Integrand size = 24, antiderivative size = 129 \[ \int e^{a+b x} \cos ^3(c+d x) \sin ^3(c+d x) \, dx=-\frac {3 d e^{a+b x} \cos (2 c+2 d x)}{16 \left (b^2+4 d^2\right )}+\frac {3 d e^{a+b x} \cos (6 c+6 d x)}{16 \left (b^2+36 d^2\right )}+\frac {3 b e^{a+b x} \sin (2 c+2 d x)}{32 \left (b^2+4 d^2\right )}-\frac {b e^{a+b x} \sin (6 c+6 d x)}{32 \left (b^2+36 d^2\right )} \]
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Time = 0.12 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4557, 4517} \[ \int e^{a+b x} \cos ^3(c+d x) \sin ^3(c+d x) \, dx=\frac {3 b e^{a+b x} \sin (2 c+2 d x)}{32 \left (b^2+4 d^2\right )}-\frac {b e^{a+b x} \sin (6 c+6 d x)}{32 \left (b^2+36 d^2\right )}-\frac {3 d e^{a+b x} \cos (2 c+2 d x)}{16 \left (b^2+4 d^2\right )}+\frac {3 d e^{a+b x} \cos (6 c+6 d x)}{16 \left (b^2+36 d^2\right )} \]
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Rule 4517
Rule 4557
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3}{32} e^{a+b x} \sin (2 c+2 d x)-\frac {1}{32} e^{a+b x} \sin (6 c+6 d x)\right ) \, dx \\ & = -\left (\frac {1}{32} \int e^{a+b x} \sin (6 c+6 d x) \, dx\right )+\frac {3}{32} \int e^{a+b x} \sin (2 c+2 d x) \, dx \\ & = -\frac {3 d e^{a+b x} \cos (2 c+2 d x)}{16 \left (b^2+4 d^2\right )}+\frac {3 d e^{a+b x} \cos (6 c+6 d x)}{16 \left (b^2+36 d^2\right )}+\frac {3 b e^{a+b x} \sin (2 c+2 d x)}{32 \left (b^2+4 d^2\right )}-\frac {b e^{a+b x} \sin (6 c+6 d x)}{32 \left (b^2+36 d^2\right )} \\ \end{align*}
Time = 0.79 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.86 \[ \int e^{a+b x} \cos ^3(c+d x) \sin ^3(c+d x) \, dx=\frac {e^{a+b x} \left (-6 d \left (b^2+36 d^2\right ) \cos (2 (c+d x))+6 d \left (b^2+4 d^2\right ) \cos (6 (c+d x))-2 b \left (-b^2-52 d^2+\left (b^2+4 d^2\right ) \cos (4 (c+d x))\right ) \sin (2 (c+d x))\right )}{32 \left (b^4+40 b^2 d^2+144 d^4\right )} \]
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Time = 1.68 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.84
method | result | size |
parallelrisch | \(-\frac {\left (\left (b^{3}+4 b \,d^{2}\right ) \sin \left (6 d x +6 c \right )+\left (-6 b^{2} d -24 d^{3}\right ) \cos \left (6 d x +6 c \right )-3 \left (b^{2}+36 d^{2}\right ) \left (b \sin \left (2 d x +2 c \right )-2 d \cos \left (2 d x +2 c \right )\right )\right ) {\mathrm e}^{x b +a}}{32 b^{4}+1280 b^{2} d^{2}+4608 d^{4}}\) | \(108\) |
default | \(-\frac {3 d \,{\mathrm e}^{x b +a} \cos \left (2 d x +2 c \right )}{16 \left (b^{2}+4 d^{2}\right )}+\frac {3 d \,{\mathrm e}^{x b +a} \cos \left (6 d x +6 c \right )}{16 \left (b^{2}+36 d^{2}\right )}+\frac {3 b \,{\mathrm e}^{x b +a} \sin \left (2 d x +2 c \right )}{32 \left (b^{2}+4 d^{2}\right )}-\frac {b \,{\mathrm e}^{x b +a} \sin \left (6 d x +6 c \right )}{32 \left (b^{2}+36 d^{2}\right )}\) | \(118\) |
risch | \(\frac {i {\mathrm e}^{x b +a} \left (-12 i d \left (b^{2}+4 d^{2}\right ) \cos \left (6 d x +6 c \right )-i \left (-2 b^{3}-8 b \,d^{2}\right ) \sin \left (6 d x +6 c \right )+12 i d \left (b^{2}+36 d^{2}\right ) \cos \left (2 d x +2 c \right )-i \left (6 b^{3}+216 b \,d^{2}\right ) \sin \left (2 d x +2 c \right )\right )}{64 \left (6 i d +b \right ) \left (2 i d +b \right ) \left (2 i d -b \right ) \left (6 i d -b \right )}\) | \(139\) |
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Time = 0.26 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.21 \[ \int e^{a+b x} \cos ^3(c+d x) \sin ^3(c+d x) \, dx=-\frac {{\left ({\left (b^{3} + 4 \, b d^{2}\right )} \cos \left (d x + c\right )^{5} - 6 \, b d^{2} \cos \left (d x + c\right ) - {\left (b^{3} + 4 \, b d^{2}\right )} \cos \left (d x + c\right )^{3}\right )} e^{\left (b x + a\right )} \sin \left (d x + c\right ) - 3 \, {\left (2 \, {\left (b^{2} d + 4 \, d^{3}\right )} \cos \left (d x + c\right )^{6} + b^{2} d \cos \left (d x + c\right )^{2} - 3 \, {\left (b^{2} d + 4 \, d^{3}\right )} \cos \left (d x + c\right )^{4} + 2 \, d^{3}\right )} e^{\left (b x + a\right )}}{b^{4} + 40 \, b^{2} d^{2} + 144 \, d^{4}} \]
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Result contains complex when optimal does not.
Time = 71.62 (sec) , antiderivative size = 1991, normalized size of antiderivative = 15.43 \[ \int e^{a+b x} \cos ^3(c+d x) \sin ^3(c+d x) \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 550 vs. \(2 (117) = 234\).
Time = 0.22 (sec) , antiderivative size = 550, normalized size of antiderivative = 4.26 \[ \int e^{a+b x} \cos ^3(c+d x) \sin ^3(c+d x) \, dx=\frac {{\left (6 \, b^{2} d \cos \left (6 \, c\right ) e^{a} + 24 \, d^{3} \cos \left (6 \, c\right ) e^{a} - b^{3} e^{a} \sin \left (6 \, c\right ) - 4 \, b d^{2} e^{a} \sin \left (6 \, c\right )\right )} \cos \left (6 \, d x\right ) e^{\left (b x\right )} + {\left (6 \, b^{2} d \cos \left (6 \, c\right ) e^{a} + 24 \, d^{3} \cos \left (6 \, c\right ) e^{a} + b^{3} e^{a} \sin \left (6 \, c\right ) + 4 \, b d^{2} e^{a} \sin \left (6 \, c\right )\right )} \cos \left (6 \, d x + 12 \, c\right ) e^{\left (b x\right )} - 3 \, {\left (2 \, b^{2} d \cos \left (6 \, c\right ) e^{a} + 72 \, d^{3} \cos \left (6 \, c\right ) e^{a} + b^{3} e^{a} \sin \left (6 \, c\right ) + 36 \, b d^{2} e^{a} \sin \left (6 \, c\right )\right )} \cos \left (2 \, d x + 8 \, c\right ) e^{\left (b x\right )} - 3 \, {\left (2 \, b^{2} d \cos \left (6 \, c\right ) e^{a} + 72 \, d^{3} \cos \left (6 \, c\right ) e^{a} - b^{3} e^{a} \sin \left (6 \, c\right ) - 36 \, b d^{2} e^{a} \sin \left (6 \, c\right )\right )} \cos \left (2 \, d x - 4 \, c\right ) e^{\left (b x\right )} - {\left (b^{3} \cos \left (6 \, c\right ) e^{a} + 4 \, b d^{2} \cos \left (6 \, c\right ) e^{a} + 6 \, b^{2} d e^{a} \sin \left (6 \, c\right ) + 24 \, d^{3} e^{a} \sin \left (6 \, c\right )\right )} e^{\left (b x\right )} \sin \left (6 \, d x\right ) - {\left (b^{3} \cos \left (6 \, c\right ) e^{a} + 4 \, b d^{2} \cos \left (6 \, c\right ) e^{a} - 6 \, b^{2} d e^{a} \sin \left (6 \, c\right ) - 24 \, d^{3} e^{a} \sin \left (6 \, c\right )\right )} e^{\left (b x\right )} \sin \left (6 \, d x + 12 \, c\right ) + 3 \, {\left (b^{3} \cos \left (6 \, c\right ) e^{a} + 36 \, b d^{2} \cos \left (6 \, c\right ) e^{a} - 2 \, b^{2} d e^{a} \sin \left (6 \, c\right ) - 72 \, d^{3} e^{a} \sin \left (6 \, c\right )\right )} e^{\left (b x\right )} \sin \left (2 \, d x + 8 \, c\right ) + 3 \, {\left (b^{3} \cos \left (6 \, c\right ) e^{a} + 36 \, b d^{2} \cos \left (6 \, c\right ) e^{a} + 2 \, b^{2} d e^{a} \sin \left (6 \, c\right ) + 72 \, d^{3} e^{a} \sin \left (6 \, c\right )\right )} e^{\left (b x\right )} \sin \left (2 \, d x - 4 \, c\right )}{64 \, {\left (b^{4} \cos \left (6 \, c\right )^{2} + b^{4} \sin \left (6 \, c\right )^{2} + 144 \, {\left (\cos \left (6 \, c\right )^{2} + \sin \left (6 \, c\right )^{2}\right )} d^{4} + 40 \, {\left (b^{2} \cos \left (6 \, c\right )^{2} + b^{2} \sin \left (6 \, c\right )^{2}\right )} d^{2}\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.86 \[ \int e^{a+b x} \cos ^3(c+d x) \sin ^3(c+d x) \, dx=\frac {1}{32} \, {\left (\frac {6 \, d \cos \left (6 \, d x + 6 \, c\right )}{b^{2} + 36 \, d^{2}} - \frac {b \sin \left (6 \, d x + 6 \, c\right )}{b^{2} + 36 \, d^{2}}\right )} e^{\left (b x + a\right )} - \frac {3}{32} \, {\left (\frac {2 \, d \cos \left (2 \, d x + 2 \, c\right )}{b^{2} + 4 \, d^{2}} - \frac {b \sin \left (2 \, d x + 2 \, c\right )}{b^{2} + 4 \, d^{2}}\right )} e^{\left (b x + a\right )} \]
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Time = 1.25 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.38 \[ \int e^{a+b x} \cos ^3(c+d x) \sin ^3(c+d x) \, dx=-\frac {3\,{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (2\,d\,x\right )-\sin \left (2\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (2\,c\right )-\sin \left (2\,c\right )\,1{}\mathrm {i}\right )}{64\,\left (2\,d+b\,1{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (6\,d\,x\right )-\sin \left (6\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (6\,c\right )-\sin \left (6\,c\right )\,1{}\mathrm {i}\right )}{64\,\left (6\,d+b\,1{}\mathrm {i}\right )}-\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (2\,d\,x\right )+\sin \left (2\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (2\,c\right )+\sin \left (2\,c\right )\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{64\,\left (b+d\,2{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (6\,d\,x\right )+\sin \left (6\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (6\,c\right )+\sin \left (6\,c\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{64\,\left (b+d\,6{}\mathrm {i}\right )} \]
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