Optimal. Leaf size=62 \[ \frac{(B+2 C) \tan (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{(B-C) \tan (c+d x)}{3 d (a \sec (c+d x)+a)^2} \]
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Rubi [A] time = 0.073853, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094, Rules used = {4052, 12, 3794} \[ \frac{(B+2 C) \tan (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{(B-C) \tan (c+d x)}{3 d (a \sec (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 4052
Rule 12
Rule 3794
Rubi steps
\begin{align*} \int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=\frac{(B-C) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac{\int \frac{a (B+2 C) \sec (c+d x)}{a+a \sec (c+d x)} \, dx}{3 a^2}\\ &=\frac{(B-C) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac{(B+2 C) \int \frac{\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{3 a}\\ &=\frac{(B-C) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac{(B+2 C) \tan (c+d x)}{3 d \left (a^2+a^2 \sec (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.287011, size = 46, normalized size = 0.74 \[ \frac{\tan \left (\frac{1}{2} (c+d x)\right ) \left ((C-B) \sec ^2\left (\frac{1}{2} (c+d x)\right )+2 (2 B+C)\right )}{6 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 60, normalized size = 1. \begin{align*}{\frac{1}{2\,d{a}^{2}} \left ( -{\frac{B}{3} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{C}{3} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+B\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +C\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.947883, size = 126, normalized size = 2.03 \begin{align*} \frac{\frac{C{\left (\frac{3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{2}} + \frac{B{\left (\frac{3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{2}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.466989, size = 144, normalized size = 2.32 \begin{align*} \frac{{\left ({\left (2 \, B + C\right )} \cos \left (d x + c\right ) + B + 2 \, C\right )} \sin \left (d x + c\right )}{3 \,{\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\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*} \frac{\int \frac{B \sec{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec{\left (c + d x \right )} + 1}\, dx + \int \frac{C \sec ^{2}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec{\left (c + d x \right )} + 1}\, dx}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1259, size = 81, normalized size = 1.31 \begin{align*} -\frac{B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 3 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 3 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{6 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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