Optimal. Leaf size=143 \[ \frac{2 \tan (c+d x)}{63 d \left (a^5 \sec (c+d x)+a^5\right )}+\frac{2 \tan (c+d x)}{63 a d \left (a^2 \sec (c+d x)+a^2\right )^2}+\frac{\tan (c+d x)}{21 a^2 d (a \sec (c+d x)+a)^3}+\frac{5 \tan (c+d x)}{63 a d (a \sec (c+d x)+a)^4}-\frac{\tan (c+d x)}{9 d (a \sec (c+d x)+a)^5} \]
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Rubi [A] time = 0.157324, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3797, 3796, 3794} \[ \frac{2 \tan (c+d x)}{63 d \left (a^5 \sec (c+d x)+a^5\right )}+\frac{2 \tan (c+d x)}{63 a d \left (a^2 \sec (c+d x)+a^2\right )^2}+\frac{\tan (c+d x)}{21 a^2 d (a \sec (c+d x)+a)^3}+\frac{5 \tan (c+d x)}{63 a d (a \sec (c+d x)+a)^4}-\frac{\tan (c+d x)}{9 d (a \sec (c+d x)+a)^5} \]
Antiderivative was successfully verified.
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Rule 3797
Rule 3796
Rule 3794
Rubi steps
\begin{align*} \int \frac{\sec ^2(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{5 \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{5 \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}+\frac{5 \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{5 \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}+\frac{\tan (c+d x)}{21 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}{21 a^3}\\ &=-\frac{\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}+\frac{5 \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}+\frac{\tan (c+d x)}{21 a^2 d (a+a \sec (c+d x))^3}+\frac{2 \tan (c+d x)}{63 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}{63 a^4}\\ &=-\frac{\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}+\frac{5 \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}+\frac{\tan (c+d x)}{21 a^2 d (a+a \sec (c+d x))^3}+\frac{2 \tan (c+d x)}{63 a^3 d (a+a \sec (c+d x))^2}+\frac{2 \tan (c+d x)}{63 d \left (a^5+a^5 \sec (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.22972, size = 125, normalized size = 0.87 \[ \frac{\sec \left (\frac{c}{2}\right ) \left (-315 \sin \left (c+\frac{d x}{2}\right )+273 \sin \left (c+\frac{3 d x}{2}\right )-147 \sin \left (2 c+\frac{3 d x}{2}\right )+117 \sin \left (2 c+\frac{5 d x}{2}\right )-63 \sin \left (3 c+\frac{5 d x}{2}\right )+45 \sin \left (3 c+\frac{7 d x}{2}\right )+5 \sin \left (4 c+\frac{9 d x}{2}\right )+315 \sin \left (\frac{d x}{2}\right )\right ) \sec ^9\left (\frac{1}{2} (c+d x)\right )}{16128 a^5 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 58, normalized size = 0.4 \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}{7} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{2}{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 = 1.05075, size = 117, normalized size = 0.82 \begin{align*} \frac{\frac{63 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{42 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{18 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{7 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{1008 \, a^{5} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61194, size = 312, normalized size = 2.18 \begin{align*} \frac{{\left (5 \, \cos \left (d x + c\right )^{4} + 25 \, \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 )}{63 \,{\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 ^{2}{\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 = 1.4709, size = 80, normalized size = 0.56 \begin{align*} -\frac{7 \, \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 )^{7} + 42 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 63 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{1008 \, a^{5} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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