Optimal. Leaf size=137 \[ -\frac{2^{-m-3} e^{2 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (-\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 i b (c+d x)}{d}\right )}{b}-\frac{2^{-m-3} e^{-2 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 i b (c+d x)}{d}\right )}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.155546, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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.
[In]
[Out]
Rule 4406
Rule 12
Rule 3308
Rule 2181
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*}
Mathematica [A] time = 0.0851013, size = 138, normalized size = 1.01 \[ -\frac{2^{-m-3} e^{-\frac{2 i (a d+b c)}{d}} (c+d x)^m \left (\frac{b^2 (c+d x)^2}{d^2}\right )^{-m} \left (e^{4 i a} \left (\frac{i b (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,-\frac{2 i b (c+d x)}{d}\right )+e^{\frac{4 i b c}{d}} \left (-\frac{i b (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,\frac{2 i b (c+d x)}{d}\right )\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.209, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{m}\cos \left ( bx+a \right ) \sin \left ( bx+a \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.515529, size = 246, normalized size = 1.8 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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 )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]