Optimal. Leaf size=102 \[ \frac {1}{8} a (3 A+4 (B+C)) x+\frac {a (A+B+C) \sin (c+d x)}{d}+\frac {a (3 A+4 (B+C)) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a A \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {a (A+B) \sin ^3(c+d x)}{3 d} \]
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Rubi [A]
time = 0.16, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4159, 4132,
2715, 8, 4129, 3092} \begin {gather*} \frac {a (A+B+C) \sin (c+d x)}{d}+\frac {a (3 A+4 (B+C)) \sin (c+d x) \cos (c+d x)}{8 d}-\frac {a (A+B) \sin ^3(c+d x)}{3 d}+\frac {1}{8} a x (3 A+4 (B+C))+\frac {a A \sin (c+d x) \cos ^3(c+d x)}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 3092
Rule 4129
Rule 4132
Rule 4159
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {a A \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {1}{4} \int \cos ^3(c+d x) \left (-4 a (A+B)-a (3 A+4 (B+C)) \sec (c+d x)-4 a C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a A \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {1}{4} \int \cos ^3(c+d x) \left (-4 a (A+B)-4 a C \sec ^2(c+d x)\right ) \, dx+\frac {1}{4} (a (3 A+4 (B+C))) \int \cos ^2(c+d x) \, dx\\ &=\frac {a (3 A+4 (B+C)) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a A \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {1}{4} \int \cos (c+d x) \left (-4 a C-4 a (A+B) \cos ^2(c+d x)\right ) \, dx+\frac {1}{8} (a (3 A+4 (B+C))) \int 1 \, dx\\ &=\frac {1}{8} a (3 A+4 (B+C)) x+\frac {a (3 A+4 (B+C)) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a A \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {\text {Subst}\left (\int \left (-4 a (A+B)-4 a C+4 a (A+B) x^2\right ) \, dx,x,-\sin (c+d x)\right )}{4 d}\\ &=\frac {1}{8} a (3 A+4 (B+C)) x+\frac {a (A+B+C) \sin (c+d x)}{d}+\frac {a (3 A+4 (B+C)) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a A \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {a (A+B) \sin ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.44, size = 97, normalized size = 0.95 \begin {gather*} \frac {a (24 A c+48 B c+36 A d x+48 B d x+48 C d x+24 (3 A+3 B+4 C) \sin (c+d x)+24 (A+B+C) \sin (2 (c+d x))+8 A \sin (3 (c+d x))+8 B \sin (3 (c+d x))+3 A \sin (4 (c+d x)))}{96 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.95, size = 141, normalized size = 1.38
method | result | size |
derivativedivides | \(\frac {A a \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {A a \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {B a \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+B a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a C \sin \left (d x +c \right )}{d}\) | \(141\) |
default | \(\frac {A a \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {A a \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {B a \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+B a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a C \sin \left (d x +c \right )}{d}\) | \(141\) |
risch | \(\frac {3 a A x}{8}+\frac {a B x}{2}+\frac {a x C}{2}+\frac {3 a A \sin \left (d x +c \right )}{4 d}+\frac {3 \sin \left (d x +c \right ) B a}{4 d}+\frac {\sin \left (d x +c \right ) a C}{d}+\frac {A a \sin \left (4 d x +4 c \right )}{32 d}+\frac {A a \sin \left (3 d x +3 c \right )}{12 d}+\frac {\sin \left (3 d x +3 c \right ) B a}{12 d}+\frac {A a \sin \left (2 d x +2 c \right )}{4 d}+\frac {\sin \left (2 d x +2 c \right ) B a}{4 d}+\frac {\sin \left (2 d x +2 c \right ) a C}{4 d}\) | \(151\) |
norman | \(\frac {\left (\frac {3}{8} A a +\frac {1}{2} B a +\frac {1}{2} a C \right ) x +\left (-\frac {3}{2} A a -2 B a -2 a C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {3}{8} A a -\frac {1}{2} B a -\frac {1}{2} a C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {3}{8} A a -\frac {1}{2} B a -\frac {1}{2} a C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {3}{4} A a +B a +a C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {3}{4} A a +B a +a C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {3}{8} A a +\frac {1}{2} B a +\frac {1}{2} a C \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {a \left (3 A +4 B +4 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {a \left (13 A -20 B -36 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {a \left (13 A +12 B +12 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {a \left (29 A -4 B +12 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {a \left (31 A +4 B +36 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {a \left (47 A +20 B -12 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}\) | \(360\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 132, normalized size = 1.29 \begin {gather*} -\frac {32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a - 3 \, {\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 + 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a - 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a - 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a - 96 \, C a \sin \left (d x + c\right )}{96 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.43, size = 87, normalized size = 0.85 \begin {gather*} \frac {3 \, {\left (3 \, A + 4 \, B + 4 \, C\right )} a d x + {\left (6 \, A a \cos \left (d x + c\right )^{3} + 8 \, {\left (A + B\right )} a \cos \left (d x + c\right )^{2} + 3 \, {\left (3 \, A + 4 \, B + 4 \, C\right )} a \cos \left (d x + c\right ) + 8 \, {\left (2 \, A + 2 \, B + 3 \, C\right )} a\right )} \sin \left (d x + c\right )}{24 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 218 vs.
\(2 (96) = 192\).
time = 0.44, size = 218, normalized size = 2.14 \begin {gather*} \frac {3 \, {\left (3 \, A a + 4 \, B a + 4 \, C a\right )} {\left (d x + c\right )} + \frac {2 \, {\left (9 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 49 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 28 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 60 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 31 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 52 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 84 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 39 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, C a \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 )}^{4}}}{24 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.55, size = 209, normalized size = 2.05 \begin {gather*} \frac {\left (\frac {3\,A\,a}{4}+B\,a+C\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {49\,A\,a}{12}+\frac {7\,B\,a}{3}+5\,C\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {31\,A\,a}{12}+\frac {13\,B\,a}{3}+7\,C\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {13\,A\,a}{4}+3\,B\,a+3\,C\,a\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 )}^8+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,A+4\,B+4\,C\right )}{4\,\left (\frac {3\,A\,a}{4}+B\,a+C\,a\right )}\right )\,\left (3\,A+4\,B+4\,C\right )}{4\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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