Optimal. Leaf size=64 \[ \frac {4 e^{n \cos \left (\frac {a}{2}+\frac {b x}{2}\right )}}{b n^2}-\frac {4 \cos \left (\frac {a}{2}+\frac {b x}{2}\right ) e^{n \cos \left (\frac {a}{2}+\frac {b x}{2}\right )}}{b n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 2176, 2194} \[ \frac {4 e^{n \cos \left (\frac {a}{2}+\frac {b x}{2}\right )}}{b n^2}-\frac {4 \cos \left (\frac {a}{2}+\frac {b x}{2}\right ) e^{n \cos \left (\frac {a}{2}+\frac {b x}{2}\right )}}{b n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2176
Rule 2194
Rubi steps
\begin {align*} \int e^{n \cos \left (\frac {1}{2} (a+b x)\right )} \sin (a+b x) \, dx &=-\frac {2 \operatorname {Subst}\left (\int 2 e^{n x} x \, dx,x,\cos \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b}\\ &=-\frac {4 \operatorname {Subst}\left (\int e^{n x} x \, dx,x,\cos \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b}\\ &=-\frac {4 e^{n \cos \left (\frac {a}{2}+\frac {b x}{2}\right )} \cos \left (\frac {a}{2}+\frac {b x}{2}\right )}{b n}+\frac {4 \operatorname {Subst}\left (\int e^{n x} \, dx,x,\cos \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b n}\\ &=\frac {4 e^{n \cos \left (\frac {a}{2}+\frac {b x}{2}\right )}}{b n^2}-\frac {4 e^{n \cos \left (\frac {a}{2}+\frac {b x}{2}\right )} \cos \left (\frac {a}{2}+\frac {b x}{2}\right )}{b n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 36, normalized size = 0.56 \[ -\frac {4 e^{n \cos \left (\frac {1}{2} (a+b x)\right )} \left (n \cos \left (\frac {1}{2} (a+b x)\right )-1\right )}{b n^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 33, normalized size = 0.52 \[ -\frac {4 \, {\left (n \cos \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) - 1\right )} e^{\left (n \cos \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )\right )}}{b n^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.22, size = 195, normalized size = 3.05 \[ \frac {4 \, {\left (n e^{\left (-\frac {n \tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} - n}{\tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} + 1}\right )} \tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} + e^{\left (-\frac {n \tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} - n}{\tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} + 1}\right )} \tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} - n e^{\left (-\frac {n \tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} - n}{\tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} + 1}\right )} + e^{\left (-\frac {n \tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} - n}{\tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} + 1}\right )}\right )}}{b n^{2} \tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} + b n^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.00, size = 123, normalized size = 1.92 \[ -\frac {2 \,{\mathrm e}^{n \cos \left (\frac {b x}{2}\right ) \cos \left (\frac {a}{2}\right )-n \sin \left (\frac {b x}{2}\right ) \sin \left (\frac {a}{2}\right )} {\mathrm e}^{\frac {i b x}{2}} {\mathrm e}^{\frac {i a}{2}}}{b n}-\frac {2 \,{\mathrm e}^{n \cos \left (\frac {b x}{2}\right ) \cos \left (\frac {a}{2}\right )-n \sin \left (\frac {b x}{2}\right ) \sin \left (\frac {a}{2}\right )} {\mathrm e}^{-\frac {i b x}{2}} {\mathrm e}^{-\frac {i a}{2}}}{b n}+\frac {4 \,{\mathrm e}^{n \left (\cos \left (\frac {b x}{2}\right ) \cos \left (\frac {a}{2}\right )-\sin \left (\frac {b x}{2}\right ) \sin \left (\frac {a}{2}\right )\right )}}{b \,n^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\left (n \cos \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )\right )} \sin \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.00, size = 33, normalized size = 0.52 \[ -\frac {4\,{\mathrm {e}}^{n\,\cos \left (\frac {a}{2}+\frac {b\,x}{2}\right )}\,\left (n\,\cos \left (\frac {a}{2}+\frac {b\,x}{2}\right )-1\right )}{b\,n^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{n \cos {\left (\frac {a}{2} + \frac {b x}{2} \right )}} \sin {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________