Optimal. Leaf size=87 \[ -\frac {a \cos ^5(c+d x)}{5 d}-\frac {b \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {b \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {b \sin (c+d x) \cos (c+d x)}{16 d}+\frac {b x}{16} \]
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Rubi [A] time = 0.11, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2838, 2565, 30, 2568, 2635, 8} \[ -\frac {a \cos ^5(c+d x)}{5 d}-\frac {b \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {b \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {b \sin (c+d x) \cos (c+d x)}{16 d}+\frac {b x}{16} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2565
Rule 2568
Rule 2635
Rule 2838
Rubi steps
\begin {align*} \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x)) \, dx &=a \int \cos ^4(c+d x) \sin (c+d x) \, dx+b \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx\\ &=-\frac {b \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac {1}{6} b \int \cos ^4(c+d x) \, dx-\frac {a \operatorname {Subst}\left (\int x^4 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a \cos ^5(c+d x)}{5 d}+\frac {b \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {b \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac {1}{8} b \int \cos ^2(c+d x) \, dx\\ &=-\frac {a \cos ^5(c+d x)}{5 d}+\frac {b \cos (c+d x) \sin (c+d x)}{16 d}+\frac {b \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {b \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac {1}{16} b \int 1 \, dx\\ &=\frac {b x}{16}-\frac {a \cos ^5(c+d x)}{5 d}+\frac {b \cos (c+d x) \sin (c+d x)}{16 d}+\frac {b \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {b \cos ^5(c+d x) \sin (c+d x)}{6 d}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 77, normalized size = 0.89 \[ -\frac {120 a \cos (c+d x)+60 a \cos (3 (c+d x))+12 a \cos (5 (c+d x))-15 b \sin (2 (c+d x))+15 b \sin (4 (c+d x))+5 b \sin (6 (c+d x))-60 b d x}{960 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 62, normalized size = 0.71 \[ -\frac {48 \, a \cos \left (d x + c\right )^{5} - 15 \, b d x + 5 \, {\left (8 \, b \cos \left (d x + c\right )^{5} - 2 \, b \cos \left (d x + c\right )^{3} - 3 \, b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 92, normalized size = 1.06 \[ \frac {1}{16} \, b x - \frac {a \cos \left (5 \, d x + 5 \, c\right )}{80 \, d} - \frac {a \cos \left (3 \, d x + 3 \, c\right )}{16 \, d} - \frac {a \cos \left (d x + c\right )}{8 \, d} - \frac {b \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac {b \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {b \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 68, normalized size = 0.78 \[ \frac {-\frac {a \left (\cos ^{5}\left (d x +c \right )\right )}{5}+b \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 52, normalized size = 0.60 \[ -\frac {192 \, a \cos \left (d x + c\right )^{5} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} b}{960 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.86, size = 181, normalized size = 2.08 \[ \frac {b\,x}{16}-\frac {-\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {47\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-\frac {13\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {13\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {47\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {2\,a}{5}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.24, size = 167, normalized size = 1.92 \[ \begin {cases} - \frac {a \cos ^{5}{\left (c + d x \right )}}{5 d} + \frac {b x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {3 b x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 b x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {b x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {b \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {b \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} - \frac {b \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\relax (c )}\right ) \sin {\relax (c )} \cos ^{4}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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