Optimal. Leaf size=120 \[ \frac{\tan (c+d x)}{5 d \left (a^4 \sec (c+d x)+a^4\right )}-\frac{8 \tan (c+d x)}{35 d \left (a^2 \sec (c+d x)+a^2\right )^2}+\frac{\tan (c+d x) \sec ^3(c+d x)}{7 d (a \sec (c+d x)+a)^4}+\frac{3 \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3} \]
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Rubi [A] time = 0.168655, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3810, 3799, 4000, 3794} \[ \frac{\tan (c+d x)}{5 d \left (a^4 \sec (c+d x)+a^4\right )}-\frac{8 \tan (c+d x)}{35 d \left (a^2 \sec (c+d x)+a^2\right )^2}+\frac{\tan (c+d x) \sec ^3(c+d x)}{7 d (a \sec (c+d x)+a)^4}+\frac{3 \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 3810
Rule 3799
Rule 4000
Rule 3794
Rubi steps
\begin{align*} \int \frac{\sec ^4(c+d x)}{(a+a \sec (c+d x))^4} \, dx &=\frac{\sec ^3(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac{3 \int \frac{\sec ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx}{7 a}\\ &=\frac{\sec ^3(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac{3 \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac{3 \int \frac{\sec (c+d x) (-3 a+5 a \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx}{35 a^3}\\ &=\frac{\sec ^3(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac{3 \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac{8 \tan (c+d x)}{35 d \left (a^2+a^2 \sec (c+d x)\right )^2}+\frac{\int \frac{\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{5 a^3}\\ &=\frac{\sec ^3(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac{3 \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac{8 \tan (c+d x)}{35 d \left (a^2+a^2 \sec (c+d x)\right )^2}+\frac{\tan (c+d x)}{5 d \left (a^4+a^4 \sec (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.206315, size = 69, normalized size = 0.57 \[ \frac{\left (35 \sin \left (\frac{1}{2} (c+d x)\right )+21 \sin \left (\frac{3}{2} (c+d x)\right )+7 \sin \left (\frac{5}{2} (c+d x)\right )+\sin \left (\frac{7}{2} (c+d x)\right )\right ) \sec ^7\left (\frac{1}{2} (c+d x)\right )}{1120 a^4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 56, normalized size = 0.5 \begin{align*}{\frac{1}{8\,d{a}^{4}} \left ({\frac{1}{7} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{3}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}+\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 = 1.2047, size = 117, normalized size = 0.98 \begin{align*} \frac{\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}}}{280 \, a^{4} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58351, size = 248, normalized size = 2.07 \begin{align*} \frac{{\left (2 \, \cos \left (d x + c\right )^{3} + 8 \, \cos \left (d x + c\right )^{2} + 13 \, \cos \left (d x + c\right ) + 12\right )} \sin \left (d x + c\right )}{35 \,{\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{\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 = 1.32315, size = 80, normalized size = 0.67 \begin{align*} \frac{5 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 21 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 35 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 35 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{280 \, a^{4} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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