Optimal. Leaf size=129 \[ \frac {b e^{a+b x} \sin (2 c+2 d x)}{4 \left (b^2+4 d^2\right )}-\frac {b e^{a+b x} \sin (4 c+4 d x)}{8 \left (b^2+16 d^2\right )}-\frac {d e^{a+b x} \cos (2 c+2 d x)}{2 \left (b^2+4 d^2\right )}+\frac {d e^{a+b x} \cos (4 c+4 d x)}{2 \left (b^2+16 d^2\right )} \]
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Rubi [A] time = 0.09, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4469, 4432} \[ \frac {b e^{a+b x} \sin (2 c+2 d x)}{4 \left (b^2+4 d^2\right )}-\frac {b e^{a+b x} \sin (4 c+4 d x)}{8 \left (b^2+16 d^2\right )}-\frac {d e^{a+b x} \cos (2 c+2 d x)}{2 \left (b^2+4 d^2\right )}+\frac {d e^{a+b x} \cos (4 c+4 d x)}{2 \left (b^2+16 d^2\right )} \]
Antiderivative was successfully verified.
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Rule 4432
Rule 4469
Rubi steps
\begin {align*} \int e^{a+b x} \cos (c+d x) \sin ^3(c+d x) \, dx &=\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*}
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Mathematica [A] time = 0.90, size = 82, normalized size = 0.64 \[ \frac {1}{8} e^{a+b x} \left (\frac {2 (b \sin (2 (c+d x))-2 d \cos (2 (c+d x)))}{b^2+4 d^2}+\frac {4 d \cos (4 (c+d x))-b \sin (4 (c+d x))}{b^2+16 d^2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 135, normalized size = 1.05 \[ -\frac {{\left ({\left (b^{3} + 4 \, b d^{2}\right )} \cos \left (d x + c\right )^{3} - {\left (b^{3} + 10 \, b d^{2}\right )} \cos \left (d x + c\right )\right )} e^{\left (b x + a\right )} \sin \left (d x + c\right ) - {\left (4 \, {\left (b^{2} d + 4 \, d^{3}\right )} \cos \left (d x + c\right )^{4} + b^{2} d + 10 \, d^{3} - {\left (5 \, b^{2} d + 32 \, d^{3}\right )} \cos \left (d x + c\right )^{2}\right )} e^{\left (b x + a\right )}}{b^{4} + 20 \, b^{2} d^{2} + 64 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 111, normalized size = 0.86 \[ \frac {1}{8} \, {\left (\frac {4 \, d \cos \left (4 \, d x + 4 \, c\right )}{b^{2} + 16 \, d^{2}} - \frac {b \sin \left (4 \, d x + 4 \, c\right )}{b^{2} + 16 \, d^{2}}\right )} e^{\left (b x + a\right )} - \frac {1}{4} \, {\left (\frac {2 \, d \cos \left (2 \, d x + 2 \, c\right )}{b^{2} + 4 \, d^{2}} - \frac {b \sin \left (2 \, d x + 2 \, c\right )}{b^{2} + 4 \, d^{2}}\right )} e^{\left (b x + a\right )} \]
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 \[ -\frac {d \,{\mathrm e}^{b x +a} \cos \left (2 d x +2 c \right )}{2 \left (b^{2}+4 d^{2}\right )}+\frac {d \,{\mathrm e}^{b x +a} \cos \left (4 d x +4 c \right )}{2 b^{2}+32 d^{2}}+\frac {b \,{\mathrm e}^{b x +a} \sin \left (2 d x +2 c \right )}{4 b^{2}+16 d^{2}}-\frac {b \,{\mathrm e}^{b x +a} \sin \left (4 d x +4 c \right )}{8 \left (b^{2}+16 d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 550, normalized size = 4.26 \[ \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 )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.03, size = 178, normalized size = 1.38 \[ -\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (2\,d\,x\right )-\sin \left (2\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (2\,c\right )-\sin \left (2\,c\right )\,1{}\mathrm {i}\right )}{8\,\left (2\,d+b\,1{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (4\,d\,x\right )-\sin \left (4\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (4\,c\right )-\sin \left (4\,c\right )\,1{}\mathrm {i}\right )}{16\,\left (4\,d+b\,1{}\mathrm {i}\right )}-\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (2\,d\,x\right )+\sin \left (2\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (2\,c\right )+\sin \left (2\,c\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{8\,\left (b+d\,2{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (4\,d\,x\right )+\sin \left (4\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (4\,c\right )+\sin \left (4\,c\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{16\,\left (b+d\,4{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 148.00, size = 1353, normalized size = 10.49 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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