Optimal. Leaf size=43 \[ \frac{2 e^{n \cos (a+b x)}}{b n^2}-\frac{2 \cos (a+b x) e^{n \cos (a+b x)}}{b n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0405534, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {12, 2176, 2194} \[ \frac{2 e^{n \cos (a+b x)}}{b n^2}-\frac{2 \cos (a+b x) e^{n \cos (a+b x)}}{b n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2176
Rule 2194
Rubi steps
\begin{align*} \int e^{n \cos (a+b x)} \sin (2 a+2 b x) \, dx &=-\frac{\operatorname{Subst}\left (\int 2 e^{n x} x \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{2 \operatorname{Subst}\left (\int e^{n x} x \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{2 e^{n \cos (a+b x)} \cos (a+b x)}{b n}+\frac{2 \operatorname{Subst}\left (\int e^{n x} \, dx,x,\cos (a+b x)\right )}{b n}\\ &=\frac{2 e^{n \cos (a+b x)}}{b n^2}-\frac{2 e^{n \cos (a+b x)} \cos (a+b x)}{b n}\\ \end{align*}
Mathematica [A] time = 0.148723, size = 28, normalized size = 0.65 \[ -\frac{2 e^{n \cos (a+b x)} (n \cos (a+b x)-1)}{b n^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.07, size = 106, normalized size = 2.5 \begin{align*} -{\frac{{{\rm e}^{n\cos \left ( bx \right ) \cos \left ( a \right ) -n\sin \left ( bx \right ) \sin \left ( a \right ) }}{{\rm e}^{-ibx}}{{\rm e}^{-ia}}}{bn}}-{\frac{{{\rm e}^{n\cos \left ( bx \right ) \cos \left ( a \right ) -n\sin \left ( bx \right ) \sin \left ( a \right ) }}{{\rm e}^{ibx}}{{\rm e}^{ia}}}{bn}}+2\,{\frac{{{\rm e}^{-n \left ( \sin \left ( bx \right ) \sin \left ( a \right ) -\cos \left ( bx \right ) \cos \left ( a \right ) \right ) }}}{{n}^{2}b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.0477, size = 50, normalized size = 1.16 \begin{align*} -\frac{2 \,{\left (n \cos \left (b x + a\right ) e^{\left (n \cos \left (b x + a\right )\right )} - e^{\left (n \cos \left (b x + a\right )\right )}\right )}}{b n^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.08946, size = 70, normalized size = 1.63 \begin{align*} -\frac{2 \,{\left (n \cos \left (b x + a\right ) - 1\right )} e^{\left (n \cos \left (b x + a\right )\right )}}{b n^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{n \cos{\left (a + b x \right )}} \sin{\left (2 a + 2 b x \right )}\, dx \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 e^{\left (n \cos \left (b x + a\right )\right )} \sin \left (2 \, b x + 2 \, a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]