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 = 0.69, antiderivative size = 232, normalized size of antiderivative = 1.00, 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 8
Rule 2748
Rule 2978
Rule 3767
Rule 3768
Rule 3770
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*}
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Mathematica [B] time = 6.51, 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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fricas [A] time = 0.59, size = 360, normalized size = 1.55 \[ \frac {105 \, {\left ({\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{6} + 4 \, {\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{5} + 6 \, {\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{3} + {\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left ({\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{6} + 4 \, {\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{5} + 6 \, {\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{3} + {\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (16 \, {\left (216 \, A - 83 \, B\right )} \cos \left (d x + c\right )^{5} + {\left (11619 \, A - 4472 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (3411 \, A - 1318 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (1509 \, A - 592 \, B\right )} \cos \left (d x + c\right )^{2} + 210 \, {\left (2 \, A - B\right )} \cos \left (d x + c\right ) - 105 \, A\right )} \sin \left (d x + c\right )}{420 \, {\left (a^{4} d \cos \left (d x + c\right )^{6} + 4 \, a^{4} d \cos \left (d x + c\right )^{5} + 6 \, a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + a^{4} d \cos \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.45, size = 267, normalized size = 1.15 \[ \frac {\frac {420 \, {\left (21 \, A - 8 \, B\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {420 \, {\left (21 \, A - 8 \, B\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} + \frac {840 \, {\left (9 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{4}} - \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 189 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 147 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1365 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 805 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 11655 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5145 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 374, normalized size = 1.61 \[ -\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{56 d \,a^{4}}+\frac {B \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 d \,a^{4}}-\frac {9 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d \,a^{4}}+\frac {7 B \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d \,a^{4}}-\frac {13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{8 d \,a^{4}}+\frac {23 B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d \,a^{4}}-\frac {111 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}+\frac {49 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}-\frac {21 A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d \,a^{4}}+\frac {4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) B}{d \,a^{4}}+\frac {9 A}{2 d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {B}{d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {A}{2 d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {9 A}{2 d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {B}{d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {21 A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d \,a^{4}}-\frac {4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) B}{d \,a^{4}}-\frac {A}{2 d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 419, normalized size = 1.81 \[ -\frac {3 \, A {\left (\frac {280 \, {\left (\frac {7 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {9 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{4} - \frac {2 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {3885 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {455 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {2940 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {2940 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} - B {\left (\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} - \frac {a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 273, normalized size = 1.18 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (9\,A-2\,B\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (7\,A-2\,B\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^4\right )}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,A}{2\,a^4}+\frac {5\,\left (A-B\right )}{4\,a^4}+\frac {3\,\left (6\,A-4\,B\right )}{4\,a^4}+\frac {3\,\left (15\,A-5\,B\right )}{8\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A-B}{4\,a^4}+\frac {6\,A-4\,B}{8\,a^4}+\frac {15\,A-5\,B}{24\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {3\,\left (A-B\right )}{40\,a^4}+\frac {6\,A-4\,B}{40\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A-B\right )}{56\,a^4\,d}+\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (21\,A-8\,B\right )}{a^4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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