Optimal. Leaf size=137 \[ -\frac {2^{-m-3} e^{2 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {2 i b (c+d x)}{d}\right )}{b}-\frac {2^{-m-3} e^{-2 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {2 i b (c+d x)}{d}\right )}{b} \]
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Rubi [A] time = 0.16, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4406, 12, 3308, 2181} \[ -\frac {2^{-m-3} e^{2 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,-\frac {2 i b (c+d x)}{d}\right )}{b}-\frac {2^{-m-3} e^{-2 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,\frac {2 i b (c+d x)}{d}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2181
Rule 3308
Rule 4406
Rubi steps
\begin {align*} \int (c+d x)^m \cos (a+b x) \sin (a+b x) \, dx &=\int \frac {1}{2} (c+d x)^m \sin (2 a+2 b x) \, dx\\ &=\frac {1}{2} \int (c+d x)^m \sin (2 a+2 b x) \, dx\\ &=\frac {1}{4} i \int e^{-i (2 a+2 b x)} (c+d x)^m \, dx-\frac {1}{4} i \int e^{i (2 a+2 b x)} (c+d x)^m \, dx\\ &=-\frac {2^{-3-m} e^{2 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {2 i b (c+d x)}{d}\right )}{b}-\frac {2^{-3-m} e^{-2 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 i b (c+d x)}{d}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 138, normalized size = 1.01 \[ -\frac {2^{-m-3} e^{-\frac {2 i (a d+b c)}{d}} (c+d x)^m \left (\frac {b^2 (c+d x)^2}{d^2}\right )^{-m} \left (e^{4 i a} \left (\frac {i b (c+d x)}{d}\right )^m \Gamma \left (m+1,-\frac {2 i b (c+d x)}{d}\right )+e^{\frac {4 i b c}{d}} \left (-\frac {i b (c+d x)}{d}\right )^m \Gamma \left (m+1,\frac {2 i b (c+d x)}{d}\right )\right )}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 94, normalized size = 0.69 \[ -\frac {e^{\left (-\frac {d m \log \left (\frac {2 i \, b}{d}\right ) - 2 i \, b c + 2 i \, a d}{d}\right )} \Gamma \left (m + 1, \frac {2 i \, b d x + 2 i \, b c}{d}\right ) + e^{\left (-\frac {d m \log \left (-\frac {2 i \, b}{d}\right ) + 2 i \, b c - 2 i \, a d}{d}\right )} \Gamma \left (m + 1, \frac {-2 i \, b d x - 2 i \, b c}{d}\right )}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{m} \cos \left (b x + a\right ) \sin \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.35, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{m} \cos \left (b x +a \right ) \sin \left (b x +a \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{m} \cos \left (b x + a\right ) \sin \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \cos \left (a+b\,x\right )\,\sin \left (a+b\,x\right )\,{\left (c+d\,x\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c + d x\right )^{m} \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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