Optimal. Leaf size=43 \[ \frac{(3 A+2 C) \tan (c+d x)}{3 d}+\frac{C \tan (c+d x) \sec ^2(c+d x)}{3 d} \]
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Rubi [A] time = 0.0404934, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4046, 3767, 8} \[ \frac{(3 A+2 C) \tan (c+d x)}{3 d}+\frac{C \tan (c+d x) \sec ^2(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 4046
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{C \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{1}{3} (3 A+2 C) \int \sec ^2(c+d x) \, dx\\ &=\frac{C \sec ^2(c+d x) \tan (c+d x)}{3 d}-\frac{(3 A+2 C) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=\frac{(3 A+2 C) \tan (c+d x)}{3 d}+\frac{C \sec ^2(c+d x) \tan (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0930625, size = 36, normalized size = 0.84 \[ \frac{A \tan (c+d x)}{d}+\frac{C \left (\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 35, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( A\tan \left ( dx+c \right ) -C \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 = 0.926532, size = 46, normalized size = 1.07 \begin{align*} \frac{{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C + 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 = 0.457099, size = 95, normalized size = 2.21 \begin{align*} \frac{{\left ({\left (3 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{2} + C\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \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 = 1.2243, size = 46, normalized size = 1.07 \begin{align*} \frac{C \tan \left (d x + c\right )^{3} + 3 \, A \tan \left (d x + c\right ) + 3 \, C \tan \left (d x + c\right )}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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