Optimal. Leaf size=15 \[ \frac {\sin ^4(a+b x)}{2 b} \]
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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 = {4288, 2564, 30} \[ \frac {\sin ^4(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2564
Rule 4288
Rubi steps
\begin {align*} \int \sin ^2(a+b x) \sin (2 a+2 b x) \, dx &=2 \int \cos (a+b x) \sin ^3(a+b x) \, dx\\ &=\frac {2 \operatorname {Subst}\left (\int x^3 \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {\sin ^4(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 15, normalized size = 1.00 \[ \frac {\sin ^4(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 24, normalized size = 1.60 \[ \frac {\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.46, size = 29, normalized size = 1.93 \[ \frac {\cos \left (4 \, b x + 4 \, a\right )}{16 \, b} - \frac {\cos \left (2 \, b x + 2 \, a\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 30, normalized size = 2.00 \[ -\frac {\cos \left (2 b x +2 a \right )}{4 b}+\frac {\cos \left (4 b x +4 a \right )}{16 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 26, normalized size = 1.73 \[ \frac {\cos \left (4 \, b x + 4 \, a\right ) - 4 \, \cos \left (2 \, b x + 2 \, a\right )}{16 \, b} \]
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 \[ \frac {{\sin \left (a+b\,x\right )}^4}{2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.30, size = 133, normalized size = 8.87 \[ \begin {cases} \frac {x \sin ^{2}{\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )}}{4} + \frac {x \sin {\left (a + b x \right )} \cos {\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{2} - \frac {x \sin {\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )}}{4} - \frac {3 \sin {\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )}}{4 b} - \frac {\cos ^{2}{\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{2 b} & \text {for}\: b \neq 0 \\x \sin ^{2}{\relax (a )} \sin {\left (2 a \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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