Optimal. Leaf size=68 \[ \frac{(a+2 b) \tan ^5(c+d x)}{5 d}+\frac{(2 a+b) \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \tan ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.0505921, antiderivative size = 68, 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 = {3675, 373} \[ \frac{(a+2 b) \tan ^5(c+d x)}{5 d}+\frac{(2 a+b) \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \tan ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 3675
Rule 373
Rubi steps
\begin{align*} \int \sec ^6(c+d x) \left (a+b \tan ^2(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \left (1+x^2\right )^2 \left (a+b x^2\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a+(2 a+b) x^2+(a+2 b) x^4+b x^6\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{a \tan (c+d x)}{d}+\frac{(2 a+b) \tan ^3(c+d x)}{3 d}+\frac{(a+2 b) \tan ^5(c+d x)}{5 d}+\frac{b \tan ^7(c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.247424, size = 75, normalized size = 1.1 \[ \frac{\tan (c+d x) \left (21 a \tan ^4(c+d x)+70 a \tan ^2(c+d x)+105 a+15 b \sec ^6(c+d x)-3 b \sec ^4(c+d x)-4 b \sec ^2(c+d x)-8 b\right )}{105 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.087, size = 94, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ( b \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{7\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}+{\frac{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{35\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{8\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{105\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) -a \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 = 1.11457, size = 76, normalized size = 1.12 \begin{align*} \frac{15 \, b \tan \left (d x + c\right )^{7} + 21 \,{\left (a + 2 \, b\right )} \tan \left (d x + c\right )^{5} + 35 \,{\left (2 \, a + b\right )} \tan \left (d x + c\right )^{3} + 105 \, a \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 = 1.35736, size = 180, normalized size = 2.65 \begin{align*} \frac{{\left (8 \,{\left (7 \, a - b\right )} \cos \left (d x + c\right )^{6} + 4 \,{\left (7 \, a - b\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (7 \, a - b\right )} \cos \left (d x + c\right )^{2} + 15 \, b\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tan ^{2}{\left (c + d x \right )}\right ) \sec ^{6}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.46791, size = 95, normalized size = 1.4 \begin{align*} \frac{15 \, b \tan \left (d x + c\right )^{7} + 21 \, a \tan \left (d x + c\right )^{5} + 42 \, b \tan \left (d x + c\right )^{5} + 70 \, a \tan \left (d x + c\right )^{3} + 35 \, b \tan \left (d x + c\right )^{3} + 105 \, a \tan \left (d x + c\right )}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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