Optimal. Leaf size=74 \[ \frac{a^2 \tan (c+d x)}{d}+\frac{b (2 a+b) \tan ^5(c+d x)}{5 d}+\frac{a (a+2 b) \tan ^3(c+d x)}{3 d}+\frac{b^2 \tan ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.0693627, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {3675, 373} \[ \frac{a^2 \tan (c+d x)}{d}+\frac{b (2 a+b) \tan ^5(c+d x)}{5 d}+\frac{a (a+2 b) \tan ^3(c+d x)}{3 d}+\frac{b^2 \tan ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 3675
Rule 373
Rubi steps
\begin{align*} \int \sec ^4(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (1+x^2\right ) \left (a+b x^2\right )^2 \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^2+a (a+2 b) x^2+b (2 a+b) x^4+b^2 x^6\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{a^2 \tan (c+d x)}{d}+\frac{a (a+2 b) \tan ^3(c+d x)}{3 d}+\frac{b (2 a+b) \tan ^5(c+d x)}{5 d}+\frac{b^2 \tan ^7(c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.514383, size = 83, normalized size = 1.12 \[ \frac{\tan (c+d x) \left (\left (35 a^2-14 a b+3 b^2\right ) \sec ^2(c+d x)+70 a^2+6 b (7 a-4 b) \sec ^4(c+d x)-28 a b+15 b^2 \sec ^6(c+d x)+6 b^2\right )}{105 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 111, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ({b}^{2} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{7\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}+{\frac{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{35\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}} \right ) +2\,ab \left ( 1/5\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+2/15\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) -{a}^{2} \left ( -{\frac{2}{3}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3}} \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 = 1.13133, size = 89, normalized size = 1.2 \begin{align*} \frac{15 \, b^{2} \tan \left (d x + c\right )^{7} + 21 \,{\left (2 \, a b + b^{2}\right )} \tan \left (d x + c\right )^{5} + 35 \,{\left (a^{2} + 2 \, a b\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 = 1.51317, size = 231, normalized size = 3.12 \begin{align*} \frac{{\left (2 \,{\left (35 \, a^{2} - 14 \, a b + 3 \, b^{2}\right )} \cos \left (d x + c\right )^{6} +{\left (35 \, a^{2} - 14 \, a b + 3 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 6 \,{\left (7 \, a b - 4 \, 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 ^{2}{\left (c + d x \right )}\right )^{2} \sec ^{4}{\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 = 2.26377, size = 108, normalized size = 1.46 \begin{align*} \frac{15 \, b^{2} \tan \left (d x + c\right )^{7} + 42 \, a b \tan \left (d x + c\right )^{5} + 21 \, b^{2} \tan \left (d x + c\right )^{5} + 35 \, a^{2} \tan \left (d x + c\right )^{3} + 70 \, a b \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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