Optimal. Leaf size=138 \[ \frac{2 (3 A+4 B) \tan (c+d x)}{105 d \left (a^4 \sec (c+d x)+a^4\right )}+\frac{2 (3 A+4 B) \tan (c+d x)}{105 d \left (a^2 \sec (c+d x)+a^2\right )^2}+\frac{(3 A+4 B) \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}+\frac{(A-B) \tan (c+d x)}{7 d (a \sec (c+d x)+a)^4} \]
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Rubi [A] time = 0.150658, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {4000, 3796, 3794} \[ \frac{2 (3 A+4 B) \tan (c+d x)}{105 d \left (a^4 \sec (c+d x)+a^4\right )}+\frac{2 (3 A+4 B) \tan (c+d x)}{105 d \left (a^2 \sec (c+d x)+a^2\right )^2}+\frac{(3 A+4 B) \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}+\frac{(A-B) \tan (c+d x)}{7 d (a \sec (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 4000
Rule 3796
Rule 3794
Rubi steps
\begin{align*} \int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx &=\frac{(A-B) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac{(3 A+4 B) \int \frac{\sec (c+d x)}{(a+a \sec (c+d x))^3} \, dx}{7 a}\\ &=\frac{(A-B) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac{(3 A+4 B) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac{(2 (3 A+4 B)) \int \frac{\sec (c+d x)}{(a+a \sec (c+d x))^2} \, dx}{35 a^2}\\ &=\frac{(A-B) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac{(3 A+4 B) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac{2 (3 A+4 B) \tan (c+d x)}{105 d \left (a^2+a^2 \sec (c+d x)\right )^2}+\frac{(2 (3 A+4 B)) \int \frac{\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{105 a^3}\\ &=\frac{(A-B) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac{(3 A+4 B) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac{2 (3 A+4 B) \tan (c+d x)}{105 d \left (a^2+a^2 \sec (c+d x)\right )^2}+\frac{2 (3 A+4 B) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.444291, size = 193, normalized size = 1.4 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left (-35 (18 A+5 B) \sin \left (c+\frac{d x}{2}\right )+70 (9 A+4 B) \sin \left (\frac{d x}{2}\right )+441 A \sin \left (c+\frac{3 d x}{2}\right )-315 A \sin \left (2 c+\frac{3 d x}{2}\right )+147 A \sin \left (2 c+\frac{5 d x}{2}\right )-105 A \sin \left (3 c+\frac{5 d x}{2}\right )+36 A \sin \left (3 c+\frac{7 d x}{2}\right )+168 B \sin \left (c+\frac{3 d x}{2}\right )-105 B \sin \left (2 c+\frac{3 d x}{2}\right )+91 B \sin \left (2 c+\frac{5 d x}{2}\right )+13 B \sin \left (3 c+\frac{7 d x}{2}\right )\right )}{420 a^4 d (\cos (c+d x)+1)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 90, normalized size = 0.7 \begin{align*}{\frac{1}{8\,d{a}^{4}} \left ({\frac{-A+B}{7} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{3\,A-B}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{-3\,A-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 = 1.00135, size = 236, normalized size = 1.71 \begin{align*} \frac{\frac{B{\left (\frac{105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}} + \frac{3 \, A{\left (\frac{35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.451195, size = 308, normalized size = 2.23 \begin{align*} \frac{{\left ({\left (36 \, A + 13 \, B\right )} \cos \left (d x + c\right )^{3} + 13 \,{\left (3 \, A + 4 \, B\right )} \cos \left (d x + c\right )^{2} + 8 \,{\left (3 \, A + 4 \, B\right )} \cos \left (d x + c\right ) + 6 \, A + 8 \, B\right )} \sin \left (d x + c\right )}{105 \,{\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} 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 ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec{\left (c + d x \right )} + 1}\, dx + \int \frac{B \sec ^{2}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec{\left (c + d x \right )} + 1}\, dx}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27616, size = 158, normalized size = 1.14 \begin{align*} -\frac{15 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 15 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 63 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 21 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 105 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 35 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 105 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 105 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{840 \, a^{4} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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