Optimal. Leaf size=232 \[ -\frac{8 (216 A-83 B) \tan (c+d x)}{105 a^4 d}+\frac{(21 A-8 B) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{(21 A-8 B) \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac{4 (216 A-83 B) \tan (c+d x) \sec (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(129 A-52 B) \tan (c+d x) \sec (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(2 A-B) \tan (c+d x) \sec (c+d x)}{5 a d (a \cos (c+d x)+a)^3}-\frac{(A-B) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
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Rubi [A] time = 0.687616, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {2978, 2748, 3768, 3770, 3767, 8} \[ -\frac{8 (216 A-83 B) \tan (c+d x)}{105 a^4 d}+\frac{(21 A-8 B) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{(21 A-8 B) \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac{4 (216 A-83 B) \tan (c+d x) \sec (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(129 A-52 B) \tan (c+d x) \sec (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(2 A-B) \tan (c+d x) \sec (c+d x)}{5 a d (a \cos (c+d x)+a)^3}-\frac{(A-B) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 2978
Rule 2748
Rule 3768
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx &=-\frac{(A-B) \sec (c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{\int \frac{(a (9 A-2 B)-5 a (A-B) \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac{(A-B) \sec (c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{(2 A-B) \sec (c+d x) \tan (c+d x)}{5 a d (a+a \cos (c+d x))^3}+\frac{\int \frac{\left (a^2 (73 A-24 B)-28 a^2 (2 A-B) \cos (c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac{(129 A-52 B) \sec (c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B) \sec (c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{(2 A-B) \sec (c+d x) \tan (c+d x)}{5 a d (a+a \cos (c+d x))^3}+\frac{\int \frac{\left (a^3 (477 A-176 B)-3 a^3 (129 A-52 B) \cos (c+d x)\right ) \sec ^3(c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=-\frac{(129 A-52 B) \sec (c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B) \sec (c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{(2 A-B) \sec (c+d x) \tan (c+d x)}{5 a d (a+a \cos (c+d x))^3}-\frac{4 (216 A-83 B) \sec (c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )}+\frac{\int \left (105 a^4 (21 A-8 B)-8 a^4 (216 A-83 B) \cos (c+d x)\right ) \sec ^3(c+d x) \, dx}{105 a^8}\\ &=-\frac{(129 A-52 B) \sec (c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B) \sec (c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{(2 A-B) \sec (c+d x) \tan (c+d x)}{5 a d (a+a \cos (c+d x))^3}-\frac{4 (216 A-83 B) \sec (c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )}-\frac{(8 (216 A-83 B)) \int \sec ^2(c+d x) \, dx}{105 a^4}+\frac{(21 A-8 B) \int \sec ^3(c+d x) \, dx}{a^4}\\ &=\frac{(21 A-8 B) \sec (c+d x) \tan (c+d x)}{2 a^4 d}-\frac{(129 A-52 B) \sec (c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B) \sec (c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{(2 A-B) \sec (c+d x) \tan (c+d x)}{5 a d (a+a \cos (c+d x))^3}-\frac{4 (216 A-83 B) \sec (c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )}+\frac{(21 A-8 B) \int \sec (c+d x) \, dx}{2 a^4}+\frac{(8 (216 A-83 B)) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{105 a^4 d}\\ &=\frac{(21 A-8 B) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{8 (216 A-83 B) \tan (c+d x)}{105 a^4 d}+\frac{(21 A-8 B) \sec (c+d x) \tan (c+d x)}{2 a^4 d}-\frac{(129 A-52 B) \sec (c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B) \sec (c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{(2 A-B) \sec (c+d x) \tan (c+d x)}{5 a d (a+a \cos (c+d x))^3}-\frac{4 (216 A-83 B) \sec (c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [B] time = 6.44624, size = 798, normalized size = 3.44 \[ -\frac{8 (21 A-8 B) \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right ) \cos ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{d (\cos (c+d x) a+a)^4}+\frac{8 (21 A-8 B) \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right ) \cos ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{d (\cos (c+d x) a+a)^4}+\frac{\sec \left (\frac{c}{2}\right ) \sec (c) \sec ^2(c+d x) \left (73206 A \sin \left (\frac{d x}{2}\right )-38668 B \sin \left (\frac{d x}{2}\right )-166668 A \sin \left (\frac{3 d x}{2}\right )+64384 B \sin \left (\frac{3 d x}{2}\right )+183162 A \sin \left (c-\frac{d x}{2}\right )-70896 B \sin \left (c-\frac{d x}{2}\right )-100842 A \sin \left (c+\frac{d x}{2}\right )+50316 B \sin \left (c+\frac{d x}{2}\right )+155526 A \sin \left (2 c+\frac{d x}{2}\right )-59248 B \sin \left (2 c+\frac{d x}{2}\right )+37380 A \sin \left (c+\frac{3 d x}{2}\right )-22820 B \sin \left (c+\frac{3 d x}{2}\right )-101148 A \sin \left (2 c+\frac{3 d x}{2}\right )+48004 B \sin \left (2 c+\frac{3 d x}{2}\right )+102900 A \sin \left (3 c+\frac{3 d x}{2}\right )-39200 B \sin \left (3 c+\frac{3 d x}{2}\right )-119364 A \sin \left (c+\frac{5 d x}{2}\right )+46032 B \sin \left (c+\frac{5 d x}{2}\right )+8820 A \sin \left (2 c+\frac{5 d x}{2}\right )-8750 B \sin \left (2 c+\frac{5 d x}{2}\right )-78204 A \sin \left (3 c+\frac{5 d x}{2}\right )+35742 B \sin \left (3 c+\frac{5 d x}{2}\right )+49980 A \sin \left (4 c+\frac{5 d x}{2}\right )-19040 B \sin \left (4 c+\frac{5 d x}{2}\right )-64053 A \sin \left (2 c+\frac{7 d x}{2}\right )+24664 B \sin \left (2 c+\frac{7 d x}{2}\right )-3885 A \sin \left (3 c+\frac{7 d x}{2}\right )-1050 B \sin \left (3 c+\frac{7 d x}{2}\right )-44733 A \sin \left (4 c+\frac{7 d x}{2}\right )+19834 B \sin \left (4 c+\frac{7 d x}{2}\right )+15435 A \sin \left (5 c+\frac{7 d x}{2}\right )-5880 B \sin \left (5 c+\frac{7 d x}{2}\right )-21987 A \sin \left (3 c+\frac{9 d x}{2}\right )+8456 B \sin \left (3 c+\frac{9 d x}{2}\right )-3675 A \sin \left (4 c+\frac{9 d x}{2}\right )+630 B \sin \left (4 c+\frac{9 d x}{2}\right )-16107 A \sin \left (5 c+\frac{9 d x}{2}\right )+6986 B \sin \left (5 c+\frac{9 d x}{2}\right )+2205 A \sin \left (6 c+\frac{9 d x}{2}\right )-840 B \sin \left (6 c+\frac{9 d x}{2}\right )-3456 A \sin \left (4 c+\frac{11 d x}{2}\right )+1328 B \sin \left (4 c+\frac{11 d x}{2}\right )-840 A \sin \left (5 c+\frac{11 d x}{2}\right )+210 B \sin \left (5 c+\frac{11 d x}{2}\right )-2616 A \sin \left (6 c+\frac{11 d x}{2}\right )+1118 B \sin \left (6 c+\frac{11 d x}{2}\right )\right ) \cos \left (\frac{c}{2}+\frac{d x}{2}\right )}{6720 d (\cos (c+d x) a+a)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.115, size = 374, 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{9\,A}{40\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{7\,B}{40\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{13\,A}{8\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{23\,B}{24\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{111\,A}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{49\,B}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{21\,A}{2\,d{a}^{4}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }+4\,{\frac{\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) B}{d{a}^{4}}}+{\frac{9\,A}{2\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{B}{d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}+{\frac{A}{2\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}+{\frac{9\,A}{2\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-{\frac{B}{d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}+{\frac{21\,A}{2\,d{a}^{4}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }-4\,{\frac{\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) B}{d{a}^{4}}}-{\frac{A}{2\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03788, size = 566, normalized size = 2.44 \begin{align*} -\frac{3 \, A{\left (\frac{280 \,{\left (\frac{7 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{9 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{4} - \frac{2 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac{\frac{3885 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{455 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac{2940 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac{2940 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} - B{\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 )}}{840 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51293, size = 938, normalized size = 4.04 \begin{align*} \frac{105 \,{\left ({\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{6} + 4 \,{\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{5} + 6 \,{\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \,{\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{3} +{\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \,{\left ({\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{6} + 4 \,{\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{5} + 6 \,{\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \,{\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{3} +{\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (16 \,{\left (216 \, A - 83 \, B\right )} \cos \left (d x + c\right )^{5} +{\left (11619 \, A - 4472 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \,{\left (3411 \, A - 1318 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \,{\left (1509 \, A - 592 \, B\right )} \cos \left (d x + c\right )^{2} + 210 \,{\left (2 \, A - B\right )} \cos \left (d x + c\right ) - 105 \, A\right )} \sin \left (d x + c\right )}{420 \,{\left (a^{4} d \cos \left (d x + c\right )^{6} + 4 \, a^{4} d \cos \left (d x + c\right )^{5} + 6 \, a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + a^{4} d \cos \left (d x + c\right )^{2}\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 = 1.28189, size = 360, normalized size = 1.55 \begin{align*} \frac{\frac{420 \,{\left (21 \, A - 8 \, B\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac{420 \,{\left (21 \, A - 8 \, 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 (9 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 7 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \, B \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 )}^{2} 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} + 189 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 147 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1365 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 805 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 11655 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 5145 \, 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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