Optimal. Leaf size=133 \[ \frac{(4 A+3 C) \tan ^3(c+d x)}{3 a d}+\frac{(4 A+3 C) \tan (c+d x)}{a d}-\frac{(3 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{(3 A+2 C) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(A+C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)} \]
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Rubi [A] time = 0.183244, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {3042, 2748, 3767, 3768, 3770} \[ \frac{(4 A+3 C) \tan ^3(c+d x)}{3 a d}+\frac{(4 A+3 C) \tan (c+d x)}{a d}-\frac{(3 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{(3 A+2 C) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(A+C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)} \]
Antiderivative was successfully verified.
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Rule 3042
Rule 2748
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \frac{\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx &=-\frac{(A+C) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \cos (c+d x))}+\frac{\int (a (4 A+3 C)-a (3 A+2 C) \cos (c+d x)) \sec ^4(c+d x) \, dx}{a^2}\\ &=-\frac{(A+C) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \cos (c+d x))}-\frac{(3 A+2 C) \int \sec ^3(c+d x) \, dx}{a}+\frac{(4 A+3 C) \int \sec ^4(c+d x) \, dx}{a}\\ &=-\frac{(3 A+2 C) \sec (c+d x) \tan (c+d x)}{2 a d}-\frac{(A+C) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \cos (c+d x))}-\frac{(3 A+2 C) \int \sec (c+d x) \, dx}{2 a}-\frac{(4 A+3 C) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{a d}\\ &=-\frac{(3 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{(4 A+3 C) \tan (c+d x)}{a d}-\frac{(3 A+2 C) \sec (c+d x) \tan (c+d x)}{2 a d}-\frac{(A+C) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \cos (c+d x))}+\frac{(4 A+3 C) \tan ^3(c+d x)}{3 a d}\\ \end{align*}
Mathematica [B] time = 6.46505, size = 765, normalized size = 5.75 \[ \frac{2 \cos ^2\left (\frac{c}{2}+\frac{d x}{2}\right ) \left (5 A \sin \left (\frac{d x}{2}\right )+3 C \sin \left (\frac{d x}{2}\right )\right )}{3 d \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) (a \cos (c+d x)+a) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}+\frac{2 \cos ^2\left (\frac{c}{2}+\frac{d x}{2}\right ) \left (5 A \sin \left (\frac{d x}{2}\right )+3 C \sin \left (\frac{d x}{2}\right )\right )}{3 d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) (a \cos (c+d x)+a) \left (\sin \left (\frac{c}{2}+\frac{d x}{2}\right )+\cos \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}+\frac{(3 A+2 C) \cos ^2\left (\frac{c}{2}+\frac{d x}{2}\right ) \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d (a \cos (c+d x)+a)}+\frac{(-3 A-2 C) \cos ^2\left (\frac{c}{2}+\frac{d x}{2}\right ) \log \left (\sin \left (\frac{c}{2}+\frac{d x}{2}\right )+\cos \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d (a \cos (c+d x)+a)}+\frac{2 \sec \left (\frac{c}{2}\right ) \cos \left (\frac{c}{2}+\frac{d x}{2}\right ) \left (A \sin \left (\frac{d x}{2}\right )+C \sin \left (\frac{d x}{2}\right )\right )}{d (a \cos (c+d x)+a)}+\frac{\left (2 A \sin \left (\frac{c}{2}\right )-A \cos \left (\frac{c}{2}\right )\right ) \cos ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) (a \cos (c+d x)+a) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^2}+\frac{\left (2 A \sin \left (\frac{c}{2}\right )+A \cos \left (\frac{c}{2}\right )\right ) \cos ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) (a \cos (c+d x)+a) \left (\sin \left (\frac{c}{2}+\frac{d x}{2}\right )+\cos \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^2}+\frac{A \sin \left (\frac{d x}{2}\right ) \cos ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) (a \cos (c+d x)+a) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^3}+\frac{A \sin \left (\frac{d x}{2}\right ) \cos ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) (a \cos (c+d x)+a) \left (\sin \left (\frac{c}{2}+\frac{d x}{2}\right )+\cos \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.062, size = 294, normalized size = 2.2 \begin{align*}{\frac{A}{da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{C}{da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{A}{3\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-3}}-{\frac{A}{da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}+{\frac{3\,A}{2\,da}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }+{\frac{C}{da}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }-{\frac{5\,A}{2\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{C}{da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{A}{3\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}+{\frac{A}{da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}-{\frac{3\,A}{2\,da}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }-{\frac{C}{da}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }-{\frac{5\,A}{2\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-{\frac{C}{da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.00923, size = 439, normalized size = 3.3 \begin{align*} \frac{A{\left (\frac{2 \,{\left (\frac{9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{16 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{15 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a - \frac{3 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{3 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac{9 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} + \frac{9 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a} + \frac{6 \, \sin \left (d x + c\right )}{a{\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - 6 \, C{\left (\frac{\log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} - \frac{\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 )}{{\left (a - \frac{a \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 )}} - \frac{\sin \left (d x + c\right )}{a{\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49427, size = 429, normalized size = 3.23 \begin{align*} -\frac{3 \,{\left ({\left (3 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{4} +{\left (3 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left ({\left (3 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{4} +{\left (3 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (4 \,{\left (4 \, A + 3 \, C\right )} \cos \left (d x + c\right )^{3} +{\left (7 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{2} - A \cos \left (d x + c\right ) + 2 \, A\right )} \sin \left (d x + c\right )}{12 \,{\left (a d \cos \left (d x + c\right )^{4} + a d \cos \left (d x + c\right )^{3}\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 = 1.2504, size = 250, normalized size = 1.88 \begin{align*} -\frac{\frac{3 \,{\left (3 \, A + 2 \, C\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a} - \frac{3 \,{\left (3 \, A + 2 \, C\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a} - \frac{6 \,{\left (A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{a} + \frac{2 \,{\left (15 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 6 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 16 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 12 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 9 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, C \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 )}^{3} a}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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