Optimal. Leaf size=97 \[ -\frac {a (A+B) \sin ^3(c+d x)}{3 d}+\frac {a (A+B) \sin (c+d x)}{d}+\frac {a (3 A+4 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} a x (3 A+4 B)+\frac {a A \sin (c+d x) \cos ^3(c+d x)}{4 d} \]
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Rubi [A] time = 0.12, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3996, 3787, 2633, 2635, 8} \[ -\frac {a (A+B) \sin ^3(c+d x)}{3 d}+\frac {a (A+B) \sin (c+d x)}{d}+\frac {a (3 A+4 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} a x (3 A+4 B)+\frac {a A \sin (c+d x) \cos ^3(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 3787
Rule 3996
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx &=\frac {a A \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {1}{4} \int \cos ^3(c+d x) (-4 a (A+B)-a (3 A+4 B) \sec (c+d x)) \, dx\\ &=\frac {a A \cos ^3(c+d x) \sin (c+d x)}{4 d}+(a (A+B)) \int \cos ^3(c+d x) \, dx+\frac {1}{4} (a (3 A+4 B)) \int \cos ^2(c+d x) \, dx\\ &=\frac {a (3 A+4 B) \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}{8} (a (3 A+4 B)) \int 1 \, dx-\frac {(a (A+B)) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac {1}{8} a (3 A+4 B) x+\frac {a (A+B) \sin (c+d x)}{d}+\frac {a (3 A+4 B) \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 = 0.24, size = 75, normalized size = 0.77 \[ \frac {a \left (-32 (A+B) \sin ^3(c+d x)+96 (A+B) \sin (c+d x)+24 (A+B) \sin (2 (c+d x))+3 A \sin (4 (c+d x))+36 A c+36 A d x+48 B c+48 B d x\right )}{96 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 74, normalized size = 0.76 \[ \frac {3 \, {\left (3 \, A + 4 \, B\right )} a d x + {\left (6 \, A a \cos \left (d x + c\right )^{3} + 8 \, {\left (A + B\right )} a \cos \left (d x + c\right )^{2} + 3 \, {\left (3 \, A + 4 \, B\right )} a \cos \left (d x + c\right ) + 16 \, {\left (A + B\right )} a\right )} \sin \left (d x + c\right )}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 156, normalized size = 1.61 \[ \frac {3 \, {\left (3 \, A a + 4 \, B a\right )} {\left (d x + c\right )} + \frac {2 \, {\left (9 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 49 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 28 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 31 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 52 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 39 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.29, size = 107, normalized size = 1.10 \[ \frac {a A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {a B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 101, normalized size = 1.04 \[ -\frac {32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a - 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a + 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a - 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.67, size = 184, normalized size = 1.90 \[ \frac {\left (\frac {3\,A\,a}{4}+B\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {49\,A\,a}{12}+\frac {7\,B\,a}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {31\,A\,a}{12}+\frac {13\,B\,a}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {13\,A\,a}{4}+3\,B\,a\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,A+4\,B\right )}{4\,\left (\frac {3\,A\,a}{4}+B\,a\right )}\right )\,\left (3\,A+4\,B\right )}{4\,d} \]
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 \[ a \left (\int A \cos ^{4}{\left (c + d x \right )}\, dx + \int A \cos ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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