Optimal. Leaf size=97 \[ \frac{\tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{22 \sin (c+d x)}{15 d \left (a^3 \cos (c+d x)+a^3\right )}-\frac{7 \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3} \]
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Rubi [A] time = 0.201109, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2766, 2978, 12, 3770} \[ \frac{\tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{22 \sin (c+d x)}{15 d \left (a^3 \cos (c+d x)+a^3\right )}-\frac{7 \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2766
Rule 2978
Rule 12
Rule 3770
Rubi steps
\begin{align*} \int \frac{\sec (c+d x)}{(a+a \cos (c+d x))^3} \, dx &=-\frac{\sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac{\int \frac{(5 a-2 a \cos (c+d x)) \sec (c+d x)}{(a+a \cos (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac{\sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{7 \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac{\int \frac{\left (15 a^2-7 a^2 \cos (c+d x)\right ) \sec (c+d x)}{a+a \cos (c+d x)} \, dx}{15 a^4}\\ &=-\frac{\sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{7 \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac{22 \sin (c+d x)}{15 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac{\int 15 a^3 \sec (c+d x) \, dx}{15 a^6}\\ &=-\frac{\sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{7 \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac{22 \sin (c+d x)}{15 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac{\int \sec (c+d x) \, dx}{a^3}\\ &=\frac{\tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{\sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{7 \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac{22 \sin (c+d x)}{15 d \left (a^3+a^3 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [B] time = 0.466445, size = 201, normalized size = 2.07 \[ -\frac{2 \cos \left (\frac{1}{2} (c+d x)\right ) \left (14 \tan \left (\frac{c}{2}\right ) \cos ^3\left (\frac{1}{2} (c+d x)\right )+3 \tan \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right )+3 \sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right )+60 \cos ^5\left (\frac{1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+88 \sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \cos ^4\left (\frac{1}{2} (c+d x)\right )+14 \sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \cos ^2\left (\frac{1}{2} (c+d x)\right )\right )}{15 a^3 d (\cos (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 96, normalized size = 1. \begin{align*} -{\frac{1}{20\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{1}{3\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{7}{4\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{1}{d{a}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }+{\frac{1}{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 = 1.19147, size = 161, normalized size = 1.66 \begin{align*} -\frac{\frac{\frac{105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{20 \, \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{60 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{3}} + \frac{60 \, \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 = 1.66999, size = 424, normalized size = 4.37 \begin{align*} \frac{15 \,{\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (22 \, \cos \left (d x + c\right )^{2} + 51 \, \cos \left (d x + c\right ) + 32\right )} \sin \left (d x + c\right )}{30 \,{\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} 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 )}}{\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 = 1.40935, size = 127, normalized size = 1.31 \begin{align*} \frac{\frac{60 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} - \frac{60 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{3}} - \frac{3 \, a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 20 \, a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 105 \, 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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