Optimal. Leaf size=207 \[ -\frac {2 (11 A-36 B+76 C) \sin (c+d x)}{15 a^3 d}-\frac {(11 A-36 B+76 C) \sin (c+d x) \cos ^2(c+d x)}{15 d \left (a^3 \cos (c+d x)+a^3\right )}+\frac {(2 A-6 B+13 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac {x (2 A-6 B+13 C)}{2 a^3}-\frac {(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac {(A-6 B+11 C) \sin (c+d x) \cos ^3(c+d x)}{15 a d (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.50, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {3041, 2977, 2734} \[ -\frac {2 (11 A-36 B+76 C) \sin (c+d x)}{15 a^3 d}-\frac {(11 A-36 B+76 C) \sin (c+d x) \cos ^2(c+d x)}{15 d \left (a^3 \cos (c+d x)+a^3\right )}+\frac {(2 A-6 B+13 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac {x (2 A-6 B+13 C)}{2 a^3}-\frac {(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac {(A-6 B+11 C) \sin (c+d x) \cos ^3(c+d x)}{15 a d (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 2977
Rule 3041
Rubi steps
\begin {align*} \int \frac {\cos ^3(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 ^4(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {\int \frac {\cos ^3(c+d x) (a (A+4 B-4 C)+a (2 A-2 B+7 C) \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(A-6 B+11 C) \cos ^3(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac {\int \frac {\cos ^2(c+d x) \left (-3 a^2 (A-6 B+11 C)+a^2 (8 A-18 B+43 C) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{15 a^4}\\ &=-\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(A-6 B+11 C) \cos ^3(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {(11 A-36 B+76 C) \cos ^2(c+d x) \sin (c+d x)}{15 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac {\int \cos (c+d x) \left (-2 a^3 (11 A-36 B+76 C)+15 a^3 (2 A-6 B+13 C) \cos (c+d x)\right ) \, dx}{15 a^6}\\ &=\frac {(2 A-6 B+13 C) x}{2 a^3}-\frac {2 (11 A-36 B+76 C) \sin (c+d x)}{15 a^3 d}+\frac {(2 A-6 B+13 C) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(A-6 B+11 C) \cos ^3(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {(11 A-36 B+76 C) \cos ^2(c+d x) \sin (c+d x)}{15 d \left (a^3+a^3 \cos (c+d x)\right )}\\ \end {align*}
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Mathematica [B] time = 0.98, size = 565, normalized size = 2.73 \[ \frac {\sec \left (\frac {c}{2}\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \left (600 d x (2 A-6 B+13 C) \cos \left (c+\frac {d x}{2}\right )+600 d x (2 A-6 B+13 C) \cos \left (\frac {d x}{2}\right )+2160 A \sin \left (c+\frac {d x}{2}\right )-1840 A \sin \left (c+\frac {3 d x}{2}\right )+720 A \sin \left (2 c+\frac {3 d x}{2}\right )-512 A \sin \left (2 c+\frac {5 d x}{2}\right )+600 A d x \cos \left (c+\frac {3 d x}{2}\right )+600 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+120 A d x \cos \left (2 c+\frac {5 d x}{2}\right )+120 A d x \cos \left (3 c+\frac {5 d x}{2}\right )-2960 A \sin \left (\frac {d x}{2}\right )-4500 B \sin \left (c+\frac {d x}{2}\right )+4860 B \sin \left (c+\frac {3 d x}{2}\right )-900 B \sin \left (2 c+\frac {3 d x}{2}\right )+1452 B \sin \left (2 c+\frac {5 d x}{2}\right )+300 B \sin \left (3 c+\frac {5 d x}{2}\right )+60 B \sin \left (3 c+\frac {7 d x}{2}\right )+60 B \sin \left (4 c+\frac {7 d x}{2}\right )-1800 B d x \cos \left (c+\frac {3 d x}{2}\right )-1800 B d x \cos \left (2 c+\frac {3 d x}{2}\right )-360 B d x \cos \left (2 c+\frac {5 d x}{2}\right )-360 B d x \cos \left (3 c+\frac {5 d x}{2}\right )+7020 B \sin \left (\frac {d x}{2}\right )+7560 C \sin \left (c+\frac {d x}{2}\right )-9230 C \sin \left (c+\frac {3 d x}{2}\right )+930 C \sin \left (2 c+\frac {3 d x}{2}\right )-2782 C \sin \left (2 c+\frac {5 d x}{2}\right )-750 C \sin \left (3 c+\frac {5 d x}{2}\right )-105 C \sin \left (3 c+\frac {7 d x}{2}\right )-105 C \sin \left (4 c+\frac {7 d x}{2}\right )+15 C \sin \left (4 c+\frac {9 d x}{2}\right )+15 C \sin \left (5 c+\frac {9 d x}{2}\right )+3900 C d x \cos \left (c+\frac {3 d x}{2}\right )+3900 C d x \cos \left (2 c+\frac {3 d x}{2}\right )+780 C d x \cos \left (2 c+\frac {5 d x}{2}\right )+780 C d x \cos \left (3 c+\frac {5 d x}{2}\right )-12760 C \sin \left (\frac {d x}{2}\right )\right )}{480 a^3 d (\cos (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 211, normalized size = 1.02 \[ \frac {15 \, {\left (2 \, A - 6 \, B + 13 \, C\right )} d x \cos \left (d x + c\right )^{3} + 45 \, {\left (2 \, A - 6 \, B + 13 \, C\right )} d x \cos \left (d x + c\right )^{2} + 45 \, {\left (2 \, A - 6 \, B + 13 \, C\right )} d x \cos \left (d x + c\right ) + 15 \, {\left (2 \, A - 6 \, B + 13 \, C\right )} d x + {\left (15 \, C \cos \left (d x + c\right )^{4} + 15 \, {\left (2 \, B - 3 \, C\right )} \cos \left (d x + c\right )^{3} - {\left (64 \, A - 234 \, B + 479 \, C\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (34 \, A - 114 \, B + 239 \, C\right )} \cos \left (d x + c\right ) - 44 \, A + 144 \, B - 304 \, C\right )} \sin \left (d x + c\right )}{30 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.59, size = 252, normalized size = 1.22 \[ \frac {\frac {30 \, {\left (d x + c\right )} {\left (2 \, A - 6 \, B + 13 \, C\right )}}{a^{3}} + \frac {60 \, {\left (2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{3}} - \frac {3 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 20 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 30 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 255 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 465 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 369, normalized size = 1.78 \[ -\frac {A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d \,a^{3}}+\frac {B \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d \,a^{3}}-\frac {C \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d \,a^{3}}+\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{3 d \,a^{3}}-\frac {B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{3}}+\frac {2 C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \,a^{3}}-\frac {7 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \,a^{3}}+\frac {17 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \,a^{3}}-\frac {31 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \,a^{3}}+\frac {2 B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {7 C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {5 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {2 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{d \,a^{3}}-\frac {6 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{d \,a^{3}}+\frac {13 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 411, normalized size = 1.99 \[ -\frac {C {\left (\frac {60 \, {\left (\frac {5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {7 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{3} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {465 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {40 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {780 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} - 3 \, B {\left (\frac {40 \, \sin \left (d x + c\right )}{{\left (a^{3} + \frac {a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {85 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} + A {\left (\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {20 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.23, size = 214, normalized size = 1.03 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A-3\,B+5\,C}{12\,a^3}+\frac {A-B+C}{4\,a^3}\right )}{d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,\left (A-3\,B+5\,C\right )}{4\,a^3}-\frac {2\,A+2\,B-10\,C}{4\,a^3}+\frac {3\,\left (A-B+C\right )}{2\,a^3}\right )}{d}+\frac {\left (2\,B-7\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,B-5\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )}+\frac {x\,\left (2\,A-6\,B+13\,C\right )}{2\,a^3}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (A-B+C\right )}{20\,a^3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 23.82, size = 1445, normalized size = 6.98 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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