Optimal. Leaf size=96 \[ \frac{2 (3 A-80 B) \sin (x)}{105 a^4 (\cos (x)+1)}+\frac{(6 A-55 B) \sin (x)}{105 a^4 (\cos (x)+1)^2}+\frac{B \tanh ^{-1}(\sin (x))}{a^4}+\frac{(3 A-10 B) \sin (x)}{35 a (a \cos (x)+a)^3}+\frac{(A-B) \sin (x)}{7 (a \cos (x)+a)^4} \]
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Rubi [A] time = 0.414411, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {2828, 2978, 12, 3770} \[ \frac{2 (3 A-80 B) \sin (x)}{105 a^4 (\cos (x)+1)}+\frac{(6 A-55 B) \sin (x)}{105 a^4 (\cos (x)+1)^2}+\frac{B \tanh ^{-1}(\sin (x))}{a^4}+\frac{(3 A-10 B) \sin (x)}{35 a (a \cos (x)+a)^3}+\frac{(A-B) \sin (x)}{7 (a \cos (x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 2828
Rule 2978
Rule 12
Rule 3770
Rubi steps
\begin{align*} \int \frac{A+B \sec (x)}{(a+a \cos (x))^4} \, dx &=\int \frac{(B+A \cos (x)) \sec (x)}{(a+a \cos (x))^4} \, dx\\ &=\frac{(A-B) \sin (x)}{7 (a+a \cos (x))^4}+\frac{\int \frac{(7 a B+3 a (A-B) \cos (x)) \sec (x)}{(a+a \cos (x))^3} \, dx}{7 a^2}\\ &=\frac{(A-B) \sin (x)}{7 (a+a \cos (x))^4}+\frac{(3 A-10 B) \sin (x)}{35 a (a+a \cos (x))^3}+\frac{\int \frac{\left (35 a^2 B+2 a^2 (3 A-10 B) \cos (x)\right ) \sec (x)}{(a+a \cos (x))^2} \, dx}{35 a^4}\\ &=\frac{(6 A-55 B) \sin (x)}{105 a^4 (1+\cos (x))^2}+\frac{(A-B) \sin (x)}{7 (a+a \cos (x))^4}+\frac{(3 A-10 B) \sin (x)}{35 a (a+a \cos (x))^3}+\frac{\int \frac{\left (105 a^3 B+a^3 (6 A-55 B) \cos (x)\right ) \sec (x)}{a+a \cos (x)} \, dx}{105 a^6}\\ &=\frac{(6 A-55 B) \sin (x)}{105 a^4 (1+\cos (x))^2}+\frac{(A-B) \sin (x)}{7 (a+a \cos (x))^4}+\frac{(3 A-10 B) \sin (x)}{35 a (a+a \cos (x))^3}+\frac{2 (3 A-80 B) \sin (x)}{105 \left (a^4+a^4 \cos (x)\right )}+\frac{\int 105 a^4 B \sec (x) \, dx}{105 a^8}\\ &=\frac{(6 A-55 B) \sin (x)}{105 a^4 (1+\cos (x))^2}+\frac{(A-B) \sin (x)}{7 (a+a \cos (x))^4}+\frac{(3 A-10 B) \sin (x)}{35 a (a+a \cos (x))^3}+\frac{2 (3 A-80 B) \sin (x)}{105 \left (a^4+a^4 \cos (x)\right )}+\frac{B \int \sec (x) \, dx}{a^4}\\ &=\frac{B \tanh ^{-1}(\sin (x))}{a^4}+\frac{(6 A-55 B) \sin (x)}{105 a^4 (1+\cos (x))^2}+\frac{(A-B) \sin (x)}{7 (a+a \cos (x))^4}+\frac{(3 A-10 B) \sin (x)}{35 a (a+a \cos (x))^3}+\frac{2 (3 A-80 B) \sin (x)}{105 \left (a^4+a^4 \cos (x)\right )}\\ \end{align*}
Mathematica [A] time = 0.805181, size = 104, normalized size = 1.08 \[ \frac{\sin (x) ((87 A-1480 B) \cos (x)+(24 A-535 B) \cos (2 x)+3 A \cos (3 x)+96 A-80 B \cos (3 x)-1055 B)-3360 B \cos ^8\left (\frac{x}{2}\right ) \left (\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )\right )}{210 a^4 (\cos (x)+1)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 119, normalized size = 1.2 \begin{align*}{\frac{A}{56\,{a}^{4}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{7}}-{\frac{B}{56\,{a}^{4}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{7}}+{\frac{3\,A}{40\,{a}^{4}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{5}}-{\frac{B}{8\,{a}^{4}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{5}}+{\frac{A}{8\,{a}^{4}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}}-{\frac{11\,B}{24\,{a}^{4}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}}+{\frac{B}{{a}^{4}}\ln \left ( 1+\tan \left ({\frac{x}{2}} \right ) \right ) }-{\frac{B}{{a}^{4}}\ln \left ( \tan \left ({\frac{x}{2}} \right ) -1 \right ) }+{\frac{A}{8\,{a}^{4}}\tan \left ({\frac{x}{2}} \right ) }-{\frac{15\,B}{8\,{a}^{4}}\tan \left ({\frac{x}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09966, size = 193, normalized size = 2.01 \begin{align*} -\frac{1}{168} \, B{\left (\frac{\frac{315 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{77 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{21 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac{3 \, \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}}}{a^{4}} - \frac{168 \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right )}{a^{4}} + \frac{168 \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} - 1\right )}{a^{4}}\right )} + \frac{A{\left (\frac{35 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{35 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{21 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac{5 \, \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}}\right )}}{280 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.53879, size = 464, normalized size = 4.83 \begin{align*} \frac{105 \,{\left (B \cos \left (x\right )^{4} + 4 \, B \cos \left (x\right )^{3} + 6 \, B \cos \left (x\right )^{2} + 4 \, B \cos \left (x\right ) + B\right )} \log \left (\sin \left (x\right ) + 1\right ) - 105 \,{\left (B \cos \left (x\right )^{4} + 4 \, B \cos \left (x\right )^{3} + 6 \, B \cos \left (x\right )^{2} + 4 \, B \cos \left (x\right ) + B\right )} \log \left (-\sin \left (x\right ) + 1\right ) + 2 \,{\left (2 \,{\left (3 \, A - 80 \, B\right )} \cos \left (x\right )^{3} +{\left (24 \, A - 535 \, B\right )} \cos \left (x\right )^{2} +{\left (39 \, A - 620 \, B\right )} \cos \left (x\right ) + 36 \, A - 260 \, B\right )} \sin \left (x\right )}{210 \,{\left (a^{4} \cos \left (x\right )^{4} + 4 \, a^{4} \cos \left (x\right )^{3} + 6 \, a^{4} \cos \left (x\right )^{2} + 4 \, a^{4} \cos \left (x\right ) + a^{4}\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 = 1.15574, size = 170, normalized size = 1.77 \begin{align*} \frac{B \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) + 1 \right |}\right )}{a^{4}} - \frac{B \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) - 1 \right |}\right )}{a^{4}} + \frac{15 \, A a^{24} \tan \left (\frac{1}{2} \, x\right )^{7} - 15 \, B a^{24} \tan \left (\frac{1}{2} \, x\right )^{7} + 63 \, A a^{24} \tan \left (\frac{1}{2} \, x\right )^{5} - 105 \, B a^{24} \tan \left (\frac{1}{2} \, x\right )^{5} + 105 \, A a^{24} \tan \left (\frac{1}{2} \, x\right )^{3} - 385 \, B a^{24} \tan \left (\frac{1}{2} \, x\right )^{3} + 105 \, A a^{24} \tan \left (\frac{1}{2} \, x\right ) - 1575 \, B a^{24} \tan \left (\frac{1}{2} \, x\right )}{840 \, a^{28}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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