Optimal. Leaf size=248 \[ -\frac {4 (454 A+83 C) \sin ^3(c+d x)}{105 a^4 d}+\frac {4 (454 A+83 C) \sin (c+d x)}{35 a^4 d}-\frac {2 (11 A+2 C) \sin (c+d x) \cos (c+d x)}{a^4 d}-\frac {4 (11 A+2 C) \sin (c+d x) \cos ^2(c+d x)}{3 a^4 d (\sec (c+d x)+1)}-\frac {(178 A+31 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac {2 x (11 A+2 C)}{a^4}-\frac {2 (8 A+C) \sin (c+d x) \cos ^2(c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4} \]
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Rubi [A] time = 0.69, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4085, 4020, 3787, 2633, 2635, 8} \[ -\frac {4 (454 A+83 C) \sin ^3(c+d x)}{105 a^4 d}+\frac {4 (454 A+83 C) \sin (c+d x)}{35 a^4 d}-\frac {2 (11 A+2 C) \sin (c+d x) \cos (c+d x)}{a^4 d}-\frac {4 (11 A+2 C) \sin (c+d x) \cos ^2(c+d x)}{3 a^4 d (\sec (c+d x)+1)}-\frac {(178 A+31 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac {2 x (11 A+2 C)}{a^4}-\frac {2 (8 A+C) \sin (c+d x) \cos ^2(c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 3787
Rule 4020
Rule 4085
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx &=-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {\int \frac {\cos ^3(c+d x) (-a (10 A+3 C)+a (6 A-C) \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 (8 A+C) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {\int \frac {\cos ^3(c+d x) \left (-7 a^2 (14 A+3 C)+10 a^2 (8 A+C) \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac {(178 A+31 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 (8 A+C) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {\int \frac {\cos ^3(c+d x) \left (-12 a^3 (69 A+13 C)+4 a^3 (178 A+31 C) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{105 a^6}\\ &=-\frac {(178 A+31 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 (8 A+C) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {4 (11 A+2 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^4+a^4 \sec (c+d x)\right )}-\frac {\int \cos ^3(c+d x) \left (-12 a^4 (454 A+83 C)+420 a^4 (11 A+2 C) \sec (c+d x)\right ) \, dx}{105 a^8}\\ &=-\frac {(178 A+31 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 (8 A+C) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {4 (11 A+2 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^4+a^4 \sec (c+d x)\right )}-\frac {(4 (11 A+2 C)) \int \cos ^2(c+d x) \, dx}{a^4}+\frac {(4 (454 A+83 C)) \int \cos ^3(c+d x) \, dx}{35 a^4}\\ &=-\frac {2 (11 A+2 C) \cos (c+d x) \sin (c+d x)}{a^4 d}-\frac {(178 A+31 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 (8 A+C) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {4 (11 A+2 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^4+a^4 \sec (c+d x)\right )}-\frac {(2 (11 A+2 C)) \int 1 \, dx}{a^4}-\frac {(4 (454 A+83 C)) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 a^4 d}\\ &=-\frac {2 (11 A+2 C) x}{a^4}+\frac {4 (454 A+83 C) \sin (c+d x)}{35 a^4 d}-\frac {2 (11 A+2 C) \cos (c+d x) \sin (c+d x)}{a^4 d}-\frac {(178 A+31 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 (8 A+C) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {4 (11 A+2 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^4+a^4 \sec (c+d x)\right )}-\frac {4 (454 A+83 C) \sin ^3(c+d x)}{105 a^4 d}\\ \end {align*}
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Mathematica [B] time = 2.95, size = 575, normalized size = 2.32 \[ -\frac {\sec \left (\frac {c}{2}\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right ) \left (58800 d x (11 A+2 C) \cos \left (c+\frac {d x}{2}\right )+687260 A \sin \left (c+\frac {d x}{2}\right )-814107 A \sin \left (c+\frac {3 d x}{2}\right )+204645 A \sin \left (2 c+\frac {3 d x}{2}\right )-357609 A \sin \left (2 c+\frac {5 d x}{2}\right )-18025 A \sin \left (3 c+\frac {5 d x}{2}\right )-72522 A \sin \left (3 c+\frac {7 d x}{2}\right )-24010 A \sin \left (4 c+\frac {7 d x}{2}\right )-2310 A \sin \left (4 c+\frac {9 d x}{2}\right )-2310 A \sin \left (5 c+\frac {9 d x}{2}\right )+175 A \sin \left (5 c+\frac {11 d x}{2}\right )+175 A \sin \left (6 c+\frac {11 d x}{2}\right )-35 A \sin \left (6 c+\frac {13 d x}{2}\right )-35 A \sin \left (7 c+\frac {13 d x}{2}\right )+388080 A d x \cos \left (c+\frac {3 d x}{2}\right )+388080 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+129360 A d x \cos \left (2 c+\frac {5 d x}{2}\right )+129360 A d x \cos \left (3 c+\frac {5 d x}{2}\right )+18480 A d x \cos \left (3 c+\frac {7 d x}{2}\right )+18480 A d x \cos \left (4 c+\frac {7 d x}{2}\right )+58800 d x (11 A+2 C) \cos \left (\frac {d x}{2}\right )-1010660 A \sin \left (\frac {d x}{2}\right )+184520 C \sin \left (c+\frac {d x}{2}\right )-184464 C \sin \left (c+\frac {3 d x}{2}\right )+72240 C \sin \left (2 c+\frac {3 d x}{2}\right )-77168 C \sin \left (2 c+\frac {5 d x}{2}\right )+8400 C \sin \left (3 c+\frac {5 d x}{2}\right )-15164 C \sin \left (3 c+\frac {7 d x}{2}\right )-2940 C \sin \left (4 c+\frac {7 d x}{2}\right )-420 C \sin \left (4 c+\frac {9 d x}{2}\right )-420 C \sin \left (5 c+\frac {9 d x}{2}\right )+70560 C d x \cos \left (c+\frac {3 d x}{2}\right )+70560 C d x \cos \left (2 c+\frac {3 d x}{2}\right )+23520 C d x \cos \left (2 c+\frac {5 d x}{2}\right )+23520 C d x \cos \left (3 c+\frac {5 d x}{2}\right )+3360 C d x \cos \left (3 c+\frac {7 d x}{2}\right )+3360 C d x \cos \left (4 c+\frac {7 d x}{2}\right )-243320 C \sin \left (\frac {d x}{2}\right )\right )}{107520 a^4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 252, normalized size = 1.02 \[ -\frac {210 \, {\left (11 \, A + 2 \, C\right )} d x \cos \left (d x + c\right )^{4} + 840 \, {\left (11 \, A + 2 \, C\right )} d x \cos \left (d x + c\right )^{3} + 1260 \, {\left (11 \, A + 2 \, C\right )} d x \cos \left (d x + c\right )^{2} + 840 \, {\left (11 \, A + 2 \, C\right )} d x \cos \left (d x + c\right ) + 210 \, {\left (11 \, A + 2 \, C\right )} d x - {\left (35 \, A \cos \left (d x + c\right )^{6} - 70 \, A \cos \left (d x + c\right )^{5} + 35 \, {\left (14 \, A + 3 \, C\right )} \cos \left (d x + c\right )^{4} + 8 \, {\left (799 \, A + 148 \, C\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (3592 \, A + 659 \, C\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (6109 \, A + 1118 \, C\right )} \cos \left (d x + c\right ) + 3632 \, A + 664 \, C\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 261, normalized size = 1.05 \[ -\frac {\frac {1680 \, {\left (d x + c\right )} {\left (11 \, A + 2 \, C\right )}}{a^{4}} - \frac {560 \, {\left (39 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 62 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 27 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{4}} + \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 231 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 147 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2065 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 805 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 21945 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5145 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.41, size = 402, normalized size = 1.62 \[ -\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{56 d \,a^{4}}-\frac {C \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 d \,a^{4}}+\frac {11 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d \,a^{4}}+\frac {7 C \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d \,a^{4}}-\frac {59 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{24 d \,a^{4}}-\frac {23 C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d \,a^{4}}+\frac {209 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}+\frac {49 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}+\frac {26 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{d \,a^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {2 C \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {124 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{3 d \,a^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {4 C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {18 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {2 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {44 A \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{4}}-\frac {8 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{d \,a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 405, normalized size = 1.63 \[ \frac {A {\left (\frac {560 \, {\left (\frac {27 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {62 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {39 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{4} + \frac {3 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{4} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\frac {21945 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2065 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {231 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {36960 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} + C {\left (\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} + \frac {a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {6720 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.69, size = 198, normalized size = 0.80 \[ \frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {209\,A}{8}+\frac {49\,C}{8}\right )}{a^4\,d}-\frac {22\,A\,d\,x+4\,C\,d\,x}{a^4\,d}+\frac {\left (26\,A+2\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {124\,A}{3}+4\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (18\,A+2\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^4\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^3}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {11\,A}{40}+\frac {7\,C}{40}\right )}{a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {A}{56}+\frac {C}{56}\right )}{a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {59\,A}{24}+\frac {23\,C}{24}\right )}{a^4\,d} \]
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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