Optimal. Leaf size=97 \[ \frac{\left (a^2+2 b^2\right ) \tan ^5(c+d x)}{5 d}+\frac{\left (2 a^2+b^2\right ) \tan ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}+\frac{a b \sec ^6(c+d x)}{3 d}+\frac{b^2 \tan ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.0825734, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3506, 696, 1810} \[ \frac{\left (a^2+2 b^2\right ) \tan ^5(c+d x)}{5 d}+\frac{\left (2 a^2+b^2\right ) \tan ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}+\frac{a b \sec ^6(c+d x)}{3 d}+\frac{b^2 \tan ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 3506
Rule 696
Rule 1810
Rubi steps
\begin{align*} \int \sec ^6(c+d x) (a+b \tan (c+d x))^2 \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^2 \left (1+\frac{x^2}{b^2}\right )^2 \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac{a b \sec ^6(c+d x)}{3 d}+\frac{\operatorname{Subst}\left (\int \left (1+\frac{x^2}{b^2}\right )^2 \left (-2 a x+(a+x)^2\right ) \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac{a b \sec ^6(c+d x)}{3 d}+\frac{\operatorname{Subst}\left (\int \left (a^2+\frac{\left (2 a^2+b^2\right ) x^2}{b^2}+\frac{\left (a^2+2 b^2\right ) x^4}{b^4}+\frac{x^6}{b^4}\right ) \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac{a b \sec ^6(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}+\frac{\left (2 a^2+b^2\right ) \tan ^3(c+d x)}{3 d}+\frac{\left (a^2+2 b^2\right ) \tan ^5(c+d x)}{5 d}+\frac{b^2 \tan ^7(c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.638971, size = 104, normalized size = 1.07 \[ \frac{\tan (c+d x) \left (21 \left (a^2+2 b^2\right ) \tan ^4(c+d x)+35 \left (2 a^2+b^2\right ) \tan ^2(c+d x)+105 a^2+35 a b \tan ^5(c+d x)+105 a b \tan ^3(c+d x)+105 a b \tan (c+d x)+15 b^2 \tan ^6(c+d x)\right )}{105 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 110, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({b}^{2} \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 ) +{\frac{ab}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}-{a}^{2} \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 = 1.35239, size = 140, normalized size = 1.44 \begin{align*} \frac{15 \, b^{2} \tan \left (d x + c\right )^{7} + 35 \, a b \tan \left (d x + c\right )^{6} + 105 \, a b \tan \left (d x + c\right )^{4} + 21 \,{\left (a^{2} + 2 \, b^{2}\right )} \tan \left (d x + c\right )^{5} + 105 \, a b \tan \left (d x + c\right )^{2} + 35 \,{\left (2 \, a^{2} + b^{2}\right )} \tan \left (d x + c\right )^{3} + 105 \, a^{2} \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 = 1.95376, size = 231, normalized size = 2.38 \begin{align*} \frac{35 \, a b \cos \left (d x + c\right ) +{\left (8 \,{\left (7 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{6} + 4 \,{\left (7 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (7 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 15 \, b^{2}\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{\left (c + d x \right )}\right )^{2} \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 = 1.39152, size = 159, normalized size = 1.64 \begin{align*} \frac{15 \, b^{2} \tan \left (d x + c\right )^{7} + 35 \, a b \tan \left (d x + c\right )^{6} + 21 \, a^{2} \tan \left (d x + c\right )^{5} + 42 \, b^{2} \tan \left (d x + c\right )^{5} + 105 \, a b \tan \left (d x + c\right )^{4} + 70 \, a^{2} \tan \left (d x + c\right )^{3} + 35 \, b^{2} \tan \left (d x + c\right )^{3} + 105 \, a b \tan \left (d x + c\right )^{2} + 105 \, a^{2} \tan \left (d x + c\right )}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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