Optimal. Leaf size=75 \[ \frac{\left (a^2+b^2\right ) (a+b \tan (c+d x))^3}{3 b^3 d}+\frac{(a+b \tan (c+d x))^5}{5 b^3 d}-\frac{a (a+b \tan (c+d x))^4}{2 b^3 d} \]
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Rubi [A] time = 0.0659744, antiderivative size = 75, 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 = {3506, 697} \[ \frac{\left (a^2+b^2\right ) (a+b \tan (c+d x))^3}{3 b^3 d}+\frac{(a+b \tan (c+d x))^5}{5 b^3 d}-\frac{a (a+b \tan (c+d x))^4}{2 b^3 d} \]
Antiderivative was successfully verified.
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Rule 3506
Rule 697
Rubi steps
\begin{align*} \int \sec ^4(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 ) \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{\left (a^2+b^2\right ) (a+x)^2}{b^2}-\frac{2 a (a+x)^3}{b^2}+\frac{(a+x)^4}{b^2}\right ) \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac{\left (a^2+b^2\right ) (a+b \tan (c+d x))^3}{3 b^3 d}-\frac{a (a+b \tan (c+d x))^4}{2 b^3 d}+\frac{(a+b \tan (c+d x))^5}{5 b^3 d}\\ \end{align*}
Mathematica [A] time = 0.186638, size = 54, normalized size = 0.72 \[ \frac{(a+b \tan (c+d x))^3 \left (a^2-3 a b \tan (c+d x)+6 b^2 \tan ^2(c+d x)+10 b^2\right )}{30 b^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 82, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({b}^{2} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{15\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) +{\frac{ab}{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{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 = 1.52541, size = 96, normalized size = 1.28 \begin{align*} \frac{6 \, b^{2} \tan \left (d x + c\right )^{5} + 15 \, a b \tan \left (d x + c\right )^{4} + 30 \, a b \tan \left (d x + c\right )^{2} + 10 \,{\left (a^{2} + b^{2}\right )} \tan \left (d x + c\right )^{3} + 30 \, a^{2} \tan \left (d x + c\right )}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88768, size = 184, normalized size = 2.45 \begin{align*} \frac{15 \, a b \cos \left (d x + c\right ) + 2 \,{\left (2 \,{\left (5 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{4} +{\left (5 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, b^{2}\right )} \sin \left (d x + c\right )}{30 \, d \cos \left (d x + c\right )^{5}} \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 ^{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 = 1.32014, size = 108, normalized size = 1.44 \begin{align*} \frac{6 \, b^{2} \tan \left (d x + c\right )^{5} + 15 \, a b \tan \left (d x + c\right )^{4} + 10 \, a^{2} \tan \left (d x + c\right )^{3} + 10 \, b^{2} \tan \left (d x + c\right )^{3} + 30 \, a b \tan \left (d x + c\right )^{2} + 30 \, a^{2} \tan \left (d x + c\right )}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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