Optimal. Leaf size=139 \[ \frac{2 \tan (c+d x)}{45 d \left (a^5 \sec (c+d x)+a^5\right )}+\frac{2 \tan (c+d x)}{45 a^3 d (a \sec (c+d x)+a)^2}+\frac{\tan (c+d x)}{15 a^2 d (a \sec (c+d x)+a)^3}-\frac{2 \tan (c+d x)}{9 a d (a \sec (c+d x)+a)^4}+\frac{\tan (c+d x)}{9 d (a \sec (c+d x)+a)^5} \]
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Rubi [A] time = 0.182088, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3799, 4000, 3796, 3794} \[ \frac{2 \tan (c+d x)}{45 d \left (a^5 \sec (c+d x)+a^5\right )}+\frac{2 \tan (c+d x)}{45 a^3 d (a \sec (c+d x)+a)^2}+\frac{\tan (c+d x)}{15 a^2 d (a \sec (c+d x)+a)^3}-\frac{2 \tan (c+d x)}{9 a d (a \sec (c+d x)+a)^4}+\frac{\tan (c+d x)}{9 d (a \sec (c+d x)+a)^5} \]
Antiderivative was successfully verified.
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Rule 3799
Rule 4000
Rule 3796
Rule 3794
Rubi steps
\begin{align*} \int \frac{\sec ^3(c+d x)}{(a+a \sec (c+d x))^5} \, dx &=\frac{\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}+\frac{\int \frac{\sec (c+d x) (-5 a+9 a \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx}{9 a^2}\\ &=\frac{\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac{2 \tan (c+d x)}{9 a d (a+a \sec (c+d x))^4}+\frac{\int \frac{\sec (c+d x)}{(a+a \sec (c+d x))^3} \, dx}{3 a^2}\\ &=\frac{\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac{2 \tan (c+d x)}{9 a d (a+a \sec (c+d x))^4}+\frac{\tan (c+d x)}{15 a^2 d (a+a \sec (c+d x))^3}+\frac{2 \int \frac{\sec (c+d x)}{(a+a \sec (c+d x))^2} \, dx}{15 a^3}\\ &=\frac{\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac{2 \tan (c+d x)}{9 a d (a+a \sec (c+d x))^4}+\frac{\tan (c+d x)}{15 a^2 d (a+a \sec (c+d x))^3}+\frac{2 \tan (c+d x)}{45 a^3 d (a+a \sec (c+d x))^2}+\frac{2 \int \frac{\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{45 a^4}\\ &=\frac{\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac{2 \tan (c+d x)}{9 a d (a+a \sec (c+d x))^4}+\frac{\tan (c+d x)}{15 a^2 d (a+a \sec (c+d x))^3}+\frac{2 \tan (c+d x)}{45 a^3 d (a+a \sec (c+d x))^2}+\frac{2 \tan (c+d x)}{45 d \left (a^5+a^5 \sec (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.218125, size = 110, normalized size = 0.79 \[ \frac{\sec \left (\frac{c}{2}\right ) \left (-45 \sin \left (c+\frac{d x}{2}\right )+54 \sin \left (c+\frac{3 d x}{2}\right )-30 \sin \left (2 c+\frac{3 d x}{2}\right )+36 \sin \left (2 c+\frac{5 d x}{2}\right )+9 \sin \left (3 c+\frac{7 d x}{2}\right )+\sin \left (4 c+\frac{9 d x}{2}\right )+81 \sin \left (\frac{d x}{2}\right )\right ) \sec ^9\left (\frac{1}{2} (c+d x)\right )}{5760 a^5 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 45, normalized size = 0.3 \begin{align*}{\frac{1}{16\,d{a}^{5}} \left ({\frac{1}{9} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9}}-{\frac{2}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+\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 = 1.13959, size = 90, normalized size = 0.65 \begin{align*} \frac{\frac{45 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{18 \, \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 )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{720 \, a^{5} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64016, size = 312, normalized size = 2.24 \begin{align*} \frac{{\left (2 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3} + 21 \, \cos \left (d x + c\right )^{2} + 10 \, \cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right )}{45 \,{\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} 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 ^{3}{\left (c + d x \right )}}{\sec ^{5}{\left (c + d x \right )} + 5 \sec ^{4}{\left (c + d x \right )} + 10 \sec ^{3}{\left (c + d x \right )} + 10 \sec ^{2}{\left (c + d x \right )} + 5 \sec{\left (c + d x \right )} + 1}\, dx}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.53563, size = 62, normalized size = 0.45 \begin{align*} \frac{5 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 18 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 45 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{720 \, a^{5} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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