Optimal. Leaf size=237 \[ -\frac {4 (9 A-19 B+34 C) \sin ^3(c+d x)}{15 a^3 d}+\frac {4 (9 A-19 B+34 C) \sin (c+d x)}{5 a^3 d}-\frac {(6 A-13 B+23 C) \sin (c+d x) \cos ^3(c+d x)}{3 d \left (a^3 \cos (c+d x)+a^3\right )}-\frac {(6 A-13 B+23 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac {x (6 A-13 B+23 C)}{2 a^3}-\frac {(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac {(3 A-8 B+13 C) \sin (c+d x) \cos ^4(c+d x)}{15 a d (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.56, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {3041, 2977, 2748, 2635, 8, 2633} \[ -\frac {4 (9 A-19 B+34 C) \sin ^3(c+d x)}{15 a^3 d}+\frac {4 (9 A-19 B+34 C) \sin (c+d x)}{5 a^3 d}-\frac {(6 A-13 B+23 C) \sin (c+d x) \cos ^3(c+d x)}{3 d \left (a^3 \cos (c+d x)+a^3\right )}-\frac {(6 A-13 B+23 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac {x (6 A-13 B+23 C)}{2 a^3}-\frac {(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac {(3 A-8 B+13 C) \sin (c+d x) \cos ^4(c+d x)}{15 a d (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2748
Rule 2977
Rule 3041
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx &=-\frac {(A-B+C) \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 (B-C)+a (3 A-3 B+8 C) \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac {(A-B+C) \cos ^5(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(3 A-8 B+13 C) \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 (3 A-8 B+13 C)+3 a^2 (6 A-11 B+21 C) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{15 a^4}\\ &=-\frac {(A-B+C) \cos ^5(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(3 A-8 B+13 C) \cos ^4(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {(6 A-13 B+23 C) \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 (6 A-13 B+23 C)+12 a^3 (9 A-19 B+34 C) \cos (c+d x)\right ) \, dx}{15 a^6}\\ &=-\frac {(A-B+C) \cos ^5(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(3 A-8 B+13 C) \cos ^4(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {(6 A-13 B+23 C) \cos ^3(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac {(6 A-13 B+23 C) \int \cos ^2(c+d x) \, dx}{a^3}+\frac {(4 (9 A-19 B+34 C)) \int \cos ^3(c+d x) \, dx}{5 a^3}\\ &=-\frac {(6 A-13 B+23 C) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {(A-B+C) \cos ^5(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(3 A-8 B+13 C) \cos ^4(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {(6 A-13 B+23 C) \cos ^3(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac {(6 A-13 B+23 C) \int 1 \, dx}{2 a^3}-\frac {(4 (9 A-19 B+34 C)) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 a^3 d}\\ &=-\frac {(6 A-13 B+23 C) x}{2 a^3}+\frac {4 (9 A-19 B+34 C) \sin (c+d x)}{5 a^3 d}-\frac {(6 A-13 B+23 C) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {(A-B+C) \cos ^5(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(3 A-8 B+13 C) \cos ^4(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {(6 A-13 B+23 C) \cos ^3(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac {4 (9 A-19 B+34 C) \sin ^3(c+d x)}{15 a^3 d}\\ \end {align*}
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Mathematica [B] time = 1.99, size = 663, normalized size = 2.80 \[ \frac {\sec \left (\frac {c}{2}\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \left (-600 d x (6 A-13 B+23 C) \cos \left (c+\frac {d x}{2}\right )-600 d x (6 A-13 B+23 C) \cos \left (\frac {d x}{2}\right )-4500 A \sin \left (c+\frac {d x}{2}\right )+4860 A \sin \left (c+\frac {3 d x}{2}\right )-900 A \sin \left (2 c+\frac {3 d x}{2}\right )+1452 A \sin \left (2 c+\frac {5 d x}{2}\right )+300 A \sin \left (3 c+\frac {5 d x}{2}\right )+60 A \sin \left (3 c+\frac {7 d x}{2}\right )+60 A \sin \left (4 c+\frac {7 d x}{2}\right )-1800 A d x \cos \left (c+\frac {3 d x}{2}\right )-1800 A d x \cos \left (2 c+\frac {3 d x}{2}\right )-360 A d x \cos \left (2 c+\frac {5 d x}{2}\right )-360 A d x \cos \left (3 c+\frac {5 d x}{2}\right )+7020 A \sin \left (\frac {d x}{2}\right )+7560 B \sin \left (c+\frac {d x}{2}\right )-9230 B \sin \left (c+\frac {3 d x}{2}\right )+930 B \sin \left (2 c+\frac {3 d x}{2}\right )-2782 B \sin \left (2 c+\frac {5 d x}{2}\right )-750 B \sin \left (3 c+\frac {5 d x}{2}\right )-105 B \sin \left (3 c+\frac {7 d x}{2}\right )-105 B \sin \left (4 c+\frac {7 d x}{2}\right )+15 B \sin \left (4 c+\frac {9 d x}{2}\right )+15 B \sin \left (5 c+\frac {9 d x}{2}\right )+3900 B d x \cos \left (c+\frac {3 d x}{2}\right )+3900 B d x \cos \left (2 c+\frac {3 d x}{2}\right )+780 B d x \cos \left (2 c+\frac {5 d x}{2}\right )+780 B d x \cos \left (3 c+\frac {5 d x}{2}\right )-12760 B \sin \left (\frac {d x}{2}\right )-11110 C \sin \left (c+\frac {d x}{2}\right )+15380 C \sin \left (c+\frac {3 d x}{2}\right )-380 C \sin \left (2 c+\frac {3 d x}{2}\right )+4777 C \sin \left (2 c+\frac {5 d x}{2}\right )+1625 C \sin \left (3 c+\frac {5 d x}{2}\right )+230 C \sin \left (3 c+\frac {7 d x}{2}\right )+230 C \sin \left (4 c+\frac {7 d x}{2}\right )-20 C \sin \left (4 c+\frac {9 d x}{2}\right )-20 C \sin \left (5 c+\frac {9 d x}{2}\right )+5 C \sin \left (5 c+\frac {11 d x}{2}\right )+5 C \sin \left (6 c+\frac {11 d x}{2}\right )-6900 C d x \cos \left (c+\frac {3 d x}{2}\right )-6900 C d x \cos \left (2 c+\frac {3 d x}{2}\right )-1380 C d x \cos \left (2 c+\frac {5 d x}{2}\right )-1380 C d x \cos \left (3 c+\frac {5 d x}{2}\right )+20410 C \sin \left (\frac {d x}{2}\right )\right )}{480 a^3 d (\cos (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 229, normalized size = 0.97 \[ -\frac {15 \, {\left (6 \, A - 13 \, B + 23 \, C\right )} d x \cos \left (d x + c\right )^{3} + 45 \, {\left (6 \, A - 13 \, B + 23 \, C\right )} d x \cos \left (d x + c\right )^{2} + 45 \, {\left (6 \, A - 13 \, B + 23 \, C\right )} d x \cos \left (d x + c\right ) + 15 \, {\left (6 \, A - 13 \, B + 23 \, C\right )} d x - {\left (10 \, C \cos \left (d x + c\right )^{5} + 15 \, {\left (B - C\right )} \cos \left (d x + c\right )^{4} + 5 \, {\left (6 \, A - 9 \, B + 19 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (234 \, A - 479 \, B + 869 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (114 \, A - 239 \, B + 429 \, C\right )} \cos \left (d x + c\right ) + 144 \, A - 304 \, B + 544 \, C\right )} \sin \left (d x + c\right )}{30 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 320, normalized size = 1.35 \[ -\frac {\frac {30 \, {\left (d x + c\right )} {\left (6 \, A - 13 \, B + 23 \, C\right )}}{a^{3}} - \frac {20 \, {\left (6 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 21 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 51 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 76 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 33 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{3}} - \frac {3 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 30 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 50 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 255 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 465 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 735 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 542, normalized size = 2.29 \[ \frac {A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d \,a^{3}}-\frac {B \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d \,a^{3}}+\frac {C \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d \,a^{3}}-\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{2 d \,a^{3}}+\frac {2 B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \,a^{3}}-\frac {5 C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d \,a^{3}}+\frac {17 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \,a^{3}}-\frac {31 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \,a^{3}}+\frac {49 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \,a^{3}}+\frac {2 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {7 B \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {17 C \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {12 B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {76 C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {2 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {5 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {11 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {6 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{d \,a^{3}}+\frac {13 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{d \,a^{3}}-\frac {23 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 547, normalized size = 2.31 \[ \frac {C {\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 )} - B {\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 )} + 3 \, A {\left (\frac {40 \, \sin \left (d x + c\right )}{{\left (a^{3} + \frac {a^{3} \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 {85 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.28, size = 259, normalized size = 1.09 \[ \frac {\left (2\,A-7\,B+17\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (4\,A-12\,B+\frac {76\,C}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A-5\,B+11\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {2\,A-4\,B+6\,C}{a^3}-\frac {A+5\,B-15\,C}{4\,a^3}+\frac {5\,\left (A-B+C\right )}{2\,a^3}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {2\,A-4\,B+6\,C}{12\,a^3}+\frac {A-B+C}{3\,a^3}\right )}{d}-\frac {x\,\left (6\,A-13\,B+23\,C\right )}{2\,a^3}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (A-B+C\right )}{20\,a^3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 35.74, size = 2373, normalized size = 10.01 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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