Optimal. Leaf size=143 \[ \frac{8 \tan (c+d x)}{315 d \left (a^5 \sec (c+d x)+a^5\right )}+\frac{8 \tan (c+d x)}{315 a d \left (a^2 \sec (c+d x)+a^2\right )^2}+\frac{4 \tan (c+d x)}{105 a^2 d (a \sec (c+d x)+a)^3}+\frac{4 \tan (c+d x)}{63 a d (a \sec (c+d x)+a)^4}+\frac{\tan (c+d x)}{9 d (a \sec (c+d x)+a)^5} \]
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Rubi [A] time = 0.14238, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3796, 3794} \[ \frac{8 \tan (c+d x)}{315 d \left (a^5 \sec (c+d x)+a^5\right )}+\frac{8 \tan (c+d x)}{315 a d \left (a^2 \sec (c+d x)+a^2\right )^2}+\frac{4 \tan (c+d x)}{105 a^2 d (a \sec (c+d x)+a)^3}+\frac{4 \tan (c+d x)}{63 a d (a \sec (c+d x)+a)^4}+\frac{\tan (c+d x)}{9 d (a \sec (c+d x)+a)^5} \]
Antiderivative was successfully verified.
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Rule 3796
Rule 3794
Rubi steps
\begin{align*} \int \frac{\sec (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{4 \int \frac{\sec (c+d x)}{(a+a \sec (c+d x))^4} \, dx}{9 a}\\ &=\frac{\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}+\frac{4 \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}+\frac{4 \int \frac{\sec (c+d x)}{(a+a \sec (c+d x))^3} \, dx}{21 a^2}\\ &=\frac{\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}+\frac{4 \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}+\frac{4 \tan (c+d x)}{105 a^2 d (a+a \sec (c+d x))^3}+\frac{8 \int \frac{\sec (c+d x)}{(a+a \sec (c+d x))^2} \, dx}{105 a^3}\\ &=\frac{\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}+\frac{4 \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}+\frac{4 \tan (c+d x)}{105 a^2 d (a+a \sec (c+d x))^3}+\frac{8 \tan (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}+\frac{8 \int \frac{\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{315 a^4}\\ &=\frac{\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}+\frac{4 \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}+\frac{4 \tan (c+d x)}{105 a^2 d (a+a \sec (c+d x))^3}+\frac{8 \tan (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}+\frac{8 \tan (c+d x)}{315 d \left (a^5+a^5 \sec (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.266656, size = 138, normalized size = 0.97 \[ \frac{\sec \left (\frac{c}{2}\right ) \left (-5040 \sin \left (c+\frac{d x}{2}\right )+3612 \sin \left (c+\frac{3 d x}{2}\right )-3360 \sin \left (2 c+\frac{3 d x}{2}\right )+1728 \sin \left (2 c+\frac{5 d x}{2}\right )-1260 \sin \left (3 c+\frac{5 d x}{2}\right )+432 \sin \left (3 c+\frac{7 d x}{2}\right )-315 \sin \left (4 c+\frac{7 d x}{2}\right )+83 \sin \left (4 c+\frac{9 d x}{2}\right )+5418 \sin \left (\frac{d x}{2}\right )\right ) \sec ^9\left (\frac{1}{2} (c+d x)\right )}{80640 a^5 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 71, normalized size = 0.5 \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{4}{7} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{6}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{4}{3} \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 = 1.0955, size = 144, normalized size = 1.01 \begin{align*} \frac{\frac{315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{420 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{378 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{180 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{5040 \, a^{5} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66341, size = 316, normalized size = 2.21 \begin{align*} \frac{{\left (83 \, \cos \left (d x + c\right )^{4} + 100 \, \cos \left (d x + c\right )^{3} + 84 \, \cos \left (d x + c\right )^{2} + 40 \, \cos \left (d x + c\right ) + 8\right )} \sin \left (d x + c\right )}{315 \,{\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{\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 = 1.50382, size = 97, normalized size = 0.68 \begin{align*} \frac{35 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 180 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 378 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 420 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 315 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{5040 \, a^{5} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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