Optimal. Leaf size=15 \[ A x+\frac{C \tan (c+d x)}{d} \]
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Rubi [A] time = 0.0130266, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3767, 8} \[ A x+\frac{C \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \left (A+C \sec ^2(c+d x)\right ) \, dx &=A x+C \int \sec ^2(c+d x) \, dx\\ &=A x-\frac{C \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=A x+\frac{C \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0026345, size = 15, normalized size = 1. \[ A x+\frac{C \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 16, normalized size = 1.1 \begin{align*} Ax+{\frac{C\tan \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.920524, size = 20, normalized size = 1.33 \begin{align*} A x + \frac{C \tan \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.465923, size = 76, normalized size = 5.07 \begin{align*} \frac{A d x \cos \left (d x + c\right ) + C \sin \left (d x + c\right )}{d \cos \left (d x + c\right )} \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 )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14451, size = 20, normalized size = 1.33 \begin{align*} A x + \frac{C \tan \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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