Optimal. Leaf size=87 \[ \frac{(6 A+7 C) \tan ^5(c+d x)}{35 d}+\frac{2 (6 A+7 C) \tan ^3(c+d x)}{21 d}+\frac{(6 A+7 C) \tan (c+d x)}{7 d}+\frac{A \tan (c+d x) \sec ^6(c+d x)}{7 d} \]
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Rubi [A] time = 0.0496711, antiderivative size = 87, 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 = {3012, 3767} \[ \frac{(6 A+7 C) \tan ^5(c+d x)}{35 d}+\frac{2 (6 A+7 C) \tan ^3(c+d x)}{21 d}+\frac{(6 A+7 C) \tan (c+d x)}{7 d}+\frac{A \tan (c+d x) \sec ^6(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 3012
Rule 3767
Rubi steps
\begin{align*} \int \left (A+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx &=\frac{A \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac{1}{7} (6 A+7 C) \int \sec ^6(c+d x) \, dx\\ &=\frac{A \sec ^6(c+d x) \tan (c+d x)}{7 d}-\frac{(6 A+7 C) \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (c+d x)\right )}{7 d}\\ &=\frac{(6 A+7 C) \tan (c+d x)}{7 d}+\frac{A \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac{2 (6 A+7 C) \tan ^3(c+d x)}{21 d}+\frac{(6 A+7 C) \tan ^5(c+d x)}{35 d}\\ \end{align*}
Mathematica [A] time = 0.30241, size = 81, normalized size = 0.93 \[ \frac{A \left (\frac{1}{7} \tan ^7(c+d x)+\frac{3}{5} \tan ^5(c+d x)+\tan ^3(c+d x)+\tan (c+d x)\right )}{d}+\frac{C \left (\frac{1}{5} \tan ^5(c+d x)+\frac{2}{3} \tan ^3(c+d x)+\tan (c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 78, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ( -A \left ( -{\frac{16}{35}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{6}}{7}}-{\frac{6\, \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{35}}-{\frac{8\, \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{35}} \right ) \tan \left ( dx+c \right ) -C \left ( -{\frac{8}{15}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15}} \right ) \tan \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01291, size = 81, normalized size = 0.93 \begin{align*} \frac{15 \, A \tan \left (d x + c\right )^{7} + 21 \,{\left (3 \, A + C\right )} \tan \left (d x + c\right )^{5} + 35 \,{\left (3 \, A + 2 \, C\right )} \tan \left (d x + c\right )^{3} + 105 \,{\left (A + C\right )} \tan \left (d x + c\right )}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60903, size = 188, normalized size = 2.16 \begin{align*} \frac{{\left (8 \,{\left (6 \, A + 7 \, C\right )} \cos \left (d x + c\right )^{6} + 4 \,{\left (6 \, A + 7 \, C\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (6 \, A + 7 \, C\right )} \cos \left (d x + c\right )^{2} + 15 \, A\right )} \sin \left (d x + c\right )}{105 \, d \cos \left (d x + c\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 = 1.19337, size = 107, normalized size = 1.23 \begin{align*} \frac{15 \, A \tan \left (d x + c\right )^{7} + 63 \, A \tan \left (d x + c\right )^{5} + 21 \, C \tan \left (d x + c\right )^{5} + 105 \, A \tan \left (d x + c\right )^{3} + 70 \, C \tan \left (d x + c\right )^{3} + 105 \, A \tan \left (d x + c\right ) + 105 \, C \tan \left (d x + c\right )}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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