}+\frac{\tan ^7(a+b x)}{7 b}\\ \end{align*}
Mathematica [B] time = 0.0345766, size = 77, normalized size = 2.48 \[ \frac{2 \tan (a+b x)}{35 b}+\frac{\tan (a+b x) \sec ^6(a+b x)}{7 b}-\frac{8 \tan (a+b x) \sec ^4(a+b x)}{35 b}+\frac{\tan (a+b x) \sec ^2(a+b x)}{35 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 42, normalized size = 1.4 \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{5}}{7\, \left ( \cos \left ( bx+a \right ) \right ) ^{7}}}+{\frac{2\, \left ( \sin \left ( bx+a \right ) \right ) ^{5}}{35\, \left ( \cos \left ( bx+a \right ) \right ) ^{5}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00731, size = 35, normalized size = 1.13 \begin{align*} \frac{5 \, \tan \left (b x + a\right )^{7} + 7 \, \tan \left (b x + a\right )^{5}}{35 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62264, size = 131, normalized size = 4.23 \begin{align*} \frac{{\left (2 \, \cos \left (b x + a\right )^{6} + \cos \left (b x + a\right )^{4} - 8 \, \cos \left (b x + a\right )^{2} + 5\right )} \sin \left (b x + a\right )}{35 \, b \cos \left (b x + a\right )^{7}} \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 = 1.18599, size = 35, normalized size = 1.13 \begin{align*} \frac{5 \, \tan \left (b x + a\right )^{7} + 7 \, \tan \left (b x + a\right )^{5}}{35 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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