Optimal. Leaf size=119 \[ \frac {b e^{a+b x} \cos (c+d x)}{4 \left (b^2+d^2\right )}-\frac {b e^{a+b x} \cos (3 c+3 d x)}{4 \left (b^2+9 d^2\right )}+\frac {d e^{a+b x} \sin (c+d x)}{4 \left (b^2+d^2\right )}-\frac {3 d e^{a+b x} \sin (3 c+3 d x)}{4 \left (b^2+9 d^2\right )} \]
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Rubi [A]
time = 0.06, antiderivative size = 119, 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 = {4557, 4518}
\begin {gather*} \frac {d e^{a+b x} \sin (c+d x)}{4 \left (b^2+d^2\right )}-\frac {3 d e^{a+b x} \sin (3 c+3 d x)}{4 \left (b^2+9 d^2\right )}+\frac {b e^{a+b x} \cos (c+d x)}{4 \left (b^2+d^2\right )}-\frac {b e^{a+b x} \cos (3 c+3 d x)}{4 \left (b^2+9 d^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 4518
Rule 4557
Rubi steps
\begin {align*} \int e^{a+b x} \cos (c+d x) \sin ^2(c+d x) \, dx &=\int \left (\frac {1}{4} e^{a+b x} \cos (c+d x)-\frac {1}{4} e^{a+b x} \cos (3 c+3 d x)\right ) \, dx\\ &=\frac {1}{4} \int e^{a+b x} \cos (c+d x) \, dx-\frac {1}{4} \int e^{a+b x} \cos (3 c+3 d x) \, dx\\ &=\frac {b e^{a+b x} \cos (c+d x)}{4 \left (b^2+d^2\right )}-\frac {b e^{a+b x} \cos (3 c+3 d x)}{4 \left (b^2+9 d^2\right )}+\frac {d e^{a+b x} \sin (c+d x)}{4 \left (b^2+d^2\right )}-\frac {3 d e^{a+b x} \sin (3 c+3 d x)}{4 \left (b^2+9 d^2\right )}\\ \end {align*}
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Mathematica [A]
time = 0.70, size = 74, normalized size = 0.62 \begin {gather*} \frac {1}{4} e^{a+b x} \left (\frac {b \cos (c+d x)+d \sin (c+d x)}{b^2+d^2}-\frac {b \cos (3 (c+d x))+3 d \sin (3 (c+d x))}{b^2+9 d^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 108, normalized size = 0.91
method | result | size |
default | \(\frac {b \,{\mathrm e}^{b x +a} \cos \left (d x +c \right )}{4 b^{2}+4 d^{2}}-\frac {b \,{\mathrm e}^{b x +a} \cos \left (3 d x +3 c \right )}{4 \left (b^{2}+9 d^{2}\right )}+\frac {d \,{\mathrm e}^{b x +a} \sin \left (d x +c \right )}{4 b^{2}+4 d^{2}}-\frac {3 d \,{\mathrm e}^{b x +a} \sin \left (3 d x +3 c \right )}{4 \left (b^{2}+9 d^{2}\right )}\) | \(108\) |
risch | \(-\frac {{\mathrm e}^{b x +a} {\mathrm e}^{3 i d x} {\mathrm e}^{3 i c}}{8 \left (3 i d +b \right )}+\frac {{\mathrm e}^{b x +a} {\mathrm e}^{i d x} {\mathrm e}^{i c}}{8 i d +8 b}+\frac {{\mathrm e}^{b x +a} {\mathrm e}^{-i d x} {\mathrm e}^{-i c}}{-8 i d +8 b}-\frac {{\mathrm e}^{b x +a} {\mathrm e}^{-3 i d x} {\mathrm e}^{-3 i c}}{8 \left (-3 i d +b \right )}\) | \(110\) |
norman | \(\frac {\frac {2 b \,d^{2} {\mathrm e}^{b x +a}}{b^{4}+10 b^{2} d^{2}+9 d^{4}}-\frac {2 b \,d^{2} {\mathrm e}^{b x +a} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{4}+10 b^{2} d^{2}+9 d^{4}}+\frac {2 b \left (2 b^{2}+3 d^{2}\right ) {\mathrm e}^{b x +a} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{4}+10 b^{2} d^{2}+9 d^{4}}-\frac {2 b \left (2 b^{2}+3 d^{2}\right ) {\mathrm e}^{b x +a} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{4}+10 b^{2} d^{2}+9 d^{4}}-\frac {4 b^{2} d \,{\mathrm e}^{b x +a} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{b^{4}+10 b^{2} d^{2}+9 d^{4}}-\frac {4 b^{2} d \,{\mathrm e}^{b x +a} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{4}+10 b^{2} d^{2}+9 d^{4}}+\frac {8 d \left (2 b^{2}+3 d^{2}\right ) {\mathrm e}^{b x +a} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{4}+10 b^{2} d^{2}+9 d^{4}}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}\) | \(323\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 538 vs.
\(2 (107) = 214\).
time = 0.30, size = 538, normalized size = 4.52 \begin {gather*} -\frac {{\left (b^{3} \cos \left (3 \, c\right ) e^{a} + b d^{2} \cos \left (3 \, c\right ) e^{a} + 3 \, b^{2} d e^{a} \sin \left (3 \, c\right ) + 3 \, d^{3} e^{a} \sin \left (3 \, c\right )\right )} \cos \left (3 \, d x\right ) e^{\left (b x\right )} + {\left (b^{3} \cos \left (3 \, c\right ) e^{a} + b d^{2} \cos \left (3 \, c\right ) e^{a} - 3 \, b^{2} d e^{a} \sin \left (3 \, c\right ) - 3 \, d^{3} e^{a} \sin \left (3 \, c\right )\right )} \cos \left (3 \, d x + 6 \, c\right ) e^{\left (b x\right )} - {\left (b^{3} \cos \left (3 \, c\right ) e^{a} + 9 \, b d^{2} \cos \left (3 \, c\right ) e^{a} - b^{2} d e^{a} \sin \left (3 \, c\right ) - 9 \, d^{3} e^{a} \sin \left (3 \, c\right )\right )} \cos \left (d x + 4 \, c\right ) e^{\left (b x\right )} - {\left (b^{3} \cos \left (3 \, c\right ) e^{a} + 9 \, b d^{2} \cos \left (3 \, c\right ) e^{a} + b^{2} d e^{a} \sin \left (3 \, c\right ) + 9 \, d^{3} e^{a} \sin \left (3 \, c\right )\right )} \cos \left (d x - 2 \, c\right ) e^{\left (b x\right )} + {\left (3 \, b^{2} d \cos \left (3 \, c\right ) e^{a} + 3 \, d^{3} \cos \left (3 \, c\right ) e^{a} - b^{3} e^{a} \sin \left (3 \, c\right ) - b d^{2} e^{a} \sin \left (3 \, c\right )\right )} e^{\left (b x\right )} \sin \left (3 \, d x\right ) + {\left (3 \, b^{2} d \cos \left (3 \, c\right ) e^{a} + 3 \, d^{3} \cos \left (3 \, c\right ) e^{a} + b^{3} e^{a} \sin \left (3 \, c\right ) + b d^{2} e^{a} \sin \left (3 \, c\right )\right )} e^{\left (b x\right )} \sin \left (3 \, d x + 6 \, c\right ) - {\left (b^{2} d \cos \left (3 \, c\right ) e^{a} + 9 \, d^{3} \cos \left (3 \, c\right ) e^{a} + b^{3} e^{a} \sin \left (3 \, c\right ) + 9 \, b d^{2} e^{a} \sin \left (3 \, c\right )\right )} e^{\left (b x\right )} \sin \left (d x + 4 \, c\right ) - {\left (b^{2} d \cos \left (3 \, c\right ) e^{a} + 9 \, d^{3} \cos \left (3 \, c\right ) e^{a} - b^{3} e^{a} \sin \left (3 \, c\right ) - 9 \, b d^{2} e^{a} \sin \left (3 \, c\right )\right )} e^{\left (b x\right )} \sin \left (d x - 2 \, c\right )}{8 \, {\left (b^{4} \cos \left (3 \, c\right )^{2} + b^{4} \sin \left (3 \, c\right )^{2} + 9 \, {\left (\cos \left (3 \, c\right )^{2} + \sin \left (3 \, c\right )^{2}\right )} d^{4} + 10 \, {\left (b^{2} \cos \left (3 \, c\right )^{2} + b^{2} \sin \left (3 \, c\right )^{2}\right )} d^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.94, size = 109, normalized size = 0.92 \begin {gather*} \frac {{\left (b^{2} d + 3 \, d^{3} - 3 \, {\left (b^{2} d + d^{3}\right )} \cos \left (d x + c\right )^{2}\right )} e^{\left (b x + a\right )} \sin \left (d x + c\right ) - {\left ({\left (b^{3} + b d^{2}\right )} \cos \left (d x + c\right )^{3} - {\left (b^{3} + 3 \, b d^{2}\right )} \cos \left (d x + c\right )\right )} e^{\left (b x + a\right )}}{b^{4} + 10 \, b^{2} d^{2} + 9 \, d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 3.20, size = 1040, normalized size = 8.74 \begin {gather*} \begin {cases} x e^{a} \sin ^{2}{\left (c \right )} \cos {\left (c \right )} & \text {for}\: b = 0 \wedge d = 0 \\\frac {i x e^{a} e^{- 3 i d x} \sin ^{3}{\left (c + d x \right )}}{8} + \frac {3 x e^{a} e^{- 3 i d x} \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8} - \frac {3 i x e^{a} e^{- 3 i d x} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{8} - \frac {x e^{a} e^{- 3 i d x} \cos ^{3}{\left (c + d x \right )}}{8} - \frac {e^{a} e^{- 3 i d x} \sin ^{3}{\left (c + d x \right )}}{24 d} + \frac {i e^{a} e^{- 3 i d x} \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} + \frac {i e^{a} e^{- 3 i d x} \cos ^{3}{\left (c + d x \right )}}{24 d} & \text {for}\: b = - 3 i d \\\frac {i x e^{a} e^{- i d x} \sin ^{3}{\left (c + d x \right )}}{8} + \frac {x e^{a} e^{- i d x} \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8} + \frac {i x e^{a} e^{- i d x} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{8} + \frac {x e^{a} e^{- i d x} \cos ^{3}{\left (c + d x \right )}}{8} + \frac {3 e^{a} e^{- i d x} \sin ^{3}{\left (c + d x \right )}}{8 d} - \frac {i e^{a} e^{- i d x} \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} - \frac {i e^{a} e^{- i d x} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: b = - i d \\- \frac {i x e^{a} e^{i d x} \sin ^{3}{\left (c + d x \right )}}{8} + \frac {x e^{a} e^{i d x} \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8} - \frac {i x e^{a} e^{i d x} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{8} + \frac {x e^{a} e^{i d x} \cos ^{3}{\left (c + d x \right )}}{8} + \frac {3 e^{a} e^{i d x} \sin ^{3}{\left (c + d x \right )}}{8 d} + \frac {i e^{a} e^{i d x} \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} + \frac {i e^{a} e^{i d x} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: b = i d \\- \frac {i x e^{a} e^{3 i d x} \sin ^{3}{\left (c + d x \right )}}{8} + \frac {3 x e^{a} e^{3 i d x} \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8} + \frac {3 i x e^{a} e^{3 i d x} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{8} - \frac {x e^{a} e^{3 i d x} \cos ^{3}{\left (c + d x \right )}}{8} - \frac {e^{a} e^{3 i d x} \sin ^{3}{\left (c + d x \right )}}{24 d} - \frac {i e^{a} e^{3 i d x} \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} - \frac {i e^{a} e^{3 i d x} \cos ^{3}{\left (c + d x \right )}}{24 d} & \text {for}\: b = 3 i d \\\frac {b^{3} e^{a} e^{b x} \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{b^{4} + 10 b^{2} d^{2} + 9 d^{4}} + \frac {b^{2} d e^{a} e^{b x} \sin ^{3}{\left (c + d x \right )}}{b^{4} + 10 b^{2} d^{2} + 9 d^{4}} - \frac {2 b^{2} d e^{a} e^{b x} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{b^{4} + 10 b^{2} d^{2} + 9 d^{4}} + \frac {3 b d^{2} e^{a} e^{b x} \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{b^{4} + 10 b^{2} d^{2} + 9 d^{4}} + \frac {2 b d^{2} e^{a} e^{b x} \cos ^{3}{\left (c + d x \right )}}{b^{4} + 10 b^{2} d^{2} + 9 d^{4}} + \frac {3 d^{3} e^{a} e^{b x} \sin ^{3}{\left (c + d x \right )}}{b^{4} + 10 b^{2} d^{2} + 9 d^{4}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 98, normalized size = 0.82 \begin {gather*} -\frac {1}{4} \, {\left (\frac {b \cos \left (3 \, d x + 3 \, c\right )}{b^{2} + 9 \, d^{2}} + \frac {3 \, d \sin \left (3 \, d x + 3 \, c\right )}{b^{2} + 9 \, d^{2}}\right )} e^{\left (b x + a\right )} + \frac {1}{4} \, {\left (\frac {b \cos \left (d x + c\right )}{b^{2} + d^{2}} + \frac {d \sin \left (d x + c\right )}{b^{2} + d^{2}}\right )} e^{\left (b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.01, size = 166, normalized size = 1.39 \begin {gather*} \frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (d\,x\right )-\sin \left (d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (c\right )-\sin \left (c\right )\,1{}\mathrm {i}\right )}{8\,\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 )\,1{}\mathrm {i}}{8\,\left (-3\,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 \left (c\right )+\sin \left (c\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{8\,\left (-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 )}{8\,\left (b-d\,3{}\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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