Optimal. Leaf size=156 \[ -\frac{152 \tan (c+d x)}{15 a^3 d}+\frac{13 \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{13 \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{76 \tan (c+d x) \sec (c+d x)}{15 d \left (a^3 \cos (c+d x)+a^3\right )}-\frac{11 \tan (c+d x) \sec (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{\tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3} \]
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Rubi [A] time = 0.304536, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2766, 2978, 2748, 3768, 3770, 3767, 8} \[ -\frac{152 \tan (c+d x)}{15 a^3 d}+\frac{13 \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{13 \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{76 \tan (c+d x) \sec (c+d x)}{15 d \left (a^3 \cos (c+d x)+a^3\right )}-\frac{11 \tan (c+d x) \sec (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{\tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2766
Rule 2978
Rule 2748
Rule 3768
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{\sec ^3(c+d x)}{(a+a \cos (c+d x))^3} \, dx &=-\frac{\sec (c+d x) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac{\int \frac{(7 a-4 a \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac{\sec (c+d x) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{11 \sec (c+d x) \tan (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac{\int \frac{\left (43 a^2-33 a^2 \cos (c+d x)\right ) \sec ^3(c+d x)}{a+a \cos (c+d x)} \, dx}{15 a^4}\\ &=-\frac{\sec (c+d x) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{11 \sec (c+d x) \tan (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac{76 \sec (c+d x) \tan (c+d x)}{15 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac{\int \left (195 a^3-152 a^3 \cos (c+d x)\right ) \sec ^3(c+d x) \, dx}{15 a^6}\\ &=-\frac{\sec (c+d x) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{11 \sec (c+d x) \tan (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac{76 \sec (c+d x) \tan (c+d x)}{15 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac{152 \int \sec ^2(c+d x) \, dx}{15 a^3}+\frac{13 \int \sec ^3(c+d x) \, dx}{a^3}\\ &=\frac{13 \sec (c+d x) \tan (c+d x)}{2 a^3 d}-\frac{\sec (c+d x) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{11 \sec (c+d x) \tan (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac{76 \sec (c+d x) \tan (c+d x)}{15 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac{13 \int \sec (c+d x) \, dx}{2 a^3}+\frac{152 \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{15 a^3 d}\\ &=\frac{13 \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{152 \tan (c+d x)}{15 a^3 d}+\frac{13 \sec (c+d x) \tan (c+d x)}{2 a^3 d}-\frac{\sec (c+d x) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{11 \sec (c+d x) \tan (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac{76 \sec (c+d x) \tan (c+d x)}{15 d \left (a^3+a^3 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [B] time = 3.83344, size = 343, normalized size = 2.2 \[ -\frac{24960 \cos ^6\left (\frac{1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+\sec \left (\frac{c}{2}\right ) \sec (c) \left (-4329 \sin \left (c-\frac{d x}{2}\right )+1989 \sin \left (c+\frac{d x}{2}\right )-3575 \sin \left (2 c+\frac{d x}{2}\right )-475 \sin \left (c+\frac{3 d x}{2}\right )+2005 \sin \left (2 c+\frac{3 d x}{2}\right )-2275 \sin \left (3 c+\frac{3 d x}{2}\right )+2673 \sin \left (c+\frac{5 d x}{2}\right )+105 \sin \left (2 c+\frac{5 d x}{2}\right )+1593 \sin \left (3 c+\frac{5 d x}{2}\right )-975 \sin \left (4 c+\frac{5 d x}{2}\right )+1325 \sin \left (2 c+\frac{7 d x}{2}\right )+255 \sin \left (3 c+\frac{7 d x}{2}\right )+875 \sin \left (4 c+\frac{7 d x}{2}\right )-195 \sin \left (5 c+\frac{7 d x}{2}\right )+304 \sin \left (3 c+\frac{9 d x}{2}\right )+90 \sin \left (4 c+\frac{9 d x}{2}\right )+214 \sin \left (5 c+\frac{9 d x}{2}\right )-1235 \sin \left (\frac{d x}{2}\right )+3805 \sin \left (\frac{3 d x}{2}\right )\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \sec ^2(c+d x)}{480 a^3 d (\cos (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.077, size = 181, normalized size = 1.2 \begin{align*} -{\frac{1}{20\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{2}{3\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{31}{4\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{1}{2\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}+{\frac{7}{2\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{13}{2\,d{a}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }-{\frac{1}{2\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}+{\frac{7}{2\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}+{\frac{13}{2\,d{a}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.3512, size = 285, normalized size = 1.83 \begin{align*} -\frac{\frac{60 \,{\left (\frac{5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{7 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{3} - \frac{2 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac{\frac{465 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{40 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac{390 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{3}} + \frac{390 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{3}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72531, size = 548, normalized size = 3.51 \begin{align*} \frac{195 \,{\left (\cos \left (d x + c\right )^{5} + 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 195 \,{\left (\cos \left (d x + c\right )^{5} + 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (304 \, \cos \left (d x + c\right )^{4} + 717 \, \cos \left (d x + c\right )^{3} + 479 \, \cos \left (d x + c\right )^{2} + 45 \, \cos \left (d x + c\right ) - 15\right )} \sin \left (d x + c\right )}{60 \,{\left (a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2}\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 )}}{\cos ^{3}{\left (c + d x \right )} + 3 \cos ^{2}{\left (c + d x \right )} + 3 \cos{\left (c + d x \right )} + 1}\, dx}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3276, size = 188, normalized size = 1.21 \begin{align*} \frac{\frac{390 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} - \frac{390 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{3}} + \frac{60 \,{\left (7 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 5 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{2} a^{3}} - \frac{3 \, a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 40 \, a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 465 \, a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{15}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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