Optimal. Leaf size=183 \[ \frac {b e^{a+b x} \sin (c+d x)}{8 \left (b^2+d^2\right )}+\frac {b e^{a+b x} \sin (3 c+3 d x)}{16 \left (b^2+9 d^2\right )}-\frac {b e^{a+b x} \sin (5 c+5 d x)}{16 \left (b^2+25 d^2\right )}-\frac {d e^{a+b x} \cos (c+d x)}{8 \left (b^2+d^2\right )}-\frac {3 d e^{a+b x} \cos (3 c+3 d x)}{16 \left (b^2+9 d^2\right )}+\frac {5 d e^{a+b x} \cos (5 c+5 d x)}{16 \left (b^2+25 d^2\right )} \]
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Rubi [A] time = 0.13, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4469, 4432} \[ \frac {b e^{a+b x} \sin (c+d x)}{8 \left (b^2+d^2\right )}+\frac {b e^{a+b x} \sin (3 c+3 d x)}{16 \left (b^2+9 d^2\right )}-\frac {b e^{a+b x} \sin (5 c+5 d x)}{16 \left (b^2+25 d^2\right )}-\frac {d e^{a+b x} \cos (c+d x)}{8 \left (b^2+d^2\right )}-\frac {3 d e^{a+b x} \cos (3 c+3 d x)}{16 \left (b^2+9 d^2\right )}+\frac {5 d e^{a+b x} \cos (5 c+5 d x)}{16 \left (b^2+25 d^2\right )} \]
Antiderivative was successfully verified.
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Rule 4432
Rule 4469
Rubi steps
\begin {align*} \int e^{a+b x} \cos ^2(c+d x) \sin ^3(c+d x) \, dx &=\int \left (\frac {1}{8} e^{a+b x} \sin (c+d x)+\frac {1}{16} e^{a+b x} \sin (3 c+3 d x)-\frac {1}{16} e^{a+b x} \sin (5 c+5 d x)\right ) \, dx\\ &=\frac {1}{16} \int e^{a+b x} \sin (3 c+3 d x) \, dx-\frac {1}{16} \int e^{a+b x} \sin (5 c+5 d x) \, dx+\frac {1}{8} \int e^{a+b x} \sin (c+d x) \, dx\\ &=-\frac {d e^{a+b x} \cos (c+d x)}{8 \left (b^2+d^2\right )}-\frac {3 d e^{a+b x} \cos (3 c+3 d x)}{16 \left (b^2+9 d^2\right )}+\frac {5 d e^{a+b x} \cos (5 c+5 d x)}{16 \left (b^2+25 d^2\right )}+\frac {b e^{a+b x} \sin (c+d x)}{8 \left (b^2+d^2\right )}+\frac {b e^{a+b x} \sin (3 c+3 d x)}{16 \left (b^2+9 d^2\right )}-\frac {b e^{a+b x} \sin (5 c+5 d x)}{16 \left (b^2+25 d^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.92, size = 110, normalized size = 0.60 \[ \frac {1}{16} e^{a+b x} \left (\frac {2 (b \sin (c+d x)-d \cos (c+d x))}{b^2+d^2}+\frac {b \sin (3 (c+d x))-3 d \cos (3 (c+d x))}{b^2+9 d^2}+\frac {5 d \cos (5 (c+d x))-b \sin (5 (c+d x))}{b^2+25 d^2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 201, normalized size = 1.10 \[ \frac {{\left (2 \, b^{3} d^{2} + 26 \, b d^{4} - {\left (b^{5} + 10 \, b^{3} d^{2} + 9 \, b d^{4}\right )} \cos \left (d x + c\right )^{4} + {\left (b^{5} + 14 \, b^{3} d^{2} + 13 \, b d^{4}\right )} \cos \left (d x + c\right )^{2}\right )} e^{\left (b x + a\right )} \sin \left (d x + c\right ) + {\left (5 \, {\left (b^{4} d + 10 \, b^{2} d^{3} + 9 \, d^{5}\right )} \cos \left (d x + c\right )^{5} - {\left (7 \, b^{4} d + 82 \, b^{2} d^{3} + 75 \, d^{5}\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (b^{4} d + 13 \, b^{2} d^{3}\right )} \cos \left (d x + c\right )\right )} e^{\left (b x + a\right )}}{b^{6} + 35 \, b^{4} d^{2} + 259 \, b^{2} d^{4} + 225 \, d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 155, normalized size = 0.85 \[ \frac {1}{16} \, {\left (\frac {5 \, d \cos \left (5 \, d x + 5 \, c\right )}{b^{2} + 25 \, d^{2}} - \frac {b \sin \left (5 \, d x + 5 \, c\right )}{b^{2} + 25 \, d^{2}}\right )} e^{\left (b x + a\right )} - \frac {1}{16} \, {\left (\frac {3 \, d \cos \left (3 \, d x + 3 \, c\right )}{b^{2} + 9 \, d^{2}} - \frac {b \sin \left (3 \, d x + 3 \, c\right )}{b^{2} + 9 \, d^{2}}\right )} e^{\left (b x + a\right )} - \frac {1}{8} \, {\left (\frac {d \cos \left (d x + c\right )}{b^{2} + d^{2}} - \frac {b \sin \left (d x + c\right )}{b^{2} + d^{2}}\right )} e^{\left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 166, normalized size = 0.91 \[ -\frac {d \,{\mathrm e}^{b x +a} \cos \left (d x +c \right )}{8 \left (b^{2}+d^{2}\right )}-\frac {3 d \,{\mathrm e}^{b x +a} \cos \left (3 d x +3 c \right )}{16 \left (b^{2}+9 d^{2}\right )}+\frac {5 d \,{\mathrm e}^{b x +a} \cos \left (5 d x +5 c \right )}{16 \left (b^{2}+25 d^{2}\right )}+\frac {b \,{\mathrm e}^{b x +a} \sin \left (d x +c \right )}{8 b^{2}+8 d^{2}}+\frac {b \,{\mathrm e}^{b x +a} \sin \left (3 d x +3 c \right )}{16 b^{2}+144 d^{2}}-\frac {b \,{\mathrm e}^{b x +a} \sin \left (5 d x +5 c \right )}{16 \left (b^{2}+25 d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 1148, normalized size = 6.27 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.74, size = 255, normalized size = 1.39 \[ -\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (d\,x\right )-\sin \left (d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \relax (c)-\sin \relax (c)\,1{}\mathrm {i}\right )}{16\,\left (d+b\,1{}\mathrm {i}\right )}-\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (d\,x\right )+\sin \left (d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \relax (c)+\sin \relax (c)\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{16\,\left (b+d\,1{}\mathrm {i}\right )}-\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (3\,d\,x\right )-\sin \left (3\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (3\,c\right )-\sin \left (3\,c\right )\,1{}\mathrm {i}\right )}{32\,\left (3\,d+b\,1{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (5\,d\,x\right )-\sin \left (5\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (5\,c\right )-\sin \left (5\,c\right )\,1{}\mathrm {i}\right )}{32\,\left (5\,d+b\,1{}\mathrm {i}\right )}-\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (3\,d\,x\right )+\sin \left (3\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (3\,c\right )+\sin \left (3\,c\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{32\,\left (b+d\,3{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (5\,d\,x\right )+\sin \left (5\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (5\,c\right )+\sin \left (5\,c\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{32\,\left (b+d\,5{}\mathrm {i}\right )} \]
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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