Optimal. Leaf size=35 \[ a x+\frac{b \tan ^3(c+d x)}{3 d}-\frac{b \tan (c+d x)}{d}+b x \]
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Rubi [A] time = 0.0251603, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3473, 8} \[ a x+\frac{b \tan ^3(c+d x)}{3 d}-\frac{b \tan (c+d x)}{d}+b x \]
Antiderivative was successfully verified.
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Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \left (a+b \tan ^4(c+d x)\right ) \, dx &=a x+b \int \tan ^4(c+d x) \, dx\\ &=a x+\frac{b \tan ^3(c+d x)}{3 d}-b \int \tan ^2(c+d x) \, dx\\ &=a x-\frac{b \tan (c+d x)}{d}+\frac{b \tan ^3(c+d x)}{3 d}+b \int 1 \, dx\\ &=a x+b x-\frac{b \tan (c+d x)}{d}+\frac{b \tan ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0263659, size = 44, normalized size = 1.26 \[ a x+\frac{b \tan ^3(c+d x)}{3 d}+\frac{b \tan ^{-1}(\tan (c+d x))}{d}-\frac{b \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 43, normalized size = 1.2 \begin{align*} ax+{\frac{b \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{b\tan \left ( dx+c \right ) }{d}}+{\frac{b\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49651, size = 46, normalized size = 1.31 \begin{align*} a x + \frac{{\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} b}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39265, size = 82, normalized size = 2.34 \begin{align*} \frac{b \tan \left (d x + c\right )^{3} + 3 \,{\left (a + b\right )} d x - 3 \, b \tan \left (d x + c\right )}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.261537, size = 32, normalized size = 0.91 \begin{align*} a x + b \left (\begin{cases} x + \frac{\tan ^{3}{\left (c + d x \right )}}{3 d} - \frac{\tan{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \tan ^{4}{\left (c \right )} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.25889, size = 797, normalized size = 22.77 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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