Optimal. Leaf size=129 \[ \frac {3 b e^{a+b x} \sin (2 c+2 d x)}{32 \left (b^2+4 d^2\right )}-\frac {b e^{a+b x} \sin (6 c+6 d x)}{32 \left (b^2+36 d^2\right )}-\frac {3 d e^{a+b x} \cos (2 c+2 d x)}{16 \left (b^2+4 d^2\right )}+\frac {3 d e^{a+b x} \cos (6 c+6 d x)}{16 \left (b^2+36 d^2\right )} \]
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Rubi [A] time = 0.10, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4469, 4432} \[ \frac {3 b e^{a+b x} \sin (2 c+2 d x)}{32 \left (b^2+4 d^2\right )}-\frac {b e^{a+b x} \sin (6 c+6 d x)}{32 \left (b^2+36 d^2\right )}-\frac {3 d e^{a+b x} \cos (2 c+2 d x)}{16 \left (b^2+4 d^2\right )}+\frac {3 d e^{a+b x} \cos (6 c+6 d x)}{16 \left (b^2+36 d^2\right )} \]
Antiderivative was successfully verified.
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Rule 4432
Rule 4469
Rubi steps
\begin {align*} \int e^{a+b x} \cos ^3(c+d x) \sin ^3(c+d x) \, dx &=\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*}
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Mathematica [A] time = 0.94, size = 111, normalized size = 0.86 \[ \frac {e^{a+b x} \left (-6 d \left (b^2+36 d^2\right ) \cos (2 (c+d x))+6 d \left (b^2+4 d^2\right ) \cos (6 (c+d x))-2 b \sin (2 (c+d x)) \left (\left (b^2+4 d^2\right ) \cos (4 (c+d x))-b^2-52 d^2\right )\right )}{32 \left (b^4+40 b^2 d^2+144 d^4\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.46, size = 156, normalized size = 1.21 \[ -\frac {{\left ({\left (b^{3} + 4 \, b d^{2}\right )} \cos \left (d x + c\right )^{5} - 6 \, b d^{2} \cos \left (d x + c\right ) - {\left (b^{3} + 4 \, b d^{2}\right )} \cos \left (d x + c\right )^{3}\right )} e^{\left (b x + a\right )} \sin \left (d x + c\right ) - 3 \, {\left (2 \, {\left (b^{2} d + 4 \, d^{3}\right )} \cos \left (d x + c\right )^{6} + b^{2} d \cos \left (d x + c\right )^{2} - 3 \, {\left (b^{2} d + 4 \, d^{3}\right )} \cos \left (d x + c\right )^{4} + 2 \, d^{3}\right )} e^{\left (b x + a\right )}}{b^{4} + 40 \, b^{2} d^{2} + 144 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 111, normalized size = 0.86 \[ \frac {1}{32} \, {\left (\frac {6 \, d \cos \left (6 \, d x + 6 \, c\right )}{b^{2} + 36 \, d^{2}} - \frac {b \sin \left (6 \, d x + 6 \, c\right )}{b^{2} + 36 \, d^{2}}\right )} e^{\left (b x + a\right )} - \frac {3}{32} \, {\left (\frac {2 \, d \cos \left (2 \, d x + 2 \, c\right )}{b^{2} + 4 \, d^{2}} - \frac {b \sin \left (2 \, d x + 2 \, c\right )}{b^{2} + 4 \, d^{2}}\right )} e^{\left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 118, normalized size = 0.91 \[ -\frac {3 d \,{\mathrm e}^{b x +a} \cos \left (2 d x +2 c \right )}{16 \left (b^{2}+4 d^{2}\right )}+\frac {3 d \,{\mathrm e}^{b x +a} \cos \left (6 d x +6 c \right )}{16 \left (b^{2}+36 d^{2}\right )}+\frac {3 b \,{\mathrm e}^{b x +a} \sin \left (2 d x +2 c \right )}{32 \left (b^{2}+4 d^{2}\right )}-\frac {b \,{\mathrm e}^{b x +a} \sin \left (6 d x +6 c \right )}{32 \left (b^{2}+36 d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 550, normalized size = 4.26 \[ \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 )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.01, size = 178, normalized size = 1.38 \[ -\frac {3\,{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (2\,d\,x\right )-\sin \left (2\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (2\,c\right )-\sin \left (2\,c\right )\,1{}\mathrm {i}\right )}{64\,\left (2\,d+b\,1{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (6\,d\,x\right )-\sin \left (6\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (6\,c\right )-\sin \left (6\,c\right )\,1{}\mathrm {i}\right )}{64\,\left (6\,d+b\,1{}\mathrm {i}\right )}-\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (2\,d\,x\right )+\sin \left (2\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (2\,c\right )+\sin \left (2\,c\right )\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{64\,\left (b+d\,2{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (6\,d\,x\right )+\sin \left (6\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (6\,c\right )+\sin \left (6\,c\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{64\,\left (b+d\,6{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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