Integrand size = 21, antiderivative size = 183 \[ \int f^{a+c x^2} \cos \left (d+e x+f x^2\right ) \, dx=\frac {e^{-i d-\frac {e^2}{4 i f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {i e+2 x (i f-c \log (f))}{2 \sqrt {i f-c \log (f)}}\right )}{4 \sqrt {i f-c \log (f)}}+\frac {e^{i d+\frac {e^2}{4 i f+4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {i e+2 x (i f+c \log (f))}{2 \sqrt {i f+c \log (f)}}\right )}{4 \sqrt {i f+c \log (f)}} \]
1/4*exp(-I*d-e^2/(4*I*f-4*c*ln(f)))*f^a*erf(1/2*(I*e+2*x*(I*f-c*ln(f)))/(I *f-c*ln(f))^(1/2))*Pi^(1/2)/(I*f-c*ln(f))^(1/2)+1/4*exp(I*d+e^2/(4*I*f+4*c *ln(f)))*f^a*erfi(1/2*(I*e+2*x*(I*f+c*ln(f)))/(I*f+c*ln(f))^(1/2))*Pi^(1/2 )/(I*f+c*ln(f))^(1/2)
Time = 0.68 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.19 \[ \int f^{a+c x^2} \cos \left (d+e x+f x^2\right ) \, dx=\frac {\sqrt [4]{-1} e^{\frac {e^2}{4 i f+4 c \log (f)}} f^a \sqrt {\pi } \left (-e^{\frac {i e^2 f}{2 \left (f^2+c^2 \log ^2(f)\right )}} \text {erfi}\left (\frac {(-1)^{3/4} (e+2 f x+2 i c x \log (f))}{2 \sqrt {f+i c \log (f)}}\right ) (f-i c \log (f)) \sqrt {f+i c \log (f)} (\cos (d)-i \sin (d))+\text {erfi}\left (\frac {\sqrt [4]{-1} (e+2 f x-2 i c x \log (f))}{2 \sqrt {f-i c \log (f)}}\right ) \sqrt {f-i c \log (f)} (-i f+c \log (f)) (\cos (d)+i \sin (d))\right )}{4 \left (f^2+c^2 \log ^2(f)\right )} \]
((-1)^(1/4)*E^(e^2/((4*I)*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*(-(E^(((I/2)*e^2*f )/(f^2 + c^2*Log[f]^2))*Erfi[((-1)^(3/4)*(e + 2*f*x + (2*I)*c*x*Log[f]))/( 2*Sqrt[f + I*c*Log[f]])]*(f - I*c*Log[f])*Sqrt[f + I*c*Log[f]]*(Cos[d] - I *Sin[d])) + Erfi[((-1)^(1/4)*(e + 2*f*x - (2*I)*c*x*Log[f]))/(2*Sqrt[f - I *c*Log[f]])]*Sqrt[f - I*c*Log[f]]*((-I)*f + c*Log[f])*(Cos[d] + I*Sin[d])) )/(4*(f^2 + c^2*Log[f]^2))
Time = 0.53 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {4976, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int f^{a+c x^2} \cos \left (d+e x+f x^2\right ) \, dx\) |
| \(\Big \downarrow \) 4976 |
| \(\displaystyle \int \left (\frac {1}{2} f^{a+c x^2} e^{-i d-i e x-i f x^2}+\frac {1}{2} f^{a+c x^2} e^{i d+i e x+i f x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {\pi } f^a e^{-\frac {e^2}{-4 c \log (f)+4 i f}-i d} \text {erf}\left (\frac {2 x (-c \log (f)+i f)+i e}{2 \sqrt {-c \log (f)+i f}}\right )}{4 \sqrt {-c \log (f)+i f}}+\frac {\sqrt {\pi } f^a e^{\frac {e^2}{4 c \log (f)+4 i f}+i d} \text {erfi}\left (\frac {2 x (c \log (f)+i f)+i e}{2 \sqrt {c \log (f)+i f}}\right )}{4 \sqrt {c \log (f)+i f}}\) |
(E^((-I)*d - e^2/((4*I)*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(I*e + 2*x*(I*f - c*Log[f]))/(2*Sqrt[I*f - c*Log[f]])])/(4*Sqrt[I*f - c*Log[f]]) + (E^(I*d + e^2/((4*I)*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + 2*x*(I*f + c*Log[f ]))/(2*Sqrt[I*f + c*Log[f]])])/(4*Sqrt[I*f + c*Log[f]])
3.2.22.3.1 Defintions of rubi rules used
Int[Cos[v_]^(n_.)*(F_)^(u_), x_Symbol] :> Int[ExpandTrigToExp[F^u, Cos[v]^n , x], x] /; FreeQ[F, x] && (LinearQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[n, 0]
Time = 0.46 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.91
| method | result | size |
| risch | \(\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {4 i d \ln \left (f \right ) c +4 d f -e^{2}}{4 \left (c \ln \left (f \right )-i f \right )}} \operatorname {erf}\left (x \sqrt {i f -c \ln \left (f \right )}+\frac {i e}{2 \sqrt {i f -c \ln \left (f \right )}}\right )}{4 \sqrt {i f -c \ln \left (f \right )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {4 i d \ln \left (f \right ) c -4 d f +e^{2}}{4 i f +4 c \ln \left (f \right )}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )-i f}\, x +\frac {i e}{2 \sqrt {-c \ln \left (f \right )-i f}}\right )}{4 \sqrt {-c \ln \left (f \right )-i f}}\) | \(167\) |
1/4*Pi^(1/2)*f^a*exp(-1/4*(4*I*d*ln(f)*c+4*d*f-e^2)/(c*ln(f)-I*f))/(I*f-c* ln(f))^(1/2)*erf(x*(I*f-c*ln(f))^(1/2)+1/2*I*e/(I*f-c*ln(f))^(1/2))-1/4*Pi ^(1/2)*f^a*exp(1/4*(4*I*d*ln(f)*c-4*d*f+e^2)/(I*f+c*ln(f)))/(-c*ln(f)-I*f) ^(1/2)*erf(-(-c*ln(f)-I*f)^(1/2)*x+1/2*I*e/(-c*ln(f)-I*f)^(1/2))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (135) = 270\).
Time = 0.26 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.64 \[ \int f^{a+c x^2} \cos \left (d+e x+f x^2\right ) \, dx=-\frac {\sqrt {\pi } {\left (c \log \left (f\right ) - i \, f\right )} \sqrt {-c \log \left (f\right ) - i \, f} \operatorname {erf}\left (\frac {{\left (2 \, c^{2} x \log \left (f\right )^{2} + 2 \, f^{2} x + i \, c e \log \left (f\right ) + e f\right )} \sqrt {-c \log \left (f\right ) - i \, f}}{2 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right ) e^{\left (\frac {4 \, a c^{2} \log \left (f\right )^{3} + 4 i \, c^{2} d \log \left (f\right )^{2} - i \, e^{2} f + 4 i \, d f^{2} + {\left (c e^{2} + 4 \, a f^{2}\right )} \log \left (f\right )}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )} + \sqrt {\pi } {\left (c \log \left (f\right ) + i \, f\right )} \sqrt {-c \log \left (f\right ) + i \, f} \operatorname {erf}\left (\frac {{\left (2 \, c^{2} x \log \left (f\right )^{2} + 2 \, f^{2} x - i \, c e \log \left (f\right ) + e f\right )} \sqrt {-c \log \left (f\right ) + i \, f}}{2 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right ) e^{\left (\frac {4 \, a c^{2} \log \left (f\right )^{3} - 4 i \, c^{2} d \log \left (f\right )^{2} + i \, e^{2} f - 4 i \, d f^{2} + {\left (c e^{2} + 4 \, a f^{2}\right )} \log \left (f\right )}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}} \]
-1/4*(sqrt(pi)*(c*log(f) - I*f)*sqrt(-c*log(f) - I*f)*erf(1/2*(2*c^2*x*log (f)^2 + 2*f^2*x + I*c*e*log(f) + e*f)*sqrt(-c*log(f) - I*f)/(c^2*log(f)^2 + f^2))*e^(1/4*(4*a*c^2*log(f)^3 + 4*I*c^2*d*log(f)^2 - I*e^2*f + 4*I*d*f^ 2 + (c*e^2 + 4*a*f^2)*log(f))/(c^2*log(f)^2 + f^2)) + sqrt(pi)*(c*log(f) + I*f)*sqrt(-c*log(f) + I*f)*erf(1/2*(2*c^2*x*log(f)^2 + 2*f^2*x - I*c*e*lo g(f) + e*f)*sqrt(-c*log(f) + I*f)/(c^2*log(f)^2 + f^2))*e^(1/4*(4*a*c^2*lo g(f)^3 - 4*I*c^2*d*log(f)^2 + I*e^2*f - 4*I*d*f^2 + (c*e^2 + 4*a*f^2)*log( f))/(c^2*log(f)^2 + f^2)))/(c^2*log(f)^2 + f^2)
\[ \int f^{a+c x^2} \cos \left (d+e x+f x^2\right ) \, dx=\int f^{a + c x^{2}} \cos {\left (d + e x + f x^{2} \right )}\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 761 vs. \(2 (135) = 270\).
Time = 0.25 (sec) , antiderivative size = 761, normalized size of antiderivative = 4.16 \[ \int f^{a+c x^2} \cos \left (d+e x+f x^2\right ) \, dx=\frac {\sqrt {\pi } \sqrt {2 \, c^{2} \log \left (f\right )^{2} + 2 \, f^{2}} {\left ({\left (i \, f^{\frac {c e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}} f^{a} \cos \left (\frac {4 \, c^{2} d \log \left (f\right )^{2} - e^{2} f + 4 \, d f^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right ) + f^{\frac {c e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}} f^{a} \sin \left (\frac {4 \, c^{2} d \log \left (f\right )^{2} - e^{2} f + 4 \, d f^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )\right )} \operatorname {erf}\left (\frac {2 \, {\left (c \log \left (f\right ) - i \, f\right )} x - i \, e}{2 \, \sqrt {-c \log \left (f\right ) + i \, f}}\right ) + {\left (-i \, f^{\frac {c e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}} f^{a} \cos \left (\frac {4 \, c^{2} d \log \left (f\right )^{2} - e^{2} f + 4 \, d f^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right ) + f^{\frac {c e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}} f^{a} \sin \left (\frac {4 \, c^{2} d \log \left (f\right )^{2} - e^{2} f + 4 \, d f^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )\right )} \operatorname {erf}\left (\frac {2 \, {\left (c \log \left (f\right ) + i \, f\right )} x + i \, e}{2 \, \sqrt {-c \log \left (f\right ) - i \, f}}\right )\right )} \sqrt {c \log \left (f\right ) + \sqrt {c^{2} \log \left (f\right )^{2} + f^{2}}} - \sqrt {\pi } \sqrt {2 \, c^{2} \log \left (f\right )^{2} + 2 \, f^{2}} {\left ({\left (f^{\frac {c e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}} f^{a} \cos \left (\frac {4 \, c^{2} d \log \left (f\right )^{2} - e^{2} f + 4 \, d f^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right ) - i \, f^{\frac {c e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}} f^{a} \sin \left (\frac {4 \, c^{2} d \log \left (f\right )^{2} - e^{2} f + 4 \, d f^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )\right )} \operatorname {erf}\left (\frac {2 \, {\left (c \log \left (f\right ) - i \, f\right )} x - i \, e}{2 \, \sqrt {-c \log \left (f\right ) + i \, f}}\right ) + {\left (f^{\frac {c e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}} f^{a} \cos \left (\frac {4 \, c^{2} d \log \left (f\right )^{2} - e^{2} f + 4 \, d f^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right ) + i \, f^{\frac {c e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}} f^{a} \sin \left (\frac {4 \, c^{2} d \log \left (f\right )^{2} - e^{2} f + 4 \, d f^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )\right )} \operatorname {erf}\left (\frac {2 \, {\left (c \log \left (f\right ) + i \, f\right )} x + i \, e}{2 \, \sqrt {-c \log \left (f\right ) - i \, f}}\right )\right )} \sqrt {-c \log \left (f\right ) + \sqrt {c^{2} \log \left (f\right )^{2} + f^{2}}}}{8 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}} \]
1/8*(sqrt(pi)*sqrt(2*c^2*log(f)^2 + 2*f^2)*((I*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*f^a*cos(1/4*(4*c^2*d*log(f)^2 - e^2*f + 4*d*f^2)/(c^2*log(f)^2 + f ^2)) + f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*f^a*sin(1/4*(4*c^2*d*log(f)^2 - e^2*f + 4*d*f^2)/(c^2*log(f)^2 + f^2)))*erf(1/2*(2*(c*log(f) - I*f)*x - I* e)/sqrt(-c*log(f) + I*f)) + (-I*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*f^a*cos (1/4*(4*c^2*d*log(f)^2 - e^2*f + 4*d*f^2)/(c^2*log(f)^2 + f^2)) + f^(1/4*c *e^2/(c^2*log(f)^2 + f^2))*f^a*sin(1/4*(4*c^2*d*log(f)^2 - e^2*f + 4*d*f^2 )/(c^2*log(f)^2 + f^2)))*erf(1/2*(2*(c*log(f) + I*f)*x + I*e)/sqrt(-c*log( f) - I*f)))*sqrt(c*log(f) + sqrt(c^2*log(f)^2 + f^2)) - sqrt(pi)*sqrt(2*c^ 2*log(f)^2 + 2*f^2)*((f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*f^a*cos(1/4*(4*c^ 2*d*log(f)^2 - e^2*f + 4*d*f^2)/(c^2*log(f)^2 + f^2)) - I*f^(1/4*c*e^2/(c^ 2*log(f)^2 + f^2))*f^a*sin(1/4*(4*c^2*d*log(f)^2 - e^2*f + 4*d*f^2)/(c^2*l og(f)^2 + f^2)))*erf(1/2*(2*(c*log(f) - I*f)*x - I*e)/sqrt(-c*log(f) + I*f )) + (f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*f^a*cos(1/4*(4*c^2*d*log(f)^2 - e ^2*f + 4*d*f^2)/(c^2*log(f)^2 + f^2)) + I*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2 ))*f^a*sin(1/4*(4*c^2*d*log(f)^2 - e^2*f + 4*d*f^2)/(c^2*log(f)^2 + f^2))) *erf(1/2*(2*(c*log(f) + I*f)*x + I*e)/sqrt(-c*log(f) - I*f)))*sqrt(-c*log( f) + sqrt(c^2*log(f)^2 + f^2)))/(c^2*log(f)^2 + f^2)
\[ \int f^{a+c x^2} \cos \left (d+e x+f x^2\right ) \, dx=\int { f^{c x^{2} + a} \cos \left (f x^{2} + e x + d\right ) \,d x } \]
Timed out. \[ \int f^{a+c x^2} \cos \left (d+e x+f x^2\right ) \, dx=\int f^{c\,x^2+a}\,\cos \left (f\,x^2+e\,x+d\right ) \,d x \]