Optimal. Leaf size=200 \[ \frac{181 \tan (c+d x)}{63 a^5 d}-\frac{5 \tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{29 \tan (c+d x) \sec ^3(c+d x)}{63 a^2 d (a \sec (c+d x)+a)^3}-\frac{67 \tan (c+d x) \sec ^2(c+d x)}{63 a^3 d (a \sec (c+d x)+a)^2}+\frac{5 \tan (c+d x)}{d \left (a^5 \sec (c+d x)+a^5\right )}-\frac{\tan (c+d x) \sec ^5(c+d x)}{9 d (a \sec (c+d x)+a)^5}-\frac{5 \tan (c+d x) \sec ^4(c+d x)}{21 a d (a \sec (c+d x)+a)^4} \]
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Rubi [A] time = 0.480232, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3816, 4019, 4008, 3787, 3770, 3767, 8} \[ \frac{181 \tan (c+d x)}{63 a^5 d}-\frac{5 \tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{29 \tan (c+d x) \sec ^3(c+d x)}{63 a^2 d (a \sec (c+d x)+a)^3}-\frac{67 \tan (c+d x) \sec ^2(c+d x)}{63 a^3 d (a \sec (c+d x)+a)^2}+\frac{5 \tan (c+d x)}{d \left (a^5 \sec (c+d x)+a^5\right )}-\frac{\tan (c+d x) \sec ^5(c+d x)}{9 d (a \sec (c+d x)+a)^5}-\frac{5 \tan (c+d x) \sec ^4(c+d x)}{21 a d (a \sec (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 3816
Rule 4019
Rule 4008
Rule 3787
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{\sec ^7(c+d x)}{(a+a \sec (c+d x))^5} \, dx &=-\frac{\sec ^5(c+d x) \tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac{\int \frac{\sec ^5(c+d x) (5 a-10 a \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx}{9 a^2}\\ &=-\frac{\sec ^5(c+d x) \tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac{5 \sec ^4(c+d x) \tan (c+d x)}{21 a d (a+a \sec (c+d x))^4}-\frac{\int \frac{\sec ^4(c+d x) \left (60 a^2-85 a^2 \sec (c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx}{63 a^4}\\ &=-\frac{\sec ^5(c+d x) \tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac{5 \sec ^4(c+d x) \tan (c+d x)}{21 a d (a+a \sec (c+d x))^4}-\frac{29 \sec ^3(c+d x) \tan (c+d x)}{63 a^2 d (a+a \sec (c+d x))^3}-\frac{\int \frac{\sec ^3(c+d x) \left (435 a^3-570 a^3 \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{315 a^6}\\ &=-\frac{\sec ^5(c+d x) \tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac{5 \sec ^4(c+d x) \tan (c+d x)}{21 a d (a+a \sec (c+d x))^4}-\frac{29 \sec ^3(c+d x) \tan (c+d x)}{63 a^2 d (a+a \sec (c+d x))^3}-\frac{67 \sec ^2(c+d x) \tan (c+d x)}{63 a^3 d (a+a \sec (c+d x))^2}-\frac{\int \frac{\sec ^2(c+d x) \left (2010 a^4-2715 a^4 \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{945 a^8}\\ &=-\frac{\sec ^5(c+d x) \tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac{5 \sec ^4(c+d x) \tan (c+d x)}{21 a d (a+a \sec (c+d x))^4}-\frac{29 \sec ^3(c+d x) \tan (c+d x)}{63 a^2 d (a+a \sec (c+d x))^3}-\frac{67 \sec ^2(c+d x) \tan (c+d x)}{63 a^3 d (a+a \sec (c+d x))^2}+\frac{5 \tan (c+d x)}{d \left (a^5+a^5 \sec (c+d x)\right )}+\frac{\int \sec (c+d x) \left (-4725 a^5+2715 a^5 \sec (c+d x)\right ) \, dx}{945 a^{10}}\\ &=-\frac{\sec ^5(c+d x) \tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac{5 \sec ^4(c+d x) \tan (c+d x)}{21 a d (a+a \sec (c+d x))^4}-\frac{29 \sec ^3(c+d x) \tan (c+d x)}{63 a^2 d (a+a \sec (c+d x))^3}-\frac{67 \sec ^2(c+d x) \tan (c+d x)}{63 a^3 d (a+a \sec (c+d x))^2}+\frac{5 \tan (c+d x)}{d \left (a^5+a^5 \sec (c+d x)\right )}+\frac{181 \int \sec ^2(c+d x) \, dx}{63 a^5}-\frac{5 \int \sec (c+d x) \, dx}{a^5}\\ &=-\frac{5 \tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{\sec ^5(c+d x) \tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac{5 \sec ^4(c+d x) \tan (c+d x)}{21 a d (a+a \sec (c+d x))^4}-\frac{29 \sec ^3(c+d x) \tan (c+d x)}{63 a^2 d (a+a \sec (c+d x))^3}-\frac{67 \sec ^2(c+d x) \tan (c+d x)}{63 a^3 d (a+a \sec (c+d x))^2}+\frac{5 \tan (c+d x)}{d \left (a^5+a^5 \sec (c+d x)\right )}-\frac{181 \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{63 a^5 d}\\ &=-\frac{5 \tanh ^{-1}(\sin (c+d x))}{a^5 d}+\frac{181 \tan (c+d x)}{63 a^5 d}-\frac{\sec ^5(c+d x) \tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac{5 \sec ^4(c+d x) \tan (c+d x)}{21 a d (a+a \sec (c+d x))^4}-\frac{29 \sec ^3(c+d x) \tan (c+d x)}{63 a^2 d (a+a \sec (c+d x))^3}-\frac{67 \sec ^2(c+d x) \tan (c+d x)}{63 a^3 d (a+a \sec (c+d x))^2}+\frac{5 \tan (c+d x)}{d \left (a^5+a^5 \sec (c+d x)\right )}\\ \end{align*}
Mathematica [B] time = 1.84508, size = 401, normalized size = 2. \[ \frac{\cos \left (\frac{1}{2} (c+d x)\right ) \sec ^5(c+d x) \left (\sec \left (\frac{c}{2}\right ) \sec (c) \left (-56952 \sin \left (c-\frac{d x}{2}\right )+43722 \sin \left (c+\frac{d x}{2}\right )-47208 \sin \left (2 c+\frac{d x}{2}\right )-18144 \sin \left (c+\frac{3 d x}{2}\right )+41796 \sin \left (2 c+\frac{3 d x}{2}\right )-28350 \sin \left (3 c+\frac{3 d x}{2}\right )+34578 \sin \left (c+\frac{5 d x}{2}\right )-5691 \sin \left (2 c+\frac{5 d x}{2}\right )+28719 \sin \left (3 c+\frac{5 d x}{2}\right )-11550 \sin \left (4 c+\frac{5 d x}{2}\right )+15517 \sin \left (2 c+\frac{7 d x}{2}\right )-504 \sin \left (3 c+\frac{7 d x}{2}\right )+13186 \sin \left (4 c+\frac{7 d x}{2}\right )-2835 \sin \left (5 c+\frac{7 d x}{2}\right )+4149 \sin \left (3 c+\frac{9 d x}{2}\right )+252 \sin \left (4 c+\frac{9 d x}{2}\right )+3582 \sin \left (5 c+\frac{9 d x}{2}\right )-315 \sin \left (6 c+\frac{9 d x}{2}\right )+496 \sin \left (4 c+\frac{11 d x}{2}\right )+63 \sin \left (5 c+\frac{11 d x}{2}\right )+433 \sin \left (6 c+\frac{11 d x}{2}\right )-33978 \sin \left (\frac{d x}{2}\right )+52002 \sin \left (\frac{3 d x}{2}\right )\right ) \sec (c+d x)+322560 \cos ^9\left (\frac{1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{2016 a^5 d (\sec (c+d x)+1)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 177, normalized size = 0.9 \begin{align*}{\frac{1}{144\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9}}+{\frac{1}{14\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{3}{8\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{3}{2\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{129}{16\,d{a}^{5}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{1}{d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-5\,{\frac{\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) }{d{a}^{5}}}-{\frac{1}{d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}+5\,{\frac{\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) }{d{a}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12283, size = 278, normalized size = 1.39 \begin{align*} \frac{\frac{2016 \, \sin \left (d x + c\right )}{{\left (a^{5} - \frac{a^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}} + \frac{\frac{8127 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{1512 \, \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{72 \, \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}}}{a^{5}} - \frac{5040 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{5}} + \frac{5040 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{5}}}{1008 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76029, size = 755, normalized size = 3.78 \begin{align*} -\frac{315 \,{\left (\cos \left (d x + c\right )^{6} + 5 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3} + 5 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 315 \,{\left (\cos \left (d x + c\right )^{6} + 5 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3} + 5 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (496 \, \cos \left (d x + c\right )^{5} + 2165 \, \cos \left (d x + c\right )^{4} + 3633 \, \cos \left (d x + c\right )^{3} + 2840 \, \cos \left (d x + c\right )^{2} + 946 \, \cos \left (d x + c\right ) + 63\right )} \sin \left (d x + c\right )}{126 \,{\left (a^{5} d \cos \left (d x + c\right )^{6} + 5 \, a^{5} d \cos \left (d x + c\right )^{5} + 10 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 5 \, a^{5} d \cos \left (d x + c\right )^{2} + a^{5} d \cos \left (d x + c\right )\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 ^{7}{\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 = 1.47177, size = 209, normalized size = 1.04 \begin{align*} -\frac{\frac{5040 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{5}} - \frac{5040 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{5}} + \frac{2016 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} a^{5}} - \frac{7 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 72 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 378 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1512 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 8127 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{45}}}{1008 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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