Optimal. Leaf size=49 \[ \frac{a^2 \tan (c+d x)}{d}+\frac{2 a b \tan ^3(c+d x)}{3 d}+\frac{b^2 \tan ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.0535059, antiderivative size = 49, 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, 194} \[ \frac{a^2 \tan (c+d x)}{d}+\frac{2 a b \tan ^3(c+d x)}{3 d}+\frac{b^2 \tan ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 3675
Rule 194
Rubi steps
\begin{align*} \int \sec ^2(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b x^2\right )^2 \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^2+2 a b x^2+b^2 x^4\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{a^2 \tan (c+d x)}{d}+\frac{2 a b \tan ^3(c+d x)}{3 d}+\frac{b^2 \tan ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.140486, size = 49, normalized size = 1. \[ \frac{a^2 \tan (c+d x)}{d}+\frac{2 a b \tan ^3(c+d x)}{3 d}+\frac{b^2 \tan ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 57, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({\frac{{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{2\,ab \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{a}^{2}\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 = 1.02884, size = 57, normalized size = 1.16 \begin{align*} \frac{3 \, b^{2} \tan \left (d x + c\right )^{5} + 10 \, a b \tan \left (d x + c\right )^{3} + 15 \, a^{2} \tan \left (d x + c\right )}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44614, size = 167, normalized size = 3.41 \begin{align*} \frac{{\left ({\left (15 \, a^{2} - 10 \, a b + 3 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 2 \,{\left (5 \, a b - 3 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, b^{2}\right )} \sin \left (d x + c\right )}{15 \, 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 ^{2}{\left (c + d x \right )}\right )^{2} \sec ^{2}{\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 = 1.77604, size = 57, normalized size = 1.16 \begin{align*} \frac{3 \, b^{2} \tan \left (d x + c\right )^{5} + 10 \, a b \tan \left (d x + c\right )^{3} + 15 \, a^{2} \tan \left (d x + c\right )}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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