Optimal. Leaf size=63 \[ \frac {b e^{a+b x} \sin (2 c+2 d x)}{2 \left (b^2+4 d^2\right )}-\frac {d e^{a+b x} \cos (2 c+2 d x)}{b^2+4 d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4469, 12, 4432} \[ \frac {b e^{a+b x} \sin (2 c+2 d x)}{2 \left (b^2+4 d^2\right )}-\frac {d e^{a+b x} \cos (2 c+2 d x)}{b^2+4 d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 4432
Rule 4469
Rubi steps
\begin {align*} \int e^{a+b x} \cos (c+d x) \sin (c+d x) \, dx &=\int \frac {1}{2} e^{a+b x} \sin (2 c+2 d x) \, dx\\ &=\frac {1}{2} \int e^{a+b x} \sin (2 c+2 d x) \, dx\\ &=-\frac {d e^{a+b x} \cos (2 c+2 d x)}{b^2+4 d^2}+\frac {b e^{a+b x} \sin (2 c+2 d x)}{2 \left (b^2+4 d^2\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 44, normalized size = 0.70 \[ \frac {e^{a+b x} (b \sin (2 (c+d x))-2 d \cos (2 (c+d x)))}{2 \left (b^2+4 d^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.61, size = 56, normalized size = 0.89 \[ \frac {b \cos \left (d x + c\right ) e^{\left (b x + a\right )} \sin \left (d x + c\right ) - {\left (2 \, d \cos \left (d x + c\right )^{2} - d\right )} e^{\left (b x + a\right )}}{b^{2} + 4 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.13, size = 55, normalized size = 0.87 \[ -\frac {1}{2} \, {\left (\frac {2 \, d \cos \left (2 \, d x + 2 \, c\right )}{b^{2} + 4 \, d^{2}} - \frac {b \sin \left (2 \, d x + 2 \, c\right )}{b^{2} + 4 \, d^{2}}\right )} e^{\left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 60, normalized size = 0.95 \[ -\frac {d \,{\mathrm e}^{b x +a} \cos \left (2 d x +2 c \right )}{b^{2}+4 d^{2}}+\frac {b \,{\mathrm e}^{b x +a} \sin \left (2 d x +2 c \right )}{2 b^{2}+8 d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 44, normalized size = 0.70 \[ -\frac {{\left (2 \, d \cos \left (2 \, d x + 2 \, c\right ) - b \sin \left (2 \, d x + 2 \, c\right )\right )} e^{\left (b x + a\right )}}{2 \, {\left (b^{2} + 4 \, d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.50, size = 46, normalized size = 0.73 \[ -\frac {{\mathrm {e}}^{a+b\,x}\,\left (2\,d\,\cos \left (2\,c+2\,d\,x\right )-b\,\sin \left (2\,c+2\,d\,x\right )\right )}{2\,\left (b^2+4\,d^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 8.52, size = 325, normalized size = 5.16 \[ \begin {cases} x e^{a} \sin {\relax (c )} \cos {\relax (c )} & \text {for}\: b = 0 \wedge d = 0 \\\frac {i x e^{a} e^{- 2 i d x} \sin ^{2}{\left (c + d x \right )}}{4} + \frac {x e^{a} e^{- 2 i d x} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2} - \frac {i x e^{a} e^{- 2 i d x} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {i e^{a} e^{- 2 i d x} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} & \text {for}\: b = - 2 i d \\- \frac {i x e^{a} e^{2 i d x} \sin ^{2}{\left (c + d x \right )}}{4} + \frac {x e^{a} e^{2 i d x} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2} + \frac {i x e^{a} e^{2 i d x} \cos ^{2}{\left (c + d x \right )}}{4} - \frac {i e^{a} e^{2 i d x} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} & \text {for}\: b = 2 i d \\\frac {b e^{a} e^{b x} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{b^{2} + 4 d^{2}} + \frac {d e^{a} e^{b x} \sin ^{2}{\left (c + d x \right )}}{b^{2} + 4 d^{2}} - \frac {d e^{a} e^{b x} \cos ^{2}{\left (c + d x \right )}}{b^{2} + 4 d^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________