Optimal. Leaf size=97 \[ \frac {3 \sin (a+x (b-d)-c)}{8 (b-d)}-\frac {\sin (3 a+x (3 b-d)-c)}{8 (3 b-d)}-\frac {3 \sin (a+x (b+d)+c)}{8 (b+d)}+\frac {\sin (3 a+x (3 b+d)+c)}{8 (3 b+d)} \]
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Rubi [A] time = 0.07, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4569, 2637} \[ \frac {3 \sin (a+x (b-d)-c)}{8 (b-d)}-\frac {\sin (3 a+x (3 b-d)-c)}{8 (3 b-d)}-\frac {3 \sin (a+x (b+d)+c)}{8 (b+d)}+\frac {\sin (3 a+x (3 b+d)+c)}{8 (3 b+d)} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 4569
Rubi steps
\begin {align*} \int \sin ^3(a+b x) \sin (c+d x) \, dx &=\int \left (\frac {3}{8} \cos (a-c+(b-d) x)-\frac {1}{8} \cos (3 a-c+(3 b-d) x)-\frac {3}{8} \cos (a+c+(b+d) x)+\frac {1}{8} \cos (3 a+c+(3 b+d) x)\right ) \, dx\\ &=-\left (\frac {1}{8} \int \cos (3 a-c+(3 b-d) x) \, dx\right )+\frac {1}{8} \int \cos (3 a+c+(3 b+d) x) \, dx+\frac {3}{8} \int \cos (a-c+(b-d) x) \, dx-\frac {3}{8} \int \cos (a+c+(b+d) x) \, dx\\ &=\frac {3 \sin (a-c+(b-d) x)}{8 (b-d)}-\frac {\sin (3 a-c+(3 b-d) x)}{8 (3 b-d)}-\frac {3 \sin (a+c+(b+d) x)}{8 (b+d)}+\frac {\sin (3 a+c+(3 b+d) x)}{8 (3 b+d)}\\ \end {align*}
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Mathematica [A] time = 0.54, size = 91, normalized size = 0.94 \[ \frac {1}{8} \left (\frac {3 \sin (a+b x-c-d x)}{b-d}-\frac {\sin (3 a+3 b x-c-d x)}{3 b-d}+\frac {\sin (3 a+3 b x+c+d x)}{3 b+d}-\frac {3 \sin (a+x (b+d)+c)}{b+d}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 115, normalized size = 1.19 \[ \frac {{\left (7 \, b^{2} d - d^{3} - {\left (b^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{2}\right )} \cos \left (d x + c\right ) \sin \left (b x + a\right ) + 3 \, {\left ({\left (b^{3} - b d^{2}\right )} \cos \left (b x + a\right )^{3} - {\left (3 \, b^{3} - b d^{2}\right )} \cos \left (b x + a\right )\right )} \sin \left (d x + c\right )}{9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.46, size = 89, normalized size = 0.92 \[ \frac {\sin \left (3 \, b x + d x + 3 \, a + c\right )}{8 \, {\left (3 \, b + d\right )}} - \frac {\sin \left (3 \, b x - d x + 3 \, a - c\right )}{8 \, {\left (3 \, b - d\right )}} - \frac {3 \, \sin \left (b x + d x + a + c\right )}{8 \, {\left (b + d\right )}} + \frac {3 \, \sin \left (b x - d x + a - c\right )}{8 \, {\left (b - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.99, size = 90, normalized size = 0.93 \[ \frac {3 \sin \left (a -c +\left (b -d \right ) x \right )}{8 \left (b -d \right )}-\frac {\sin \left (3 a -c +\left (3 b -d \right ) x \right )}{8 \left (3 b -d \right )}-\frac {3 \sin \left (a +c +\left (b +d \right ) x \right )}{8 \left (b +d \right )}+\frac {\sin \left (3 a +c +\left (3 b +d \right ) x \right )}{24 b +8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 789, normalized size = 8.13 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.72, size = 494, normalized size = 5.09 \[ {\mathrm {e}}^{a\,3{}\mathrm {i}-c\,1{}\mathrm {i}+b\,x\,3{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\left (\frac {-3\,b^3-b^2\,d+3\,b\,d^2+d^3}{b^4\,144{}\mathrm {i}-b^2\,d^2\,160{}\mathrm {i}+d^4\,16{}\mathrm {i}}+\frac {{\mathrm {e}}^{-a\,6{}\mathrm {i}-b\,x\,6{}\mathrm {i}}\,\left (-3\,b^3+b^2\,d+3\,b\,d^2-d^3\right )}{b^4\,144{}\mathrm {i}-b^2\,d^2\,160{}\mathrm {i}+d^4\,16{}\mathrm {i}}-\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (-27\,b^3-27\,b^2\,d+3\,b\,d^2+3\,d^3\right )}{b^4\,144{}\mathrm {i}-b^2\,d^2\,160{}\mathrm {i}+d^4\,16{}\mathrm {i}}-\frac {{\mathrm {e}}^{-a\,4{}\mathrm {i}-b\,x\,4{}\mathrm {i}}\,\left (-27\,b^3+27\,b^2\,d+3\,b\,d^2-3\,d^3\right )}{b^4\,144{}\mathrm {i}-b^2\,d^2\,160{}\mathrm {i}+d^4\,16{}\mathrm {i}}\right )-{\mathrm {e}}^{a\,3{}\mathrm {i}+c\,1{}\mathrm {i}+b\,x\,3{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {-3\,b^3+b^2\,d+3\,b\,d^2-d^3}{b^4\,144{}\mathrm {i}-b^2\,d^2\,160{}\mathrm {i}+d^4\,16{}\mathrm {i}}+\frac {{\mathrm {e}}^{-a\,6{}\mathrm {i}-b\,x\,6{}\mathrm {i}}\,\left (-3\,b^3-b^2\,d+3\,b\,d^2+d^3\right )}{b^4\,144{}\mathrm {i}-b^2\,d^2\,160{}\mathrm {i}+d^4\,16{}\mathrm {i}}-\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (-27\,b^3+27\,b^2\,d+3\,b\,d^2-3\,d^3\right )}{b^4\,144{}\mathrm {i}-b^2\,d^2\,160{}\mathrm {i}+d^4\,16{}\mathrm {i}}-\frac {{\mathrm {e}}^{-a\,4{}\mathrm {i}-b\,x\,4{}\mathrm {i}}\,\left (-27\,b^3-27\,b^2\,d+3\,b\,d^2+3\,d^3\right )}{b^4\,144{}\mathrm {i}-b^2\,d^2\,160{}\mathrm {i}+d^4\,16{}\mathrm {i}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 31.74, size = 933, normalized size = 9.62 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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