Optimal. Leaf size=218 \[ \frac {(13 A-23 B) x}{2 a^3}-\frac {4 (19 A-34 B) \sin (c+d x)}{5 a^3 d}+\frac {(13 A-23 B) \cos (c+d x) \sin (c+d x)}{2 a^3 d}+\frac {(A-B) \cos ^5(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {(8 A-13 B) \cos ^4(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac {(13 A-23 B) \cos ^3(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac {4 (19 A-34 B) \sin ^3(c+d x)}{15 a^3 d} \]
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Rubi [A]
time = 0.34, antiderivative size = 218, normalized size of antiderivative = 1.00, 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 = {3056, 2827,
2715, 8, 2713} \begin {gather*} \frac {4 (19 A-34 B) \sin ^3(c+d x)}{15 a^3 d}-\frac {4 (19 A-34 B) \sin (c+d x)}{5 a^3 d}+\frac {(13 A-23 B) \sin (c+d x) \cos ^3(c+d x)}{3 d \left (a^3 \cos (c+d x)+a^3\right )}+\frac {(13 A-23 B) \sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac {x (13 A-23 B)}{2 a^3}+\frac {(A-B) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac {(8 A-13 B) \sin (c+d x) \cos ^4(c+d x)}{15 a d (a \cos (c+d x)+a)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2713
Rule 2715
Rule 2827
Rule 3056
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx &=\frac {(A-B) \cos ^5(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {\int \frac {\cos ^4(c+d x) (5 a (A-B)-a (3 A-8 B) \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx}{5 a^2}\\ &=\frac {(A-B) \cos ^5(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {(8 A-13 B) \cos ^4(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac {\int \frac {\cos ^3(c+d x) \left (4 a^2 (8 A-13 B)-3 a^2 (11 A-21 B) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{15 a^4}\\ &=\frac {(A-B) \cos ^5(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {(8 A-13 B) \cos ^4(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac {(13 A-23 B) \cos ^3(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac {\int \cos ^2(c+d x) \left (15 a^3 (13 A-23 B)-12 a^3 (19 A-34 B) \cos (c+d x)\right ) \, dx}{15 a^6}\\ &=\frac {(A-B) \cos ^5(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {(8 A-13 B) \cos ^4(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac {(13 A-23 B) \cos ^3(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac {(4 (19 A-34 B)) \int \cos ^3(c+d x) \, dx}{5 a^3}+\frac {(13 A-23 B) \int \cos ^2(c+d x) \, dx}{a^3}\\ &=\frac {(13 A-23 B) \cos (c+d x) \sin (c+d x)}{2 a^3 d}+\frac {(A-B) \cos ^5(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {(8 A-13 B) \cos ^4(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac {(13 A-23 B) \cos ^3(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac {(13 A-23 B) \int 1 \, dx}{2 a^3}+\frac {(4 (19 A-34 B)) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 a^3 d}\\ &=\frac {(13 A-23 B) x}{2 a^3}-\frac {4 (19 A-34 B) \sin (c+d x)}{5 a^3 d}+\frac {(13 A-23 B) \cos (c+d x) \sin (c+d x)}{2 a^3 d}+\frac {(A-B) \cos ^5(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {(8 A-13 B) \cos ^4(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac {(13 A-23 B) \cos ^3(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac {4 (19 A-34 B) \sin ^3(c+d x)}{15 a^3 d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(491\) vs. \(2(218)=436\).
time = 0.96, size = 491, normalized size = 2.25 \begin {gather*} \frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (600 (13 A-23 B) d x \cos \left (\frac {d x}{2}\right )+600 (13 A-23 B) d x \cos \left (c+\frac {d x}{2}\right )+3900 A d x \cos \left (c+\frac {3 d x}{2}\right )-6900 B d x \cos \left (c+\frac {3 d x}{2}\right )+3900 A d x \cos \left (2 c+\frac {3 d x}{2}\right )-6900 B d x \cos \left (2 c+\frac {3 d x}{2}\right )+780 A d x \cos \left (2 c+\frac {5 d x}{2}\right )-1380 B d x \cos \left (2 c+\frac {5 d x}{2}\right )+780 A d x \cos \left (3 c+\frac {5 d x}{2}\right )-1380 B d x \cos \left (3 c+\frac {5 d x}{2}\right )-12760 A \sin \left (\frac {d x}{2}\right )+20410 B \sin \left (\frac {d x}{2}\right )+7560 A \sin \left (c+\frac {d x}{2}\right )-11110 B \sin \left (c+\frac {d x}{2}\right )-9230 A \sin \left (c+\frac {3 d x}{2}\right )+15380 B \sin \left (c+\frac {3 d x}{2}\right )+930 A \sin \left (2 c+\frac {3 d x}{2}\right )-380 B \sin \left (2 c+\frac {3 d x}{2}\right )-2782 A \sin \left (2 c+\frac {5 d x}{2}\right )+4777 B \sin \left (2 c+\frac {5 d x}{2}\right )-750 A \sin \left (3 c+\frac {5 d x}{2}\right )+1625 B \sin \left (3 c+\frac {5 d x}{2}\right )-105 A \sin \left (3 c+\frac {7 d x}{2}\right )+230 B \sin \left (3 c+\frac {7 d x}{2}\right )-105 A \sin \left (4 c+\frac {7 d x}{2}\right )+230 B \sin \left (4 c+\frac {7 d x}{2}\right )+15 A \sin \left (4 c+\frac {9 d x}{2}\right )-20 B \sin \left (4 c+\frac {9 d x}{2}\right )+15 A \sin \left (5 c+\frac {9 d x}{2}\right )-20 B \sin \left (5 c+\frac {9 d x}{2}\right )+5 B \sin \left (5 c+\frac {11 d x}{2}\right )+5 B \sin \left (6 c+\frac {11 d x}{2}\right )\right )}{480 a^3 d (1+\cos (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 182, normalized size = 0.83
method | result | size |
derivativedivides | \(\frac {-\frac {A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {B \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{3}-\frac {10 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{3}-31 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+49 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {16 \left (-\frac {7 A}{4}+\frac {17 B}{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+16 \left (-3 A +\frac {19 B}{3}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+16 \left (-\frac {5 A}{4}+\frac {11 B}{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+4 \left (13 A -23 B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(182\) |
default | \(\frac {-\frac {A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {B \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{3}-\frac {10 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{3}-31 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+49 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {16 \left (-\frac {7 A}{4}+\frac {17 B}{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+16 \left (-3 A +\frac {19 B}{3}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+16 \left (-\frac {5 A}{4}+\frac {11 B}{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+4 \left (13 A -23 B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(182\) |
risch | \(\frac {13 x A}{2 a^{3}}-\frac {23 B x}{2 a^{3}}-\frac {i B \,{\mathrm e}^{3 i \left (d x +c \right )}}{24 a^{3} d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} A}{8 a^{3} d}+\frac {3 i {\mathrm e}^{2 i \left (d x +c \right )} B}{8 a^{3} d}+\frac {3 i {\mathrm e}^{i \left (d x +c \right )} A}{2 a^{3} d}-\frac {27 i {\mathrm e}^{i \left (d x +c \right )} B}{8 a^{3} d}-\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} A}{2 a^{3} d}+\frac {27 i {\mathrm e}^{-i \left (d x +c \right )} B}{8 a^{3} d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} A}{8 a^{3} d}-\frac {3 i {\mathrm e}^{-2 i \left (d x +c \right )} B}{8 a^{3} d}+\frac {i B \,{\mathrm e}^{-3 i \left (d x +c \right )}}{24 a^{3} d}-\frac {2 i \left (150 A \,{\mathrm e}^{4 i \left (d x +c \right )}-225 B \,{\mathrm e}^{4 i \left (d x +c \right )}+525 A \,{\mathrm e}^{3 i \left (d x +c \right )}-810 B \,{\mathrm e}^{3 i \left (d x +c \right )}+745 A \,{\mathrm e}^{2 i \left (d x +c \right )}-1160 B \,{\mathrm e}^{2 i \left (d x +c \right )}+485 A \,{\mathrm e}^{i \left (d x +c \right )}-760 B \,{\mathrm e}^{i \left (d x +c \right )}+127 A -197 B \right )}{15 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5}}\) | \(331\) |
norman | \(\frac {\frac {\left (13 A -23 B \right ) x}{2 a}-\frac {\left (A -B \right ) \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d a}-\frac {\left (9 A -16 B \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}+\frac {\left (11 A -16 B \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{30 d a}+\frac {3 \left (13 A -23 B \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {15 \left (13 A -23 B \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {10 \left (13 A -23 B \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {15 \left (13 A -23 B \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {3 \left (13 A -23 B \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {\left (13 A -23 B \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {3 \left (17 A -31 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d a}-\frac {\left (89 A -158 B \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}-\frac {5 \left (164 A -293 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}-\frac {\left (437 A -786 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}-\frac {3 \left (691 A -1236 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10 d a}-\frac {\left (1703 A -3048 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6} a^{2}}\) | \(408\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 412 vs.
\(2 (204) = 408\).
time = 0.50, size = 412, normalized size = 1.89 \begin {gather*} \frac {B {\left (\frac {20 \, {\left (\frac {33 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {76 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {51 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{3} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\frac {735 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {50 \, \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}}}{a^{3}} - \frac {1380 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} - A {\left (\frac {60 \, {\left (\frac {5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {7 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{3} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {465 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {40 \, \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}}}{a^{3}} - \frac {780 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{60 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 205, normalized size = 0.94 \begin {gather*} \frac {15 \, {\left (13 \, A - 23 \, B\right )} d x \cos \left (d x + c\right )^{3} + 45 \, {\left (13 \, A - 23 \, B\right )} d x \cos \left (d x + c\right )^{2} + 45 \, {\left (13 \, A - 23 \, B\right )} d x \cos \left (d x + c\right ) + 15 \, {\left (13 \, A - 23 \, B\right )} d x + {\left (10 \, B \cos \left (d x + c\right )^{5} + 15 \, {\left (A - B\right )} \cos \left (d x + c\right )^{4} - 5 \, {\left (9 \, A - 19 \, B\right )} \cos \left (d x + c\right )^{3} - {\left (479 \, A - 869 \, B\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (239 \, A - 429 \, B\right )} \cos \left (d x + c\right ) - 304 \, A + 544 \, B\right )} \sin \left (d x + c\right )}{30 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1584 vs.
\(2 (206) = 412\).
time = 6.80, size = 1584, normalized size = 7.27 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 228, normalized size = 1.05 \begin {gather*} \frac {\frac {30 \, {\left (d x + c\right )} {\left (13 \, A - 23 \, B\right )}}{a^{3}} - \frac {20 \, {\left (21 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 51 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 76 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 33 \, 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 )}^{3} a^{3}} - \frac {3 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 50 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 465 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 735 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{60 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.33, size = 238, normalized size = 1.09 \begin {gather*} \frac {x\,\left (13\,A-23\,B\right )}{2\,a^3}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,\left (A-B\right )}{2\,a^3}+\frac {4\,A-6\,B}{a^3}+\frac {5\,A-15\,B}{4\,a^3}\right )}{d}-\frac {\left (7\,A-17\,B\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (12\,A-\frac {76\,B}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (5\,A-11\,B\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A-B}{3\,a^3}+\frac {4\,A-6\,B}{12\,a^3}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (A-B\right )}{20\,a^3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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