Optimal. Leaf size=81 \[ \frac{10 \tan (c+d x)}{3 a^2 d}-\frac{2 \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{2 \tan (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{\tan (c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.172821, antiderivative size = 81, 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 = {2766, 2978, 2748, 3767, 8, 3770} \[ \frac{10 \tan (c+d x)}{3 a^2 d}-\frac{2 \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{2 \tan (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{\tan (c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2766
Rule 2978
Rule 2748
Rule 3767
Rule 8
Rule 3770
Rubi steps
\begin{align*} \int \frac{\sec ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx &=-\frac{\tan (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{\int \frac{(4 a-2 a \cos (c+d x)) \sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx}{3 a^2}\\ &=-\frac{2 \tan (c+d x)}{a^2 d (1+\cos (c+d x))}-\frac{\tan (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{\int \left (10 a^2-6 a^2 \cos (c+d x)\right ) \sec ^2(c+d x) \, dx}{3 a^4}\\ &=-\frac{2 \tan (c+d x)}{a^2 d (1+\cos (c+d x))}-\frac{\tan (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac{2 \int \sec (c+d x) \, dx}{a^2}+\frac{10 \int \sec ^2(c+d x) \, dx}{3 a^2}\\ &=-\frac{2 \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{2 \tan (c+d x)}{a^2 d (1+\cos (c+d x))}-\frac{\tan (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac{10 \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 a^2 d}\\ &=-\frac{2 \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{10 \tan (c+d x)}{3 a^2 d}-\frac{2 \tan (c+d x)}{a^2 d (1+\cos (c+d x))}-\frac{\tan (c+d x)}{3 d (a+a \cos (c+d x))^2}\\ \end{align*}
Mathematica [B] time = 1.01288, size = 239, normalized size = 2.95 \[ \frac{2 \cos \left (\frac{1}{2} (c+d x)\right ) \left (\tan \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right )+\sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right )+6 \cos ^3\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\sin (d x)}{\left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}+2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-2 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+14 \sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \cos ^2\left (\frac{1}{2} (c+d x)\right )\right )}{3 a^2 d (\cos (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.064, size = 120, normalized size = 1.5 \begin{align*}{\frac{1}{6\,{a}^{2}d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{5}{2\,{a}^{2}d}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{1}{{a}^{2}d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}+2\,{\frac{\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) }{{a}^{2}d}}-{\frac{1}{{a}^{2}d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-2\,{\frac{\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) }{{a}^{2}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08842, size = 196, normalized size = 2.42 \begin{align*} \frac{\frac{\frac{15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac{12 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{2}} + \frac{12 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{2}} + \frac{12 \, \sin \left (d x + c\right )}{{\left (a^{2} - \frac{a^{2} \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 )}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70317, size = 387, normalized size = 4.78 \begin{align*} -\frac{3 \,{\left (\cos \left (d x + c\right )^{3} + 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (\cos \left (d x + c\right )^{3} + 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) -{\left (10 \, \cos \left (d x + c\right )^{2} + 14 \, \cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right )}{3 \,{\left (a^{2} d \cos \left (d x + c\right )^{3} + 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} 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 ^{2}{\left (c + d x \right )}}{\cos ^{2}{\left (c + d x \right )} + 2 \cos{\left (c + d x \right )} + 1}\, dx}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32408, size = 143, normalized size = 1.77 \begin{align*} -\frac{\frac{12 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{2}} - \frac{12 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{2}} + \frac{12 \, \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^{2}} - \frac{a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 15 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{6}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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