Optimal. Leaf size=43 \[ \frac {\sin (a+x (b-d)-c)}{2 (b-d)}-\frac {\sin (a+x (b+d)+c)}{2 (b+d)} \]
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Rubi [A] time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4569, 2637} \[ \frac {\sin (a+x (b-d)-c)}{2 (b-d)}-\frac {\sin (a+x (b+d)+c)}{2 (b+d)} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 4569
Rubi steps
\begin {align*} \int \sin (a+b x) \sin (c+d x) \, dx &=\int \left (\frac {1}{2} \cos (a-c+(b-d) x)-\frac {1}{2} \cos (a+c+(b+d) x)\right ) \, dx\\ &=\frac {1}{2} \int \cos (a-c+(b-d) x) \, dx-\frac {1}{2} \int \cos (a+c+(b+d) x) \, dx\\ &=\frac {\sin (a-c+(b-d) x)}{2 (b-d)}-\frac {\sin (a+c+(b+d) x)}{2 (b+d)}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 43, normalized size = 1.00 \[ \frac {\sin (a+x (b-d)-c)}{2 (b-d)}-\frac {\sin (a+x (b+d)+c)}{2 (b+d)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 42, normalized size = 0.98 \[ \frac {d \cos \left (d x + c\right ) \sin \left (b x + a\right ) - b \cos \left (b x + a\right ) \sin \left (d x + c\right )}{b^{2} - d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 40, normalized size = 0.93 \[ -\frac {\sin \left (b x + d x + a + c\right )}{2 \, {\left (b + d\right )}} + \frac {\sin \left (b x - d x + a - c\right )}{2 \, {\left (b - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 40, normalized size = 0.93 \[ \frac {\sin \left (a -c +\left (b -d \right ) x \right )}{2 b -2 d}-\frac {\sin \left (a +c +\left (b +d \right ) x \right )}{2 \left (b +d \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 40, normalized size = 0.93 \[ -\frac {\sin \left (b x + d x + a + c\right )}{2 \, {\left (b + d\right )}} - \frac {\sin \left (-b x + d x - a + c\right )}{2 \, {\left (b - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.06, size = 84, normalized size = 1.95 \[ \frac {d\,\left (\frac {\sin \left (a+c+b\,x+d\,x\right )}{2}+\frac {\sin \left (a-c+b\,x-d\,x\right )}{2}\right )}{b^2-d^2}-\frac {b\,\left (\frac {\sin \left (a+c+b\,x+d\,x\right )}{2}-\frac {\sin \left (a-c+b\,x-d\,x\right )}{2}\right )}{b^2-d^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.49, size = 153, normalized size = 3.56 \[ \begin {cases} x \sin {\relax (a )} \sin {\relax (c )} & \text {for}\: b = 0 \wedge d = 0 \\\frac {x \sin {\left (a - d x \right )} \sin {\left (c + d x \right )}}{2} - \frac {x \cos {\left (a - d x \right )} \cos {\left (c + d x \right )}}{2} - \frac {\sin {\left (a - d x \right )} \cos {\left (c + d x \right )}}{2 d} & \text {for}\: b = - d \\\frac {x \sin {\left (a + d x \right )} \sin {\left (c + d x \right )}}{2} + \frac {x \cos {\left (a + d x \right )} \cos {\left (c + d x \right )}}{2} - \frac {\sin {\left (c + d x \right )} \cos {\left (a + d x \right )}}{2 d} & \text {for}\: b = d \\- \frac {b \sin {\left (c + d x \right )} \cos {\left (a + b x \right )}}{b^{2} - d^{2}} + \frac {d \sin {\left (a + b x \right )} \cos {\left (c + d x \right )}}{b^{2} - d^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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