Optimal. Leaf size=175 \[ \frac{8 (83 A-20 B) \tan (c+d x)}{105 a^4 d}-\frac{(4 A-B) \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{(4 A-B) \tan (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac{(88 A-25 B) \tan (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(12 A-5 B) \tan (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A-B) \tan (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
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Rubi [A] time = 0.671646, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {2978, 2748, 3767, 8, 3770} \[ \frac{8 (83 A-20 B) \tan (c+d x)}{105 a^4 d}-\frac{(4 A-B) \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{(4 A-B) \tan (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac{(88 A-25 B) \tan (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(12 A-5 B) \tan (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A-B) \tan (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 2978
Rule 2748
Rule 3767
Rule 8
Rule 3770
Rubi steps
\begin{align*} \int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx &=-\frac{(A-B) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{\int \frac{(a (8 A-B)-4 a (A-B) \cos (c+d x)) \sec ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac{(A-B) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{(12 A-5 B) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{\int \frac{\left (2 a^2 (26 A-5 B)-3 a^2 (12 A-5 B) \cos (c+d x)\right ) \sec ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac{(88 A-25 B) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{(12 A-5 B) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{\int \frac{\left (a^3 (244 A-55 B)-2 a^3 (88 A-25 B) \cos (c+d x)\right ) \sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=-\frac{(88 A-25 B) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{(12 A-5 B) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac{(4 A-B) \tan (c+d x)}{d \left (a^4+a^4 \cos (c+d x)\right )}+\frac{\int \left (8 a^4 (83 A-20 B)-105 a^4 (4 A-B) \cos (c+d x)\right ) \sec ^2(c+d x) \, dx}{105 a^8}\\ &=-\frac{(88 A-25 B) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{(12 A-5 B) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac{(4 A-B) \tan (c+d x)}{d \left (a^4+a^4 \cos (c+d x)\right )}+\frac{(8 (83 A-20 B)) \int \sec ^2(c+d x) \, dx}{105 a^4}-\frac{(4 A-B) \int \sec (c+d x) \, dx}{a^4}\\ &=-\frac{(4 A-B) \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{(88 A-25 B) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{(12 A-5 B) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac{(4 A-B) \tan (c+d x)}{d \left (a^4+a^4 \cos (c+d x)\right )}-\frac{(8 (83 A-20 B)) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{105 a^4 d}\\ &=-\frac{(4 A-B) \tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac{8 (83 A-20 B) \tan (c+d x)}{105 a^4 d}-\frac{(88 A-25 B) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{(12 A-5 B) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac{(4 A-B) \tan (c+d x)}{d \left (a^4+a^4 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [B] time = 4.87917, size = 595, normalized size = 3.4 \[ \frac{26880 (4 A-B) \cos ^8\left (\frac{1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+\sec \left (\frac{c}{2}\right ) \sec (c) \cos \left (\frac{1}{2} (c+d x)\right ) \sec (c+d x) \left (-245 (44 A-17 B) \sin \left (\frac{d x}{2}\right )+7 (2684 A-635 B) \sin \left (\frac{3 d x}{2}\right )-20524 A \sin \left (c-\frac{d x}{2}\right )+14644 A \sin \left (c+\frac{d x}{2}\right )-16660 A \sin \left (2 c+\frac{d x}{2}\right )-4690 A \sin \left (c+\frac{3 d x}{2}\right )+14378 A \sin \left (2 c+\frac{3 d x}{2}\right )-9100 A \sin \left (3 c+\frac{3 d x}{2}\right )+11668 A \sin \left (c+\frac{5 d x}{2}\right )-630 A \sin \left (2 c+\frac{5 d x}{2}\right )+9358 A \sin \left (3 c+\frac{5 d x}{2}\right )-2940 A \sin \left (4 c+\frac{5 d x}{2}\right )+4228 A \sin \left (2 c+\frac{7 d x}{2}\right )+315 A \sin \left (3 c+\frac{7 d x}{2}\right )+3493 A \sin \left (4 c+\frac{7 d x}{2}\right )-420 A \sin \left (5 c+\frac{7 d x}{2}\right )+664 A \sin \left (3 c+\frac{9 d x}{2}\right )+105 A \sin \left (4 c+\frac{9 d x}{2}\right )+559 A \sin \left (5 c+\frac{9 d x}{2}\right )+4795 B \sin \left (c-\frac{d x}{2}\right )-4795 B \sin \left (c+\frac{d x}{2}\right )+4165 B \sin \left (2 c+\frac{d x}{2}\right )+2275 B \sin \left (c+\frac{3 d x}{2}\right )-4445 B \sin \left (2 c+\frac{3 d x}{2}\right )+2275 B \sin \left (3 c+\frac{3 d x}{2}\right )-2785 B \sin \left (c+\frac{5 d x}{2}\right )+735 B \sin \left (2 c+\frac{5 d x}{2}\right )-2785 B \sin \left (3 c+\frac{5 d x}{2}\right )+735 B \sin \left (4 c+\frac{5 d x}{2}\right )-1015 B \sin \left (2 c+\frac{7 d x}{2}\right )+105 B \sin \left (3 c+\frac{7 d x}{2}\right )-1015 B \sin \left (4 c+\frac{7 d x}{2}\right )+105 B \sin \left (5 c+\frac{7 d x}{2}\right )-160 B \sin \left (3 c+\frac{9 d x}{2}\right )-160 B \sin \left (5 c+\frac{9 d x}{2}\right )\right )}{1680 a^4 d (\cos (c+d x)+1)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.113, size = 285, normalized size = 1.6 \begin{align*}{\frac{A}{56\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{B}{56\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{7\,A}{40\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{B}{8\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{23\,A}{24\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{11\,B}{24\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{49\,A}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{15\,B}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+4\,{\frac{A\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) }{d{a}^{4}}}-{\frac{B}{d{a}^{4}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }-{\frac{A}{d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-4\,{\frac{A\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) }{d{a}^{4}}}+{\frac{B}{d{a}^{4}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }-{\frac{A}{d{a}^{4}} \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 [A] time = 1.03907, size = 440, normalized size = 2.51 \begin{align*} \frac{A{\left (\frac{1680 \, \sin \left (d x + c\right )}{{\left (a^{4} - \frac{a^{4} \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{5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac{3360 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac{3360 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} - 5 \, B{\left (\frac{\frac{315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac{168 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac{168 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )}}{840 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.41286, size = 838, normalized size = 4.79 \begin{align*} -\frac{105 \,{\left ({\left (4 \, A - B\right )} \cos \left (d x + c\right )^{5} + 4 \,{\left (4 \, A - B\right )} \cos \left (d x + c\right )^{4} + 6 \,{\left (4 \, A - B\right )} \cos \left (d x + c\right )^{3} + 4 \,{\left (4 \, A - B\right )} \cos \left (d x + c\right )^{2} +{\left (4 \, A - B\right )} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \,{\left ({\left (4 \, A - B\right )} \cos \left (d x + c\right )^{5} + 4 \,{\left (4 \, A - B\right )} \cos \left (d x + c\right )^{4} + 6 \,{\left (4 \, A - B\right )} \cos \left (d x + c\right )^{3} + 4 \,{\left (4 \, A - B\right )} \cos \left (d x + c\right )^{2} +{\left (4 \, A - B\right )} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (8 \,{\left (83 \, A - 20 \, B\right )} \cos \left (d x + c\right )^{4} +{\left (2236 \, A - 535 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \,{\left (659 \, A - 155 \, B\right )} \cos \left (d x + c\right )^{2} + 4 \,{\left (296 \, A - 65 \, B\right )} \cos \left (d x + c\right ) + 105 \, A\right )} \sin \left (d x + c\right )}{210 \,{\left (a^{4} d \cos \left (d x + c\right )^{5} + 4 \, a^{4} d \cos \left (d x + c\right )^{4} + 6 \, a^{4} d \cos \left (d x + c\right )^{3} + 4 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} 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 = 1.2903, size = 302, normalized size = 1.73 \begin{align*} -\frac{\frac{840 \,{\left (4 \, A - B\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac{840 \,{\left (4 \, A - B\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} + \frac{1680 \, 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^{4}} - \frac{15 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 147 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 105 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 805 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 385 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 5145 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1575 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{28}}}{840 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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