Optimal. Leaf size=28 \[ \frac{a \tan (c+d x)}{d}+\frac{b \tan ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0289415, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {3675} \[ \frac{a \tan (c+d x)}{d}+\frac{b \tan ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3675
Rubi steps
\begin{align*} \int \sec ^2(c+d x) \left (a+b \tan ^2(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b x^2\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{a \tan (c+d x)}{d}+\frac{b \tan ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0109921, size = 28, normalized size = 1. \[ \frac{a \tan (c+d x)}{d}+\frac{b \tan ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 33, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({\frac{b \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+a\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 = 1.12524, size = 34, normalized size = 1.21 \begin{align*} \frac{b \tan \left (d x + c\right )^{3} + 3 \, a \tan \left (d x + c\right )}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48839, size = 92, normalized size = 3.29 \begin{align*} \frac{{\left ({\left (3 \, a - b\right )} \cos \left (d x + c\right )^{2} + b\right )} \sin \left (d x + c\right )}{3 \, d \cos \left (d x + c\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.76578, size = 36, normalized size = 1.29 \begin{align*} \begin{cases} \frac{a \tan{\left (c + d x \right )} + \frac{b \tan ^{3}{\left (c + d x \right )}}{3}}{d} & \text{for}\: d \neq 0 \\x \left (a + b \tan ^{2}{\left (c \right )}\right ) \sec ^{2}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.46188, size = 34, normalized size = 1.21 \begin{align*} \frac{b \tan \left (d x + c\right )^{3} + 3 \, a \tan \left (d x + c\right )}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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