Optimal. Leaf size=91 \[ \frac{(10 A+C) \tan (c+d x)}{3 a^2 d}-\frac{2 A \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{2 A \tan (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{(A+C) \tan (c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.296305, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3042, 2978, 2748, 3767, 8, 3770} \[ \frac{(10 A+C) \tan (c+d x)}{3 a^2 d}-\frac{2 A \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{2 A \tan (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{(A+C) \tan (c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 3042
Rule 2978
Rule 2748
Rule 3767
Rule 8
Rule 3770
Rubi steps
\begin{align*} \int \frac{\left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx &=-\frac{(A+C) \tan (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{\int \frac{(a (4 A+C)-a (2 A-C) \cos (c+d x)) \sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx}{3 a^2}\\ &=-\frac{2 A \tan (c+d x)}{a^2 d (1+\cos (c+d x))}-\frac{(A+C) \tan (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{\int \left (a^2 (10 A+C)-6 a^2 A \cos (c+d x)\right ) \sec ^2(c+d x) \, dx}{3 a^4}\\ &=-\frac{2 A \tan (c+d x)}{a^2 d (1+\cos (c+d x))}-\frac{(A+C) \tan (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac{(2 A) \int \sec (c+d x) \, dx}{a^2}+\frac{(10 A+C) \int \sec ^2(c+d x) \, dx}{3 a^2}\\ &=-\frac{2 A \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{2 A \tan (c+d x)}{a^2 d (1+\cos (c+d x))}-\frac{(A+C) \tan (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac{(10 A+C) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 a^2 d}\\ &=-\frac{2 A \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{(10 A+C) \tan (c+d x)}{3 a^2 d}-\frac{2 A \tan (c+d x)}{a^2 d (1+\cos (c+d x))}-\frac{(A+C) \tan (c+d x)}{3 d (a+a \cos (c+d x))^2}\\ \end{align*}
Mathematica [B] time = 1.39357, size = 288, normalized size = 3.16 \[ \frac{4 \cos \left (\frac{1}{2} (c+d x)\right ) \cos ^2(c+d x) \left (A \sec ^2(c+d x)+C\right ) \left ((A+C) \tan \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right )+(A+C) \sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right )+2 (7 A+C) \sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \cos ^2\left (\frac{1}{2} (c+d x)\right )+6 A \cos ^3\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\sin (d x)}{\left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}+2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-2 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{3 a^2 d (\cos (c+d x)+1)^2 (2 A+C \cos (2 (c+d x))+C)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 164, normalized size = 1.8 \begin{align*}{\frac{A}{6\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{C}{6\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{5\,A}{2\,d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{C}{2\,d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{A}{d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}+2\,{\frac{A\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) }{d{a}^{2}}}-{\frac{A}{d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-2\,{\frac{A\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) }{d{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02808, size = 258, normalized size = 2.84 \begin{align*} \frac{A{\left (\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 )}}\right )} + \frac{C{\left (\frac{3 \, \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}}\right )}}{a^{2}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41459, size = 429, normalized size = 4.71 \begin{align*} -\frac{3 \,{\left (A \cos \left (d x + c\right )^{3} + 2 \, A \cos \left (d x + c\right )^{2} + A \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (A \cos \left (d x + c\right )^{3} + 2 \, A \cos \left (d x + c\right )^{2} + A \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) -{\left ({\left (10 \, A + C\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (7 \, A + C\right )} \cos \left (d x + c\right ) + 3 \, A\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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29955, size = 192, normalized size = 2.11 \begin{align*} -\frac{\frac{12 \, A \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{2}} - \frac{12 \, A \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{2}} + \frac{12 \, A \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 a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 15 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, C 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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