Optimal. Leaf size=42 \[ -\frac {\cos (a+b x)}{b}+\frac {2 \cos ^3(a+b x)}{3 b}-\frac {\cos ^5(a+b x)}{5 b} \]
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Rubi [A]
time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2713}
\begin {gather*} -\frac {\cos ^5(a+b x)}{5 b}+\frac {2 \cos ^3(a+b x)}{3 b}-\frac {\cos (a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2713
Rubi steps
\begin {align*} \int \sin ^5(a+b x) \, dx &=-\frac {\text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {\cos (a+b x)}{b}+\frac {2 \cos ^3(a+b x)}{3 b}-\frac {\cos ^5(a+b x)}{5 b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 44, normalized size = 1.05 \begin {gather*} -\frac {5 \cos (a+b x)}{8 b}+\frac {5 \cos (3 (a+b x))}{48 b}-\frac {\cos (5 (a+b x))}{80 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 32, normalized size = 0.76
method | result | size |
derivativedivides | \(-\frac {\left (\frac {8}{3}+\sin ^{4}\left (b x +a \right )+\frac {4 \left (\sin ^{2}\left (b x +a \right )\right )}{3}\right ) \cos \left (b x +a \right )}{5 b}\) | \(32\) |
default | \(-\frac {\left (\frac {8}{3}+\sin ^{4}\left (b x +a \right )+\frac {4 \left (\sin ^{2}\left (b x +a \right )\right )}{3}\right ) \cos \left (b x +a \right )}{5 b}\) | \(32\) |
risch | \(-\frac {5 \cos \left (b x +a \right )}{8 b}-\frac {\cos \left (5 b x +5 a \right )}{80 b}+\frac {5 \cos \left (3 b x +3 a \right )}{48 b}\) | \(41\) |
norman | \(\frac {-\frac {16}{15 b}-\frac {16 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}-\frac {32 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{5}}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 34, normalized size = 0.81 \begin {gather*} -\frac {3 \, \cos \left (b x + a\right )^{5} - 10 \, \cos \left (b x + a\right )^{3} + 15 \, \cos \left (b x + a\right )}{15 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 34, normalized size = 0.81 \begin {gather*} -\frac {3 \, \cos \left (b x + a\right )^{5} - 10 \, \cos \left (b x + a\right )^{3} + 15 \, \cos \left (b x + a\right )}{15 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.25, size = 60, normalized size = 1.43 \begin {gather*} \begin {cases} - \frac {\sin ^{4}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b} - \frac {4 \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{3 b} - \frac {8 \cos ^{5}{\left (a + b x \right )}}{15 b} & \text {for}\: b \neq 0 \\x \sin ^{5}{\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.25, size = 38, normalized size = 0.90 \begin {gather*} -\frac {\cos \left (b x + a\right )^{5}}{5 \, b} + \frac {2 \, \cos \left (b x + a\right )^{3}}{3 \, b} - \frac {\cos \left (b x + a\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.36, size = 32, normalized size = 0.76 \begin {gather*} -\frac {\frac {{\cos \left (a+b\,x\right )}^5}{5}-\frac {2\,{\cos \left (a+b\,x\right )}^3}{3}+\cos \left (a+b\,x\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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