Integrand size = 22, antiderivative size = 129 \[ \int e^{a+b x} \cos (c+d x) \sin ^3(c+d x) \, dx=-\frac {d e^{a+b x} \cos (2 c+2 d x)}{2 \left (b^2+4 d^2\right )}+\frac {d e^{a+b x} \cos (4 c+4 d x)}{2 \left (b^2+16 d^2\right )}+\frac {b e^{a+b x} \sin (2 c+2 d x)}{4 \left (b^2+4 d^2\right )}-\frac {b e^{a+b x} \sin (4 c+4 d x)}{8 \left (b^2+16 d^2\right )} \]
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Time = 0.19 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4557, 4517} \[ \int e^{a+b x} \cos (c+d x) \sin ^3(c+d x) \, dx=\frac {b e^{a+b x} \sin (2 c+2 d x)}{4 \left (b^2+4 d^2\right )}-\frac {b e^{a+b x} \sin (4 c+4 d x)}{8 \left (b^2+16 d^2\right )}-\frac {d e^{a+b x} \cos (2 c+2 d x)}{2 \left (b^2+4 d^2\right )}+\frac {d e^{a+b x} \cos (4 c+4 d x)}{2 \left (b^2+16 d^2\right )} \]
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Rule 4517
Rule 4557
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{4} e^{a+b x} \sin (2 c+2 d x)-\frac {1}{8} e^{a+b x} \sin (4 c+4 d x)\right ) \, dx \\ & = -\left (\frac {1}{8} \int e^{a+b x} \sin (4 c+4 d x) \, dx\right )+\frac {1}{4} \int e^{a+b x} \sin (2 c+2 d x) \, dx \\ & = -\frac {d e^{a+b x} \cos (2 c+2 d x)}{2 \left (b^2+4 d^2\right )}+\frac {d e^{a+b x} \cos (4 c+4 d x)}{2 \left (b^2+16 d^2\right )}+\frac {b e^{a+b x} \sin (2 c+2 d x)}{4 \left (b^2+4 d^2\right )}-\frac {b e^{a+b x} \sin (4 c+4 d x)}{8 \left (b^2+16 d^2\right )} \\ \end{align*}
Time = 0.65 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.64 \[ \int e^{a+b x} \cos (c+d x) \sin ^3(c+d x) \, dx=\frac {1}{8} e^{a+b x} \left (\frac {2 (-2 d \cos (2 (c+d x))+b \sin (2 (c+d x)))}{b^2+4 d^2}+\frac {4 d \cos (4 (c+d x))-b \sin (4 (c+d x))}{b^2+16 d^2}\right ) \]
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Time = 0.92 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.84
method | result | size |
parallelrisch | \(-\frac {\left (\left (b^{3}+4 b \,d^{2}\right ) \sin \left (4 d x +4 c \right )+\left (-4 b^{2} d -16 d^{3}\right ) \cos \left (4 d x +4 c \right )-2 \left (b^{2}+16 d^{2}\right ) \left (b \sin \left (2 d x +2 c \right )-2 d \cos \left (2 d x +2 c \right )\right )\right ) {\mathrm e}^{x b +a}}{8 b^{4}+160 b^{2} d^{2}+512 d^{4}}\) | \(108\) |
default | \(-\frac {d \,{\mathrm e}^{x b +a} \cos \left (2 d x +2 c \right )}{2 \left (b^{2}+4 d^{2}\right )}+\frac {d \,{\mathrm e}^{x b +a} \cos \left (4 d x +4 c \right )}{2 b^{2}+32 d^{2}}+\frac {b \,{\mathrm e}^{x b +a} \sin \left (2 d x +2 c \right )}{4 b^{2}+16 d^{2}}-\frac {b \,{\mathrm e}^{x b +a} \sin \left (4 d x +4 c \right )}{8 \left (b^{2}+16 d^{2}\right )}\) | \(118\) |
risch | \(\frac {i {\mathrm e}^{x b +a} \left (-8 i d \left (b^{2}+4 d^{2}\right ) \cos \left (4 d x +4 c \right )-i \left (-2 b^{3}-8 b \,d^{2}\right ) \sin \left (4 d x +4 c \right )+8 i d \left (b^{2}+16 d^{2}\right ) \cos \left (2 d x +2 c \right )-i \left (4 b^{3}+64 b \,d^{2}\right ) \sin \left (2 d x +2 c \right )\right )}{16 \left (4 i d +b \right ) \left (2 i d +b \right ) \left (2 i d -b \right ) \left (4 i d -b \right )}\) | \(139\) |
norman | \(\frac {-\frac {6 d^{3} {\mathrm e}^{x b +a}}{b^{4}+20 b^{2} d^{2}+64 d^{4}}-\frac {6 d^{3} {\mathrm e}^{x b +a} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{b^{4}+20 b^{2} d^{2}+64 d^{4}}+\frac {12 b \,d^{2} {\mathrm e}^{x b +a} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{b^{4}+20 b^{2} d^{2}+64 d^{4}}-\frac {12 b \,d^{2} {\mathrm e}^{x b +a} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{b^{4}+20 b^{2} d^{2}+64 d^{4}}+\frac {4 b \left (2 b^{2}+11 d^{2}\right ) {\mathrm e}^{x b +a} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{b^{4}+20 b^{2} d^{2}+64 d^{4}}-\frac {4 b \left (2 b^{2}+11 d^{2}\right ) {\mathrm e}^{x b +a} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{b^{4}+20 b^{2} d^{2}+64 d^{4}}-\frac {12 d \left (b^{2}+2 d^{2}\right ) {\mathrm e}^{x b +a} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{b^{4}+20 b^{2} d^{2}+64 d^{4}}-\frac {12 d \left (b^{2}+2 d^{2}\right ) {\mathrm e}^{x b +a} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{b^{4}+20 b^{2} d^{2}+64 d^{4}}+\frac {20 d \left (2 b^{2}+11 d^{2}\right ) {\mathrm e}^{x b +a} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{b^{4}+20 b^{2} d^{2}+64 d^{4}}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}\) | \(417\) |
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Time = 0.25 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.05 \[ \int e^{a+b x} \cos (c+d x) \sin ^3(c+d x) \, dx=-\frac {{\left ({\left (b^{3} + 4 \, b d^{2}\right )} \cos \left (d x + c\right )^{3} - {\left (b^{3} + 10 \, b d^{2}\right )} \cos \left (d x + c\right )\right )} e^{\left (b x + a\right )} \sin \left (d x + c\right ) - {\left (4 \, {\left (b^{2} d + 4 \, d^{3}\right )} \cos \left (d x + c\right )^{4} + b^{2} d + 10 \, d^{3} - {\left (5 \, b^{2} d + 32 \, d^{3}\right )} \cos \left (d x + c\right )^{2}\right )} e^{\left (b x + a\right )}}{b^{4} + 20 \, b^{2} d^{2} + 64 \, d^{4}} \]
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Result contains complex when optimal does not.
Time = 7.29 (sec) , antiderivative size = 1353, normalized size of antiderivative = 10.49 \[ \int e^{a+b x} \cos (c+d x) \sin ^3(c+d x) \, dx=\text {Too large to display} \]
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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 (c+d x) \sin ^3(c+d x) \, dx=\frac {{\left (4 \, b^{2} d \cos \left (4 \, c\right ) e^{a} + 16 \, d^{3} \cos \left (4 \, c\right ) e^{a} - b^{3} e^{a} \sin \left (4 \, c\right ) - 4 \, b d^{2} e^{a} \sin \left (4 \, c\right )\right )} \cos \left (4 \, d x\right ) e^{\left (b x\right )} + {\left (4 \, b^{2} d \cos \left (4 \, c\right ) e^{a} + 16 \, d^{3} \cos \left (4 \, c\right ) e^{a} + b^{3} e^{a} \sin \left (4 \, c\right ) + 4 \, b d^{2} e^{a} \sin \left (4 \, c\right )\right )} \cos \left (4 \, d x + 8 \, c\right ) e^{\left (b x\right )} - 2 \, {\left (2 \, b^{2} d \cos \left (4 \, c\right ) e^{a} + 32 \, d^{3} \cos \left (4 \, c\right ) e^{a} + b^{3} e^{a} \sin \left (4 \, c\right ) + 16 \, b d^{2} e^{a} \sin \left (4 \, c\right )\right )} \cos \left (2 \, d x + 6 \, c\right ) e^{\left (b x\right )} - 2 \, {\left (2 \, b^{2} d \cos \left (4 \, c\right ) e^{a} + 32 \, d^{3} \cos \left (4 \, c\right ) e^{a} - b^{3} e^{a} \sin \left (4 \, c\right ) - 16 \, b d^{2} e^{a} \sin \left (4 \, c\right )\right )} \cos \left (2 \, d x - 2 \, c\right ) e^{\left (b x\right )} - {\left (b^{3} \cos \left (4 \, c\right ) e^{a} + 4 \, b d^{2} \cos \left (4 \, c\right ) e^{a} + 4 \, b^{2} d e^{a} \sin \left (4 \, c\right ) + 16 \, d^{3} e^{a} \sin \left (4 \, c\right )\right )} e^{\left (b x\right )} \sin \left (4 \, d x\right ) - {\left (b^{3} \cos \left (4 \, c\right ) e^{a} + 4 \, b d^{2} \cos \left (4 \, c\right ) e^{a} - 4 \, b^{2} d e^{a} \sin \left (4 \, c\right ) - 16 \, d^{3} e^{a} \sin \left (4 \, c\right )\right )} e^{\left (b x\right )} \sin \left (4 \, d x + 8 \, c\right ) + 2 \, {\left (b^{3} \cos \left (4 \, c\right ) e^{a} + 16 \, b d^{2} \cos \left (4 \, c\right ) e^{a} - 2 \, b^{2} d e^{a} \sin \left (4 \, c\right ) - 32 \, d^{3} e^{a} \sin \left (4 \, c\right )\right )} e^{\left (b x\right )} \sin \left (2 \, d x + 6 \, c\right ) + 2 \, {\left (b^{3} \cos \left (4 \, c\right ) e^{a} + 16 \, b d^{2} \cos \left (4 \, c\right ) e^{a} + 2 \, b^{2} d e^{a} \sin \left (4 \, c\right ) + 32 \, d^{3} e^{a} \sin \left (4 \, c\right )\right )} e^{\left (b x\right )} \sin \left (2 \, d x - 2 \, c\right )}{16 \, {\left (b^{4} \cos \left (4 \, c\right )^{2} + b^{4} \sin \left (4 \, c\right )^{2} + 64 \, {\left (\cos \left (4 \, c\right )^{2} + \sin \left (4 \, c\right )^{2}\right )} d^{4} + 20 \, {\left (b^{2} \cos \left (4 \, c\right )^{2} + b^{2} \sin \left (4 \, c\right )^{2}\right )} d^{2}\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.86 \[ \int e^{a+b x} \cos (c+d x) \sin ^3(c+d x) \, dx=\frac {1}{8} \, {\left (\frac {4 \, d \cos \left (4 \, d x + 4 \, c\right )}{b^{2} + 16 \, d^{2}} - \frac {b \sin \left (4 \, d x + 4 \, c\right )}{b^{2} + 16 \, d^{2}}\right )} e^{\left (b x + a\right )} - \frac {1}{4} \, {\left (\frac {2 \, d \cos \left (2 \, d x + 2 \, c\right )}{b^{2} + 4 \, d^{2}} - \frac {b \sin \left (2 \, d x + 2 \, c\right )}{b^{2} + 4 \, d^{2}}\right )} e^{\left (b x + a\right )} \]
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Time = 28.12 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.38 \[ \int e^{a+b x} \cos (c+d x) \sin ^3(c+d x) \, dx=-\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (2\,d\,x\right )-\sin \left (2\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (2\,c\right )-\sin \left (2\,c\right )\,1{}\mathrm {i}\right )}{8\,\left (2\,d+b\,1{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (4\,d\,x\right )-\sin \left (4\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (4\,c\right )-\sin \left (4\,c\right )\,1{}\mathrm {i}\right )}{16\,\left (4\,d+b\,1{}\mathrm {i}\right )}-\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (2\,d\,x\right )+\sin \left (2\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (2\,c\right )+\sin \left (2\,c\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{8\,\left (b+d\,2{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (4\,d\,x\right )+\sin \left (4\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (4\,c\right )+\sin \left (4\,c\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{16\,\left (b+d\,4{}\mathrm {i}\right )} \]
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