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 = 0.0877769, antiderivative size = 129, normalized size of antiderivative = 1., 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 4469
Rule 4432
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*}
Mathematica [A] time = 0.931011, 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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Maple [A] time = 0.017, size = 118, normalized size = 0.9 \begin{align*} -{\frac{d{{\rm e}^{bx+a}}\cos \left ( 2\,dx+2\,c \right ) }{2\,{b}^{2}+8\,{d}^{2}}}+{\frac{d{{\rm e}^{bx+a}}\cos \left ( 4\,dx+4\,c \right ) }{2\,{b}^{2}+32\,{d}^{2}}}+{\frac{b{{\rm e}^{bx+a}}\sin \left ( 2\,dx+2\,c \right ) }{4\,{b}^{2}+16\,{d}^{2}}}-{\frac{b{{\rm e}^{bx+a}}\sin \left ( 4\,dx+4\,c \right ) }{8\,{b}^{2}+128\,{d}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.14362, size = 743, normalized size = 5.76 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.487336, size = 302, normalized size = 2.34 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20013, size = 150, normalized size = 1.16 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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