Optimal. Leaf size=83 \[ -\frac{2 \tan (c+d x)}{a d}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{3 \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{\tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)} \]
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Rubi [A] time = 0.0931044, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2768, 2748, 3768, 3770, 3767, 8} \[ -\frac{2 \tan (c+d x)}{a d}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{3 \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{\tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)} \]
Antiderivative was successfully verified.
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Rule 2768
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)} \, dx &=-\frac{\sec (c+d x) \tan (c+d x)}{d (a+a \cos (c+d x))}-\frac{\int (-3 a+2 a \cos (c+d x)) \sec ^3(c+d x) \, dx}{a^2}\\ &=-\frac{\sec (c+d x) \tan (c+d x)}{d (a+a \cos (c+d x))}-\frac{2 \int \sec ^2(c+d x) \, dx}{a}+\frac{3 \int \sec ^3(c+d x) \, dx}{a}\\ &=\frac{3 \sec (c+d x) \tan (c+d x)}{2 a d}-\frac{\sec (c+d x) \tan (c+d x)}{d (a+a \cos (c+d x))}+\frac{3 \int \sec (c+d x) \, dx}{2 a}+\frac{2 \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{a d}\\ &=\frac{3 \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{2 \tan (c+d x)}{a d}+\frac{3 \sec (c+d x) \tan (c+d x)}{2 a d}-\frac{\sec (c+d x) \tan (c+d x)}{d (a+a \cos (c+d x))}\\ \end{align*}
Mathematica [B] time = 1.16563, size = 244, normalized size = 2.94 \[ \frac{\cos \left (\frac{1}{2} (c+d x)\right ) \left (\cos \left (\frac{1}{2} (c+d x)\right ) \left (-\frac{4 \sin (d x)}{\left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{1}{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}-\frac{1}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}-6 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+6 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-4 \sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right )\right )}{2 a d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 143, normalized size = 1.7 \begin{align*} -{\frac{1}{da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{1}{2\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}+{\frac{3}{2\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{3}{2\,da}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }-{\frac{1}{2\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}+{\frac{3}{2\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}+{\frac{3}{2\,da}\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 [B] time = 1.15837, size = 219, normalized size = 2.64 \begin{align*} -\frac{\frac{2 \,{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a - \frac{2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac{3 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} + \frac{3 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a} + \frac{2 \, \sin \left (d x + c\right )}{a{\left (\cos \left (d x + c\right ) + 1\right )}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64609, size = 301, normalized size = 3.63 \begin{align*} \frac{3 \,{\left (\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 ) - 3 \,{\left (\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 (4 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right )}{4 \,{\left (a d \cos \left (d x + c\right )^{3} + a 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{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.52233, size = 136, normalized size = 1.64 \begin{align*} \frac{\frac{3 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a} - \frac{3 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a} - \frac{2 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a} + \frac{2 \,{\left (3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - \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}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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