Optimal. Leaf size=102 \[ \frac{(3 A+7 B) \tan (c+d x)}{15 d \left (a^3 \sec (c+d x)+a^3\right )}+\frac{(3 A-8 B) \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A-B) \tan (c+d x)}{5 d (a \sec (c+d x)+a)^3} \]
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Rubi [A] time = 0.203119, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {4008, 4000, 3794} \[ \frac{(3 A+7 B) \tan (c+d x)}{15 d \left (a^3 \sec (c+d x)+a^3\right )}+\frac{(3 A-8 B) \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A-B) \tan (c+d x)}{5 d (a \sec (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 4008
Rule 4000
Rule 3794
Rubi steps
\begin{align*} \int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx &=-\frac{(A-B) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{\int \frac{\sec (c+d x) (-3 a (A-B)-5 a B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac{(A-B) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{(3 A-8 B) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac{(3 A+7 B) \int \frac{\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{15 a^2}\\ &=-\frac{(A-B) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{(3 A-8 B) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac{(3 A+7 B) \tan (c+d x)}{15 d \left (a^3+a^3 \sec (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.290918, size = 96, normalized size = 0.94 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left ((3 A+2 B) \left (5 \sin \left (c+\frac{3 d x}{2}\right )+\sin \left (2 c+\frac{5 d x}{2}\right )\right )+5 (3 A+4 B) \sin \left (\frac{d x}{2}\right )-15 A \sin \left (c+\frac{d x}{2}\right )\right )}{30 a^3 d (\cos (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 64, normalized size = 0.6 \begin{align*}{\frac{1}{4\,d{a}^{3}} \left ({\frac{-A+B}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{2\,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 = 1.062, size = 155, normalized size = 1.52 \begin{align*} \frac{\frac{B{\left (\frac{15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{3}} + \frac{3 \, A{\left (\frac{5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{3}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.439018, size = 227, normalized size = 2.23 \begin{align*} \frac{{\left ({\left (3 \, A + 2 \, B\right )} \cos \left (d x + c\right )^{2} + 3 \,{\left (3 \, A + 2 \, B\right )} \cos \left (d x + c\right ) + 3 \, A + 7 \, B\right )} \sin \left (d x + c\right )}{15 \,{\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} 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 ^{2}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec{\left (c + d x \right )} + 1}\, dx + \int \frac{B \sec ^{3}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec{\left (c + d x \right )} + 1}\, dx}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21226, size = 101, normalized size = 0.99 \begin{align*} -\frac{3 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 10 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 15 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 15 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{60 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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