Optimal. Leaf size=15 \[ \frac {2 \sin ^5(a+b x)}{5 b} \]
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Rubi [A]
time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4373, 2644, 30}
\begin {gather*} \frac {2 \sin ^5(a+b x)}{5 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2644
Rule 4373
Rubi steps
\begin {align*} \int \sin ^3(a+b x) \sin (2 a+2 b x) \, dx &=2 \int \cos (a+b x) \sin ^4(a+b x) \, dx\\ &=\frac {2 \text {Subst}\left (\int x^4 \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {2 \sin ^5(a+b x)}{5 b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} \frac {2 \sin ^5(a+b x)}{5 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. \(40\) vs.
\(2(13)=26\).
time = 0.06, size = 41, normalized size = 2.73
method | result | size |
default | \(\frac {\sin \left (x b +a \right )}{4 b}-\frac {\sin \left (3 x b +3 a \right )}{8 b}+\frac {\sin \left (5 x b +5 a \right )}{40 b}\) | \(41\) |
risch | \(\frac {\sin \left (x b +a \right )}{4 b}-\frac {\sin \left (3 x b +3 a \right )}{8 b}+\frac {\sin \left (5 x b +5 a \right )}{40 b}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 34 vs.
\(2 (13) = 26\).
time = 0.27, size = 34, normalized size = 2.27 \begin {gather*} \frac {\sin \left (5 \, b x + 5 \, a\right ) - 5 \, \sin \left (3 \, b x + 3 \, a\right ) + 10 \, \sin \left (b x + a\right )}{40 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs.
\(2 (13) = 26\).
time = 3.38, size = 31, normalized size = 2.07 \begin {gather*} \frac {2 \, {\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \sin \left (b x + a\right )}{5 \, 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. 117 vs.
\(2 (12) = 24\).
time = 1.11, size = 117, normalized size = 7.80 \begin {gather*} \begin {cases} - \frac {2 \sin ^{3}{\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{5 b} - \frac {\sin ^{2}{\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )}}{5 b} - \frac {4 \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{5 b} + \frac {2 \sin {\left (2 a + 2 b x \right )} \cos ^{3}{\left (a + b x \right )}}{5 b} & \text {for}\: b \neq 0 \\x \sin ^{3}{\left (a \right )} \sin {\left (2 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 = 0.40, size = 13, normalized size = 0.87 \begin {gather*} \frac {2 \, \sin \left (b x + a\right )^{5}}{5 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 13, normalized size = 0.87 \begin {gather*} \frac {2\,{\sin \left (a+b\,x\right )}^5}{5\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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