Optimal. Leaf size=65 \[ \frac{(A+2 B) \tan (c+d x)}{3 d \left (a^2 \sec (c+d x)+a^2\right )}+\frac{(A-B) \tan (c+d x)}{3 d (a \sec (c+d x)+a)^2} \]
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Rubi [A] time = 0.0797186, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {4000, 3794} \[ \frac{(A+2 B) \tan (c+d x)}{3 d \left (a^2 \sec (c+d x)+a^2\right )}+\frac{(A-B) \tan (c+d x)}{3 d (a \sec (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 4000
Rule 3794
Rubi steps
\begin{align*} \int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx &=\frac{(A-B) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac{(A+2 B) \int \frac{\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{3 a}\\ &=\frac{(A-B) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac{(A+2 B) \tan (c+d x)}{3 d \left (a^2+a^2 \sec (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.199152, size = 76, normalized size = 1.17 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left ((2 A+B) \sin \left (c+\frac{3 d x}{2}\right )+3 (A+B) \sin \left (\frac{d x}{2}\right )-3 A \sin \left (c+\frac{d x}{2}\right )\right )}{3 a^2 d (\cos (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 60, normalized size = 0.9 \begin{align*}{\frac{1}{2\,d{a}^{2}} \left ( -{\frac{A}{3} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{B}{3} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+A\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +B\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 = 0.977688, size = 126, normalized size = 1.94 \begin{align*} \frac{\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}} + \frac{A{\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 = 0.43814, size = 144, normalized size = 2.22 \begin{align*} \frac{{\left ({\left (2 \, A + B\right )} \cos \left (d x + c\right ) + A + 2 \, B\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{A \sec{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec{\left (c + d x \right )} + 1}\, dx + \int \frac{B \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 = 1.23922, size = 81, normalized size = 1.25 \begin{align*} -\frac{A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 3 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 3 \, B \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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