Optimal. Leaf size=23 \[ -\frac{e^{n \cos (c (a+b x))}}{b c n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0149893, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {4335, 2194} \[ -\frac{e^{n \cos (c (a+b x))}}{b c n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4335
Rule 2194
Rubi steps
\begin{align*} \int e^{n \cos (a c+b c x)} \sin (c (a+b x)) \, dx &=-\frac{\operatorname{Subst}\left (\int e^{n x} \, dx,x,\cos (c (a+b x))\right )}{b c}\\ &=-\frac{e^{n \cos (c (a+b x))}}{b c n}\\ \end{align*}
Mathematica [A] time = 0.236888, size = 23, normalized size = 1. \[ -\frac{e^{n \cos (c (a+b x))}}{b c n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.014, size = 24, normalized size = 1. \begin{align*} -{\frac{{{\rm e}^{n\cos \left ( bcx+ac \right ) }}}{cbn}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.98337, size = 31, normalized size = 1.35 \begin{align*} -\frac{e^{\left (n \cos \left (b c x + a c\right )\right )}}{b c n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.88868, size = 45, normalized size = 1.96 \begin{align*} -\frac{e^{\left (n \cos \left (b c x + a c\right )\right )}}{b c n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 9.43634, size = 54, normalized size = 2.35 \begin{align*} \begin{cases} x e^{n \cos{\left (a c \right )}} \sin{\left (a c \right )} & \text{for}\: b = 0 \\0 & \text{for}\: c = 0 \\\begin{cases} x \sin{\left (a c \right )} & \text{for}\: b = 0 \\0 & \text{for}\: c = 0 \\- \frac{\cos{\left (a c + b c x \right )}}{b c} & \text{otherwise} \end{cases} & \text{for}\: n = 0 \\- \frac{e^{n \cos{\left (a c + b c x \right )}}}{b c n} & \text{otherwise} \end{cases} \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 c x + a c\right )\right )} \sin \left ({\left (b x + a\right )} c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]