Optimal. Leaf size=129 \[ \frac{2 (36 A+C) \tan (c+d x)}{15 a^3 d}-\frac{3 A \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{3 A \tan (c+d x)}{d \left (a^3 \cos (c+d x)+a^3\right )}-\frac{(9 A-C) \tan (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A+C) \tan (c+d x)}{5 d (a \cos (c+d x)+a)^3} \]
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Rubi [A] time = 0.442715, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{2 (36 A+C) \tan (c+d x)}{15 a^3 d}-\frac{3 A \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{3 A \tan (c+d x)}{d \left (a^3 \cos (c+d x)+a^3\right )}-\frac{(9 A-C) \tan (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A+C) \tan (c+d x)}{5 d (a \cos (c+d x)+a)^3} \]
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))^3} \, dx &=-\frac{(A+C) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac{\int \frac{(a (6 A+C)-a (3 A-2 C) \cos (c+d x)) \sec ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac{(A+C) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{(9 A-C) \tan (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac{\int \frac{\left (a^2 (27 A+2 C)-2 a^2 (9 A-C) \cos (c+d x)\right ) \sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx}{15 a^4}\\ &=-\frac{(A+C) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{(9 A-C) \tan (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac{3 A \tan (c+d x)}{d \left (a^3+a^3 \cos (c+d x)\right )}+\frac{\int \left (2 a^3 (36 A+C)-45 a^3 A \cos (c+d x)\right ) \sec ^2(c+d x) \, dx}{15 a^6}\\ &=-\frac{(A+C) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{(9 A-C) \tan (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac{3 A \tan (c+d x)}{d \left (a^3+a^3 \cos (c+d x)\right )}-\frac{(3 A) \int \sec (c+d x) \, dx}{a^3}+\frac{(2 (36 A+C)) \int \sec ^2(c+d x) \, dx}{15 a^3}\\ &=-\frac{3 A \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{(A+C) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{(9 A-C) \tan (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac{3 A \tan (c+d x)}{d \left (a^3+a^3 \cos (c+d x)\right )}-\frac{(2 (36 A+C)) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{15 a^3 d}\\ &=-\frac{3 A \tanh ^{-1}(\sin (c+d x))}{a^3 d}+\frac{2 (36 A+C) \tan (c+d x)}{15 a^3 d}-\frac{(A+C) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{(9 A-C) \tan (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac{3 A \tan (c+d x)}{d \left (a^3+a^3 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [B] time = 6.30674, size = 596, normalized size = 4.62 \[ \frac{\frac{\sec \left (\frac{c}{2}\right ) \sec (c) \cos (c+d x) \cos \left (\frac{c}{2}+\frac{d x}{2}\right ) \left (-600 A \sin \left (c-\frac{d x}{2}\right )+375 A \sin \left (c+\frac{d x}{2}\right )-480 A \sin \left (2 c+\frac{d x}{2}\right )-60 A \sin \left (c+\frac{3 d x}{2}\right )+402 A \sin \left (2 c+\frac{3 d x}{2}\right )-225 A \sin \left (3 c+\frac{3 d x}{2}\right )+315 A \sin \left (c+\frac{5 d x}{2}\right )+30 A \sin \left (2 c+\frac{5 d x}{2}\right )+240 A \sin \left (3 c+\frac{5 d x}{2}\right )-45 A \sin \left (4 c+\frac{5 d x}{2}\right )+72 A \sin \left (2 c+\frac{7 d x}{2}\right )+15 A \sin \left (3 c+\frac{7 d x}{2}\right )+57 A \sin \left (4 c+\frac{7 d x}{2}\right )-255 A \sin \left (\frac{d x}{2}\right )+567 A \sin \left (\frac{3 d x}{2}\right )-10 C \sin \left (c-\frac{d x}{2}\right )+10 C \sin \left (c+\frac{d x}{2}\right )-20 C \sin \left (2 c+\frac{d x}{2}\right )+22 C \sin \left (2 c+\frac{3 d x}{2}\right )+10 C \sin \left (c+\frac{5 d x}{2}\right )+10 C \sin \left (3 c+\frac{5 d x}{2}\right )+2 C \sin \left (2 c+\frac{7 d x}{2}\right )+2 C \sin \left (4 c+\frac{7 d x}{2}\right )-20 C \sin \left (\frac{d x}{2}\right )+22 C \sin \left (\frac{3 d x}{2}\right )\right ) \left (A \sec ^2(c+d x)+C\right )}{60 d (\cos (c+d x)+1)^3 (2 A+C \cos (2 c+2 d x)+C)}+\frac{48 A \cos ^2(c+d x) \cos ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) \left (A \sec ^2(c+d x)+C\right ) \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d (\cos (c+d x)+1)^3 (2 A+C \cos (2 c+2 d x)+C)}-\frac{48 A \cos ^2(c+d x) \cos ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) \left (A \sec ^2(c+d x)+C\right ) \log \left (\sin \left (\frac{c}{2}+\frac{d x}{2}\right )+\cos \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d (\cos (c+d x)+1)^3 (2 A+C \cos (2 c+2 d x)+C)}}{a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 204, normalized size = 1.6 \begin{align*}{\frac{A}{20\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{C}{20\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{A}{2\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{C}{6\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{17\,A}{4\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{C}{4\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{A}{d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}+3\,{\frac{A\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) }{d{a}^{3}}}-{\frac{A}{d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-3\,{\frac{A\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) }{d{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03376, size = 315, normalized size = 2.44 \begin{align*} \frac{3 \, A{\left (\frac{40 \, \sin \left (d x + c\right )}{{\left (a^{3} - \frac{a^{3} \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{\frac{85 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{\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}}\right )} + \frac{C{\left (\frac{15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{10 \, \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}}\right )}}{a^{3}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46351, size = 576, normalized size = 4.47 \begin{align*} -\frac{45 \,{\left (A \cos \left (d x + c\right )^{4} + 3 \, A \cos \left (d x + c\right )^{3} + 3 \, 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 ) - 45 \,{\left (A \cos \left (d x + c\right )^{4} + 3 \, A \cos \left (d x + c\right )^{3} + 3 \, 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 ) - 2 \,{\left (2 \,{\left (36 \, A + C\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (57 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{2} +{\left (117 \, A + 7 \, C\right )} \cos \left (d x + c\right ) + 15 \, A\right )} \sin \left (d x + c\right )}{30 \,{\left (a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} 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 = 1.33288, size = 240, normalized size = 1.86 \begin{align*} -\frac{\frac{180 \, A \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} - \frac{180 \, A \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{3}} + \frac{120 \, 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^{3}} - \frac{3 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, C a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 30 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 10 \, C a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 255 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 15 \, C 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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