Optimal. Leaf size=275 \[ -\frac{i e^{i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (-\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{i b (c+d x)}{d}\right )}{8 b}+\frac{i 3^{-m-1} e^{3 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (-\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{3 i b (c+d x)}{d}\right )}{8 b}+\frac{i e^{-i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{i b (c+d x)}{d}\right )}{8 b}-\frac{i 3^{-m-1} e^{-3 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{3 i b (c+d x)}{d}\right )}{8 b} \]
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Rubi [A] time = 0.32994, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {4406, 3307, 2181} \[ -\frac{i e^{i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (-\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{i b (c+d x)}{d}\right )}{8 b}+\frac{i 3^{-m-1} e^{3 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (-\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{3 i b (c+d x)}{d}\right )}{8 b}+\frac{i e^{-i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{i b (c+d x)}{d}\right )}{8 b}-\frac{i 3^{-m-1} e^{-3 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{3 i b (c+d x)}{d}\right )}{8 b} \]
Antiderivative was successfully verified.
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Rule 4406
Rule 3307
Rule 2181
Rubi steps
\begin{align*} \int (c+d x)^m \cos (a+b x) \sin ^2(a+b x) \, dx &=\int \left (\frac{1}{4} (c+d x)^m \cos (a+b x)-\frac{1}{4} (c+d x)^m \cos (3 a+3 b x)\right ) \, dx\\ &=\frac{1}{4} \int (c+d x)^m \cos (a+b x) \, dx-\frac{1}{4} \int (c+d x)^m \cos (3 a+3 b x) \, dx\\ &=\frac{1}{8} \int e^{-i (a+b x)} (c+d x)^m \, dx+\frac{1}{8} \int e^{i (a+b x)} (c+d x)^m \, dx-\frac{1}{8} \int e^{-i (3 a+3 b x)} (c+d x)^m \, dx-\frac{1}{8} \int e^{i (3 a+3 b x)} (c+d x)^m \, dx\\ &=-\frac{i e^{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{i b (c+d x)}{d}\right )}{8 b}+\frac{i e^{-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{i b (c+d x)}{d}\right )}{8 b}+\frac{i 3^{-1-m} e^{3 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{3 i b (c+d x)}{d}\right )}{8 b}-\frac{i 3^{-1-m} e^{-3 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{3 i b (c+d x)}{d}\right )}{8 b}\\ \end{align*}
Mathematica [A] time = 0.722672, size = 237, normalized size = 0.86 \[ -\frac{i e^{-\frac{3 i (a d+b c)}{d}} (c+d x)^m \left (\left (\frac{i b (c+d x)}{d}\right )^{-m} \left (3^{-m} \left (e^{\frac{6 i b c}{d}} \text{Gamma}\left (m+1,\frac{3 i b (c+d x)}{d}\right )-e^{6 i a} \left (\frac{i b (c+d x)}{d}\right )^{2 m} \left (\frac{b^2 (c+d x)^2}{d^2}\right )^{-m} \text{Gamma}\left (m+1,-\frac{3 i b (c+d x)}{d}\right )\right )-3 e^{2 i a+\frac{4 i b c}{d}} \text{Gamma}\left (m+1,\frac{i b (c+d x)}{d}\right )\right )+3 e^{2 i \left (2 a+\frac{b c}{d}\right )} \left (-\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{i b (c+d x)}{d}\right )\right )}{24 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.335, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{m}\cos \left ( bx+a \right ) \left ( \sin \left ( bx+a \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{m} \cos \left (b x + a\right ) \sin \left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.553723, size = 471, normalized size = 1.71 \begin{align*} \frac{-i \, e^{\left (-\frac{d m \log \left (\frac{3 i \, b}{d}\right ) - 3 i \, b c + 3 i \, a d}{d}\right )} \Gamma \left (m + 1, \frac{3 i \, b d x + 3 i \, b c}{d}\right ) + 3 i \, e^{\left (-\frac{d m \log \left (\frac{i \, b}{d}\right ) - i \, b c + i \, a d}{d}\right )} \Gamma \left (m + 1, \frac{i \, b d x + i \, b c}{d}\right ) - 3 i \, e^{\left (-\frac{d m \log \left (-\frac{i \, b}{d}\right ) + i \, b c - i \, a d}{d}\right )} \Gamma \left (m + 1, \frac{-i \, b d x - i \, b c}{d}\right ) + i \, e^{\left (-\frac{d m \log \left (-\frac{3 i \, b}{d}\right ) + 3 i \, b c - 3 i \, a d}{d}\right )} \Gamma \left (m + 1, \frac{-3 i \, b d x - 3 i \, b c}{d}\right )}{24 \, b} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{m} \cos \left (b x + a\right ) \sin \left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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