Optimal. Leaf size=54 \[ -\frac {\cos (a+b x)}{b}+\frac {\cos ^3(a+b x)}{b}-\frac {3 \cos ^5(a+b x)}{5 b}+\frac {\cos ^7(a+b x)}{7 b} \]
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Rubi [A]
time = 0.01, antiderivative size = 54, 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 ^7(a+b x)}{7 b}-\frac {3 \cos ^5(a+b x)}{5 b}+\frac {\cos ^3(a+b x)}{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 ^7(a+b x) \, dx &=-\frac {\text {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {\cos (a+b x)}{b}+\frac {\cos ^3(a+b x)}{b}-\frac {3 \cos ^5(a+b x)}{5 b}+\frac {\cos ^7(a+b x)}{7 b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 59, normalized size = 1.09 \begin {gather*} -\frac {35 \cos (a+b x)}{64 b}+\frac {7 \cos (3 (a+b x))}{64 b}-\frac {7 \cos (5 (a+b x))}{320 b}+\frac {\cos (7 (a+b x))}{448 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 42, normalized size = 0.78
method | result | size |
derivativedivides | \(-\frac {\left (\frac {16}{5}+\sin ^{6}\left (b x +a \right )+\frac {6 \left (\sin ^{4}\left (b x +a \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (b x +a \right )\right )}{5}\right ) \cos \left (b x +a \right )}{7 b}\) | \(42\) |
default | \(-\frac {\left (\frac {16}{5}+\sin ^{6}\left (b x +a \right )+\frac {6 \left (\sin ^{4}\left (b x +a \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (b x +a \right )\right )}{5}\right ) \cos \left (b x +a \right )}{7 b}\) | \(42\) |
risch | \(-\frac {35 \cos \left (b x +a \right )}{64 b}+\frac {\cos \left (7 b x +7 a \right )}{448 b}-\frac {7 \cos \left (5 b x +5 a \right )}{320 b}+\frac {7 \cos \left (3 b x +3 a \right )}{64 b}\) | \(55\) |
norman | \(\frac {-\frac {32 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {32}{35 b}-\frac {32 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{5 b}-\frac {96 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{5 b}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{7}}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 44, normalized size = 0.81 \begin {gather*} \frac {5 \, \cos \left (b x + a\right )^{7} - 21 \, \cos \left (b x + a\right )^{5} + 35 \, \cos \left (b x + a\right )^{3} - 35 \, \cos \left (b x + a\right )}{35 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 44, normalized size = 0.81 \begin {gather*} \frac {5 \, \cos \left (b x + a\right )^{7} - 21 \, \cos \left (b x + a\right )^{5} + 35 \, \cos \left (b x + a\right )^{3} - 35 \, \cos \left (b x + a\right )}{35 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.61, size = 80, normalized size = 1.48 \begin {gather*} \begin {cases} - \frac {\sin ^{6}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b} - \frac {2 \sin ^{4}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{b} - \frac {8 \sin ^{2}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{5 b} - \frac {16 \cos ^{7}{\left (a + b x \right )}}{35 b} & \text {for}\: b \neq 0 \\x \sin ^{7}{\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 = 4.34, size = 50, normalized size = 0.93 \begin {gather*} \frac {\cos \left (b x + a\right )^{7}}{7 \, b} - \frac {3 \, \cos \left (b x + a\right )^{5}}{5 \, b} + \frac {\cos \left (b x + a\right )^{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 = 0.38, size = 43, normalized size = 0.80 \begin {gather*} \frac {\cos \left (a+b\,x\right )\,\left (5\,{\cos \left (a+b\,x\right )}^6-21\,{\cos \left (a+b\,x\right )}^4+35\,{\cos \left (a+b\,x\right )}^2-35\right )}{35\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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