Optimal. Leaf size=148 \[ \frac{(8 A+13 B+36 C) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)}+\frac{(23 A-2 B-54 C) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}-\frac{(6 A+B-8 C) \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3} \]
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Rubi [A] time = 0.406298, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.098, Rules used = {4084, 4008, 4000, 3794} \[ \frac{(8 A+13 B+36 C) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)}+\frac{(23 A-2 B-54 C) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}-\frac{(6 A+B-8 C) \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 4084
Rule 4008
Rule 4000
Rule 3794
Rubi steps
\begin{align*} \int \frac{\sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx &=-\frac{(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac{\int \frac{\sec ^2(c+d x) (a (5 A+2 B-2 C)-a (A-B-6 C) \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac{(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{(6 A+B-8 C) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac{\int \frac{\sec (c+d x) \left (-3 a^2 (6 A+B-8 C)+5 a^2 (A-B-6 C) \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{35 a^4}\\ &=\frac{(23 A-2 B-54 C) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac{(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{(6 A+B-8 C) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac{(8 A+13 B+36 C) \int \frac{\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{105 a^3}\\ &=\frac{(23 A-2 B-54 C) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac{(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{(6 A+B-8 C) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac{(8 A+13 B+36 C) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.683712, size = 200, normalized size = 1.35 \[ \frac{\sec \left (\frac{c}{2}\right ) \sec ^7\left (\frac{1}{2} (c+d x)\right ) \left (-35 (5 A+4 B) \sin \left (c+\frac{d x}{2}\right )+70 (4 A+2 B+3 C) \sin \left (\frac{d x}{2}\right )+168 A \sin \left (c+\frac{3 d x}{2}\right )-105 A \sin \left (2 c+\frac{3 d x}{2}\right )+91 A \sin \left (2 c+\frac{5 d x}{2}\right )+13 A \sin \left (3 c+\frac{7 d x}{2}\right )+168 B \sin \left (c+\frac{3 d x}{2}\right )+56 B \sin \left (2 c+\frac{5 d x}{2}\right )+8 B \sin \left (3 c+\frac{7 d x}{2}\right )+126 C \sin \left (c+\frac{3 d x}{2}\right )+42 C \sin \left (2 c+\frac{5 d x}{2}\right )+6 C \sin \left (3 c+\frac{7 d x}{2}\right )\right )}{6720 a^4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.075, size = 106, normalized size = 0.7 \begin{align*}{\frac{1}{8\,d{a}^{4}} \left ({\frac{A-B+C}{7} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{-A+3\,C-B}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{-A+B+3\,C}{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 ) +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 = 0.986233, size = 350, normalized size = 2.36 \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{A{\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 \, C{\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 = 0.472272, size = 343, normalized size = 2.32 \begin{align*} \frac{{\left ({\left (13 \, A + 8 \, B + 6 \, C\right )} \cos \left (d x + c\right )^{3} + 4 \,{\left (13 \, A + 8 \, B + 6 \, C\right )} \cos \left (d x + c\right )^{2} +{\left (32 \, A + 52 \, B + 39 \, C\right )} \cos \left (d x + c\right ) + 8 \, A + 13 \, B + 36 \, C\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 ^{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 + \int \frac{B \sec ^{3}{\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{C \sec ^{4}{\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 = 1.21144, size = 231, normalized size = 1.56 \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} + 15 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 21 \, 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} + 63 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 35 \, 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 \, C \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 ) + 105 \, C \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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