Optimal. Leaf size=237 \[ -\frac {(23 A-13 B+6 C) x}{2 a^3}+\frac {4 (34 A-19 B+9 C) \sin (c+d x)}{5 a^3 d}-\frac {(23 A-13 B+6 C) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(13 A-8 B+3 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(23 A-13 B+6 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {4 (34 A-19 B+9 C) \sin ^3(c+d x)}{15 a^3 d} \]
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Rubi [A]
time = 0.38, 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 = {4169, 4105,
3872, 2713, 2715, 8} \begin {gather*} -\frac {4 (34 A-19 B+9 C) \sin ^3(c+d x)}{15 a^3 d}+\frac {4 (34 A-19 B+9 C) \sin (c+d x)}{5 a^3 d}-\frac {(23 A-13 B+6 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac {(23 A-13 B+6 C) \sin (c+d x) \cos ^2(c+d x)}{3 d \left (a^3 \sec (c+d x)+a^3\right )}-\frac {x (23 A-13 B+6 C)}{2 a^3}-\frac {(13 A-8 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2713
Rule 2715
Rule 3872
Rule 4105
Rule 4169
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx &=-\frac {(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {\int \frac {\cos ^3(c+d x) (a (8 A-3 B+3 C)-5 a (A-B) \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac {(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(13 A-8 B+3 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac {\int \frac {\cos ^3(c+d x) \left (3 a^2 (21 A-11 B+6 C)-4 a^2 (13 A-8 B+3 C) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{15 a^4}\\ &=-\frac {(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(13 A-8 B+3 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(23 A-13 B+6 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac {\int \cos ^3(c+d x) \left (12 a^3 (34 A-19 B+9 C)-15 a^3 (23 A-13 B+6 C) \sec (c+d x)\right ) \, dx}{15 a^6}\\ &=-\frac {(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(13 A-8 B+3 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(23 A-13 B+6 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {(23 A-13 B+6 C) \int \cos ^2(c+d x) \, dx}{a^3}+\frac {(4 (34 A-19 B+9 C)) \int \cos ^3(c+d x) \, dx}{5 a^3}\\ &=-\frac {(23 A-13 B+6 C) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(13 A-8 B+3 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(23 A-13 B+6 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {(23 A-13 B+6 C) \int 1 \, dx}{2 a^3}-\frac {(4 (34 A-19 B+9 C)) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 a^3 d}\\ &=-\frac {(23 A-13 B+6 C) x}{2 a^3}+\frac {4 (34 A-19 B+9 C) \sin (c+d x)}{5 a^3 d}-\frac {(23 A-13 B+6 C) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(13 A-8 B+3 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(23 A-13 B+6 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {4 (34 A-19 B+9 C) \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. \(655\) vs. \(2(237)=474\).
time = 2.77, size = 655, normalized size = 2.76 \begin {gather*} \frac {\sec \left (\frac {c}{2}\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right ) \left (-600 (23 A-13 B+6 C) d x \cos \left (\frac {d x}{2}\right )-600 (23 A-13 B+6 C) d x \cos \left (c+\frac {d x}{2}\right )-6900 A d x \cos \left (c+\frac {3 d x}{2}\right )+3900 B d x \cos \left (c+\frac {3 d x}{2}\right )-1800 C d x \cos \left (c+\frac {3 d x}{2}\right )-6900 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+3900 B d x \cos \left (2 c+\frac {3 d x}{2}\right )-1800 C d x \cos \left (2 c+\frac {3 d x}{2}\right )-1380 A d x \cos \left (2 c+\frac {5 d x}{2}\right )+780 B d x \cos \left (2 c+\frac {5 d x}{2}\right )-360 C d x \cos \left (2 c+\frac {5 d x}{2}\right )-1380 A d x \cos \left (3 c+\frac {5 d x}{2}\right )+780 B d x \cos \left (3 c+\frac {5 d x}{2}\right )-360 C d x \cos \left (3 c+\frac {5 d x}{2}\right )+20410 A \sin \left (\frac {d x}{2}\right )-12760 B \sin \left (\frac {d x}{2}\right )+7020 C \sin \left (\frac {d x}{2}\right )-11110 A \sin \left (c+\frac {d x}{2}\right )+7560 B \sin \left (c+\frac {d x}{2}\right )-4500 C \sin \left (c+\frac {d x}{2}\right )+15380 A \sin \left (c+\frac {3 d x}{2}\right )-9230 B \sin \left (c+\frac {3 d x}{2}\right )+4860 C \sin \left (c+\frac {3 d x}{2}\right )-380 A \sin \left (2 c+\frac {3 d x}{2}\right )+930 B \sin \left (2 c+\frac {3 d x}{2}\right )-900 C \sin \left (2 c+\frac {3 d x}{2}\right )+4777 A \sin \left (2 c+\frac {5 d x}{2}\right )-2782 B \sin \left (2 c+\frac {5 d x}{2}\right )+1452 C \sin \left (2 c+\frac {5 d x}{2}\right )+1625 A \sin \left (3 c+\frac {5 d x}{2}\right )-750 B \sin \left (3 c+\frac {5 d x}{2}\right )+300 C \sin \left (3 c+\frac {5 d x}{2}\right )+230 A \sin \left (3 c+\frac {7 d x}{2}\right )-105 B \sin \left (3 c+\frac {7 d x}{2}\right )+60 C \sin \left (3 c+\frac {7 d x}{2}\right )+230 A \sin \left (4 c+\frac {7 d x}{2}\right )-105 B \sin \left (4 c+\frac {7 d x}{2}\right )+60 C \sin \left (4 c+\frac {7 d x}{2}\right )-20 A \sin \left (4 c+\frac {9 d x}{2}\right )+15 B \sin \left (4 c+\frac {9 d x}{2}\right )-20 A \sin \left (5 c+\frac {9 d x}{2}\right )+15 B \sin \left (5 c+\frac {9 d x}{2}\right )+5 A \sin \left (5 c+\frac {11 d x}{2}\right )+5 A \sin \left (6 c+\frac {11 d x}{2}\right )\right )}{3840 a^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.95, size = 234, normalized size = 0.99 Too large to display
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. 547 vs.
\(2 (223) = 446\).
time = 0.52, size = 547, normalized size = 2.31 \begin {gather*} \frac {A {\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 \, C {\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.72, size = 229, normalized size = 0.97 \begin {gather*} -\frac {15 \, {\left (23 \, A - 13 \, B + 6 \, C\right )} d x \cos \left (d x + c\right )^{3} + 45 \, {\left (23 \, A - 13 \, B + 6 \, C\right )} d x \cos \left (d x + c\right )^{2} + 45 \, {\left (23 \, A - 13 \, B + 6 \, C\right )} d x \cos \left (d x + c\right ) + 15 \, {\left (23 \, A - 13 \, B + 6 \, C\right )} d x - {\left (10 \, A \cos \left (d x + c\right )^{5} - 15 \, {\left (A - B\right )} \cos \left (d x + c\right )^{4} + 5 \, {\left (19 \, A - 9 \, B + 6 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (869 \, A - 479 \, B + 234 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (429 \, A - 239 \, B + 114 \, C\right )} \cos \left (d x + c\right ) + 544 \, A - 304 \, B + 144 \, 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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {A \cos ^{3}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.50, size = 320, normalized size = 1.35 \begin {gather*} -\frac {\frac {30 \, {\left (d x + c\right )} {\left (23 \, A - 13 \, B + 6 \, C\right )}}{a^{3}} - \frac {20 \, {\left (51 \, 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} + 6 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 76 \, 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} + 12 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 33 \, 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 ) + 6 \, 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} - 50 \, 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} - 30 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 735 \, 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 ) + 255 \, C 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 = 3.39, size = 259, normalized size = 1.09 \begin {gather*} \frac {\left (17\,A-7\,B+2\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {76\,A}{3}-12\,B+4\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (11\,A-5\,B+2\,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 {6\,A-4\,B+2\,C}{a^3}-\frac {5\,B-15\,A+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 {6\,A-4\,B+2\,C}{12\,a^3}+\frac {A-B+C}{3\,a^3}\right )}{d}-\frac {x\,\left (23\,A-13\,B+6\,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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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