Optimal. Leaf size=44 \[ -\frac {4 \cos ^8(a+b x)}{b}+\frac {32 \cos ^{10}(a+b x)}{5 b}-\frac {8 \cos ^{12}(a+b x)}{3 b} \]
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Rubi [A]
time = 0.05, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4372, 2645,
272, 45} \begin {gather*} -\frac {8 \cos ^{12}(a+b x)}{3 b}+\frac {32 \cos ^{10}(a+b x)}{5 b}-\frac {4 \cos ^8(a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 2645
Rule 4372
Rubi steps
\begin {align*} \int \cos ^2(a+b x) \sin ^5(2 a+2 b x) \, dx &=32 \int \cos ^7(a+b x) \sin ^5(a+b x) \, dx\\ &=-\frac {32 \text {Subst}\left (\int x^7 \left (1-x^2\right )^2 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {16 \text {Subst}\left (\int (1-x)^2 x^3 \, dx,x,\cos ^2(a+b x)\right )}{b}\\ &=-\frac {16 \text {Subst}\left (\int \left (x^3-2 x^4+x^5\right ) \, dx,x,\cos ^2(a+b x)\right )}{b}\\ &=-\frac {4 \cos ^8(a+b x)}{b}+\frac {32 \cos ^{10}(a+b x)}{5 b}-\frac {8 \cos ^{12}(a+b x)}{3 b}\\ \end {align*}
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Mathematica [A]
time = 0.44, size = 68, normalized size = 1.55 \begin {gather*} -\frac {600 \cos (2 (a+b x))+75 \cos (4 (a+b x))-100 \cos (6 (a+b x))-30 \cos (8 (a+b x))+12 \cos (10 (a+b x))+5 \cos (12 (a+b x))}{3840 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(85\) vs.
\(2(40)=80\).
time = 0.24, size = 86, normalized size = 1.95
method | result | size |
default | \(-\frac {5 \cos \left (2 x b +2 a \right )}{32 b}-\frac {5 \cos \left (4 x b +4 a \right )}{256 b}+\frac {5 \cos \left (6 x b +6 a \right )}{192 b}+\frac {\cos \left (8 x b +8 a \right )}{128 b}-\frac {\cos \left (10 x b +10 a \right )}{320 b}-\frac {\cos \left (12 x b +12 a \right )}{768 b}\) | \(86\) |
risch | \(-\frac {5 \cos \left (2 x b +2 a \right )}{32 b}-\frac {5 \cos \left (4 x b +4 a \right )}{256 b}+\frac {5 \cos \left (6 x b +6 a \right )}{192 b}+\frac {\cos \left (8 x b +8 a \right )}{128 b}-\frac {\cos \left (10 x b +10 a \right )}{320 b}-\frac {\cos \left (12 x b +12 a \right )}{768 b}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 72, normalized size = 1.64 \begin {gather*} -\frac {5 \, \cos \left (12 \, b x + 12 \, a\right ) + 12 \, \cos \left (10 \, b x + 10 \, a\right ) - 30 \, \cos \left (8 \, b x + 8 \, a\right ) - 100 \, \cos \left (6 \, b x + 6 \, a\right ) + 75 \, \cos \left (4 \, b x + 4 \, a\right ) + 600 \, \cos \left (2 \, b x + 2 \, a\right )}{3840 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.68, size = 36, normalized size = 0.82 \begin {gather*} -\frac {4 \, {\left (10 \, \cos \left (b x + a\right )^{12} - 24 \, \cos \left (b x + a\right )^{10} + 15 \, \cos \left (b x + a\right )^{8}\right )}}{15 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 597 vs.
\(2 (37) = 74\).
time = 15.36, size = 597, normalized size = 13.57 \begin {gather*} \begin {cases} - \frac {5 x \sin ^{2}{\left (a + b x \right )} \sin ^{5}{\left (2 a + 2 b x \right )}}{32} - \frac {5 x \sin ^{2}{\left (a + b x \right )} \sin ^{3}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{16} - \frac {5 x \sin ^{2}{\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos ^{4}{\left (2 a + 2 b x \right )}}{32} - \frac {5 x \sin {\left (a + b x \right )} \sin ^{4}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{16} - \frac {5 x \sin {\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{8} - \frac {5 x \sin {\left (a + b x \right )} \cos {\left (a + b x \right )} \cos ^{5}{\left (2 a + 2 b x \right )}}{16} + \frac {5 x \sin ^{5}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )}}{32} + \frac {5 x \sin ^{3}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{16} + \frac {5 x \sin {\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )} \cos ^{4}{\left (2 a + 2 b x \right )}}{32} - \frac {125 \sin ^{2}{\left (a + b x \right )} \sin ^{4}{\left (2 a + 2 b x \right )} \cos {\left (2 a + 2 b x \right )}}{384 b} - \frac {2 \sin ^{2}{\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{3 b} - \frac {217 \sin ^{2}{\left (a + b x \right )} \cos ^{5}{\left (2 a + 2 b x \right )}}{640 b} + \frac {95 \sin {\left (a + b x \right )} \sin ^{5}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )}}{192 b} + \frac {13 \sin {\left (a + b x \right )} \sin ^{3}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{12 b} + \frac {109 \sin {\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{4}{\left (2 a + 2 b x \right )}}{192 b} - \frac {67 \sin ^{4}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{384 b} + \frac {139 \cos ^{2}{\left (a + b x \right )} \cos ^{5}{\left (2 a + 2 b x \right )}}{1920 b} & \text {for}\: b \neq 0 \\x \sin ^{5}{\left (2 a \right )} \cos ^{2}{\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 = 0.41, size = 36, normalized size = 0.82 \begin {gather*} -\frac {4 \, {\left (10 \, \cos \left (b x + a\right )^{12} - 24 \, \cos \left (b x + a\right )^{10} + 15 \, \cos \left (b x + a\right )^{8}\right )}}{15 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 35, normalized size = 0.80 \begin {gather*} -\frac {4\,{\cos \left (a+b\,x\right )}^8\,\left (10\,{\cos \left (a+b\,x\right )}^4-24\,{\cos \left (a+b\,x\right )}^2+15\right )}{15\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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