Optimal. Leaf size=169 \[ \frac {1}{4} a^2 (3 A+4 C) x+\frac {a^2 (18 A+25 C) \sin (c+d x)}{15 d}+\frac {a^2 (3 A+4 C) \cos (c+d x) \sin (c+d x)}{4 d}+\frac {a^2 (9 A+10 C) \cos ^2(c+d x) \sin (c+d x)}{30 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {A \cos ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{10 d} \]
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Rubi [A]
time = 0.26, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {4172, 4102,
4081, 3872, 2715, 8, 2717} \begin {gather*} \frac {a^2 (18 A+25 C) \sin (c+d x)}{15 d}+\frac {a^2 (9 A+10 C) \sin (c+d x) \cos ^2(c+d x)}{30 d}+\frac {a^2 (3 A+4 C) \sin (c+d x) \cos (c+d x)}{4 d}+\frac {A \sin (c+d x) \cos ^3(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{10 d}+\frac {1}{4} a^2 x (3 A+4 C)+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 2717
Rule 3872
Rule 4081
Rule 4102
Rule 4172
Rubi steps
\begin {align*} \int \cos ^5(c+d x) (a+a \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {\int \cos ^4(c+d x) (a+a \sec (c+d x))^2 (2 a A+a (2 A+5 C) \sec (c+d x)) \, dx}{5 a}\\ &=\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {A \cos ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{10 d}+\frac {\int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (2 a^2 (9 A+10 C)+4 a^2 (3 A+5 C) \sec (c+d x)\right ) \, dx}{20 a}\\ &=\frac {a^2 (9 A+10 C) \cos ^2(c+d x) \sin (c+d x)}{30 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {A \cos ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{10 d}-\frac {\int \cos ^2(c+d x) \left (-30 a^3 (3 A+4 C)-4 a^3 (18 A+25 C) \sec (c+d x)\right ) \, dx}{60 a}\\ &=\frac {a^2 (9 A+10 C) \cos ^2(c+d x) \sin (c+d x)}{30 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {A \cos ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{10 d}+\frac {1}{2} \left (a^2 (3 A+4 C)\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{15} \left (a^2 (18 A+25 C)\right ) \int \cos (c+d x) \, dx\\ &=\frac {a^2 (18 A+25 C) \sin (c+d x)}{15 d}+\frac {a^2 (3 A+4 C) \cos (c+d x) \sin (c+d x)}{4 d}+\frac {a^2 (9 A+10 C) \cos ^2(c+d x) \sin (c+d x)}{30 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {A \cos ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{10 d}+\frac {1}{4} \left (a^2 (3 A+4 C)\right ) \int 1 \, dx\\ &=\frac {1}{4} a^2 (3 A+4 C) x+\frac {a^2 (18 A+25 C) \sin (c+d x)}{15 d}+\frac {a^2 (3 A+4 C) \cos (c+d x) \sin (c+d x)}{4 d}+\frac {a^2 (9 A+10 C) \cos ^2(c+d x) \sin (c+d x)}{30 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {A \cos ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{10 d}\\ \end {align*}
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Mathematica [A]
time = 0.43, size = 97, normalized size = 0.57 \begin {gather*} \frac {a^2 (120 A c+180 A d x+240 C d x+30 (11 A+14 C) \sin (c+d x)+120 (A+C) \sin (2 (c+d x))+45 A \sin (3 (c+d x))+20 C \sin (3 (c+d x))+15 A \sin (4 (c+d x))+3 A \sin (5 (c+d x)))}{240 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.75, size = 160, normalized size = 0.95
method | result | size |
risch | \(\frac {3 a^{2} A x}{4}+a^{2} x C +\frac {11 a^{2} A \sin \left (d x +c \right )}{8 d}+\frac {7 \sin \left (d x +c \right ) a^{2} C}{4 d}+\frac {a^{2} A \sin \left (5 d x +5 c \right )}{80 d}+\frac {a^{2} A \sin \left (4 d x +4 c \right )}{16 d}+\frac {3 a^{2} A \sin \left (3 d x +3 c \right )}{16 d}+\frac {\sin \left (3 d x +3 c \right ) a^{2} C}{12 d}+\frac {a^{2} A \sin \left (2 d x +2 c \right )}{2 d}+\frac {\sin \left (2 d x +2 c \right ) a^{2} C}{2 d}\) | \(153\) |
derivativedivides | \(\frac {\frac {a^{2} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{2} C \sin \left (d x +c \right )+2 a^{2} A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {a^{2} A \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {a^{2} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(160\) |
default | \(\frac {\frac {a^{2} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{2} C \sin \left (d x +c \right )+2 a^{2} A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {a^{2} A \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {a^{2} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(160\) |
norman | \(\frac {-\frac {a^{2} \left (3 A +4 C \right ) x}{4}+\frac {5 a^{2} \left (3 A +4 C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {a^{2} \left (3 A +4 C \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {a^{2} \left (3 A +4 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a^{2} \left (3 A +4 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {3 a^{2} \left (3 A +4 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {3 a^{2} \left (3 A +4 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {a^{2} \left (3 A +4 C \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a^{2} \left (3 A +4 C \right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a^{2} \left (3 A +4 C \right ) x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {23 a^{2} \left (9 A -20 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{30 d}-\frac {a^{2} \left (13 A +12 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}+\frac {a^{2} \left (63 A +4 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {a^{2} \left (63 A +100 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{30 d}-\frac {a^{2} \left (441 A +380 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{30 d}+\frac {a^{2} \left (471 A +20 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{30 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}\) | \(420\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 156, normalized size = 0.92 \begin {gather*} \frac {16 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{2} - 80 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} - 80 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} + 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} + 240 \, C a^{2} \sin \left (d x + c\right )}{240 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.87, size = 106, normalized size = 0.63 \begin {gather*} \frac {15 \, {\left (3 \, A + 4 \, C\right )} a^{2} d x + {\left (12 \, A a^{2} \cos \left (d x + c\right )^{4} + 30 \, A a^{2} \cos \left (d x + c\right )^{3} + 4 \, {\left (9 \, A + 5 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \, {\left (3 \, A + 4 \, C\right )} a^{2} \cos \left (d x + c\right ) + 4 \, {\left (18 \, A + 25 \, C\right )} a^{2}\right )} \sin \left (d x + c\right )}{60 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 210, normalized size = 1.24 \begin {gather*} \frac {15 \, {\left (3 \, A a^{2} + 4 \, C a^{2}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (45 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 60 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 210 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 280 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 432 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 560 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 270 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 520 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 195 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 180 \, C a^{2} \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 )}^{5}}}{60 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.34, size = 247, normalized size = 1.46 \begin {gather*} \frac {\left (\frac {3\,A\,a^2}{2}+2\,C\,a^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (7\,A\,a^2+\frac {28\,C\,a^2}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {72\,A\,a^2}{5}+\frac {56\,C\,a^2}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (9\,A\,a^2+\frac {52\,C\,a^2}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {13\,A\,a^2}{2}+6\,C\,a^2\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^2\,\mathrm {atan}\left (\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,A+4\,C\right )}{2\,\left (\frac {3\,A\,a^2}{2}+2\,C\,a^2\right )}\right )\,\left (3\,A+4\,C\right )}{2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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