Integrand size = 21, antiderivative size = 231 \[ \int f^{a+b x+c x^2} \cos ^2(d+e x) \, dx=\frac {f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {e^{-2 i d+\frac {(2 e+i b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {2 i e-b \log (f)-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{2 i d-\frac {(2 i e+b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {2 i e+b \log (f)+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}} \]
1/8*exp(-2*I*d+1/4*(2*e+I*b*ln(f))^2/c/ln(f))*f^a*erfi(1/2*(-2*I*e+b*ln(f) +2*c*x*ln(f))/c^(1/2)/ln(f)^(1/2))*Pi^(1/2)/c^(1/2)/ln(f)^(1/2)+1/8*exp(2* I*d-1/4*(2*I*e+b*ln(f))^2/c/ln(f))*f^a*erfi(1/2*(2*I*e+b*ln(f)+2*c*x*ln(f) )/c^(1/2)/ln(f)^(1/2))*Pi^(1/2)/c^(1/2)/ln(f)^(1/2)+1/4*f^(a-1/4*b^2/c)*er fi(1/2*(2*c*x+b)*ln(f)^(1/2)/c^(1/2))*Pi^(1/2)/c^(1/2)/ln(f)^(1/2)
Time = 0.42 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.88 \[ \int f^{a+b x+c x^2} \cos ^2(d+e x) \, dx=\frac {e^{-\frac {i b e}{c}} f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \left (2 e^{\frac {i b e}{c}} \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )+e^{\frac {e (e+2 i b \log (f))}{c \log (f)}} \text {erfi}\left (\frac {-2 i e+(b+2 c x) \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cos (2 d)-i \sin (2 d))+e^{\frac {e^2}{c \log (f)}} \text {erfi}\left (\frac {2 i e+(b+2 c x) \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cos (2 d)+i \sin (2 d))\right )}{8 \sqrt {c} \sqrt {\log (f)}} \]
(f^(a - b^2/(4*c))*Sqrt[Pi]*(2*E^((I*b*e)/c)*Erfi[((b + 2*c*x)*Sqrt[Log[f] ])/(2*Sqrt[c])] + E^((e*(e + (2*I)*b*Log[f]))/(c*Log[f]))*Erfi[((-2*I)*e + (b + 2*c*x)*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])]*(Cos[2*d] - I*Sin[2*d]) + E ^(e^2/(c*Log[f]))*Erfi[((2*I)*e + (b + 2*c*x)*Log[f])/(2*Sqrt[c]*Sqrt[Log[ f]])]*(Cos[2*d] + I*Sin[2*d])))/(8*Sqrt[c]*E^((I*b*e)/c)*Sqrt[Log[f]])
Time = 0.53 (sec) , antiderivative size = 231, 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.
| \(\displaystyle \int \cos ^2(d+e x) f^{a+b x+c x^2} \, dx\) |
| \(\Big \downarrow \) 4976 |
| \(\displaystyle \int \left (\frac {1}{4} e^{-2 i d-2 i e x} f^{a+b x+c x^2}+\frac {1}{4} e^{2 i d+2 i e x} f^{a+b x+c x^2}+\frac {1}{2} f^{a+b x+c x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {\sqrt {\pi } f^a e^{\frac {(2 e+i b \log (f))^2}{4 c \log (f)}-2 i d} \text {erfi}\left (\frac {-b \log (f)-2 c x \log (f)+2 i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{2 i d-\frac {(b \log (f)+2 i e)^2}{4 c \log (f)}} \text {erfi}\left (\frac {b \log (f)+2 c x \log (f)+2 i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}\) |
(f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/ (4*Sqrt[c]*Sqrt[Log[f]]) - (E^((-2*I)*d + (2*e + I*b*Log[f])^2/(4*c*Log[f] ))*f^a*Sqrt[Pi]*Erfi[((2*I)*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[L og[f]])])/(8*Sqrt[c]*Sqrt[Log[f]]) + (E^((2*I)*d - ((2*I)*e + b*Log[f])^2/ (4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[((2*I)*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqr t[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]])
3.2.26.3.1 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 = 0.74 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.94
| method | result | size |
| risch | \(-\frac {\sqrt {\pi }\, f^{a} f^{-\frac {b^{2}}{4 c}} {\mathrm e}^{\frac {i \ln \left (f \right ) b e -2 i d \ln \left (f \right ) c +e^{2}}{\ln \left (f \right ) c}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )-2 i e}{2 \sqrt {-c \ln \left (f \right )}}\right )}{8 \sqrt {-c \ln \left (f \right )}}-\frac {\sqrt {\pi }\, f^{a} f^{-\frac {b^{2}}{4 c}} {\mathrm e}^{-\frac {i \ln \left (f \right ) b e -2 i d \ln \left (f \right ) c -e^{2}}{\ln \left (f \right ) c}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {2 i e +b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right )}{8 \sqrt {-c \ln \left (f \right )}}-\frac {\sqrt {\pi }\, f^{-\frac {b^{2}}{4 c}} f^{a} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {\ln \left (f \right ) b}{2 \sqrt {-c \ln \left (f \right )}}\right )}{4 \sqrt {-c \ln \left (f \right )}}\) | \(218\) |
-1/8*Pi^(1/2)*f^a*f^(-1/4*b^2/c)*exp((I*ln(f)*b*e-2*I*d*ln(f)*c+e^2)/ln(f) /c)/(-c*ln(f))^(1/2)*erf(-(-c*ln(f))^(1/2)*x+1/2*(b*ln(f)-2*I*e)/(-c*ln(f) )^(1/2))-1/8*Pi^(1/2)*f^a*f^(-1/4*b^2/c)*exp(-(I*ln(f)*b*e-2*I*d*ln(f)*c-e ^2)/ln(f)/c)/(-c*ln(f))^(1/2)*erf(-(-c*ln(f))^(1/2)*x+1/2*(2*I*e+b*ln(f))/ (-c*ln(f))^(1/2))-1/4*Pi^(1/2)*f^(-1/4*b^2/c)*f^a/(-c*ln(f))^(1/2)*erf(-(- c*ln(f))^(1/2)*x+1/2*ln(f)*b/(-c*ln(f))^(1/2))
Time = 0.25 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.97 \[ \int f^{a+b x+c x^2} \cos ^2(d+e x) \, dx=-\frac {\sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) - 2 i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 4 \, e^{2} + 4 \, {\left (2 i \, c d - i \, b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right )} + \sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) + 2 i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 4 \, e^{2} + 4 \, {\left (-2 i \, c d + i \, b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right )} + \frac {2 \, \sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c \log \left (f\right )}}{2 \, c}\right )}{f^{\frac {b^{2} - 4 \, a c}{4 \, c}}}}{8 \, c \log \left (f\right )} \]
-1/8*(sqrt(pi)*sqrt(-c*log(f))*erf(1/2*((2*c*x + b)*log(f) - 2*I*e)*sqrt(- c*log(f))/(c*log(f)))*e^(-1/4*((b^2 - 4*a*c)*log(f)^2 - 4*e^2 + 4*(2*I*c*d - I*b*e)*log(f))/(c*log(f))) + sqrt(pi)*sqrt(-c*log(f))*erf(1/2*((2*c*x + b)*log(f) + 2*I*e)*sqrt(-c*log(f))/(c*log(f)))*e^(-1/4*((b^2 - 4*a*c)*log (f)^2 - 4*e^2 + 4*(-2*I*c*d + I*b*e)*log(f))/(c*log(f))) + 2*sqrt(pi)*sqrt (-c*log(f))*erf(1/2*(2*c*x + b)*sqrt(-c*log(f))/c)/f^(1/4*(b^2 - 4*a*c)/c) )/(c*log(f))
\[ \int f^{a+b x+c x^2} \cos ^2(d+e x) \, dx=\int f^{a + b x + c x^{2}} \cos ^{2}{\left (d + e x \right )}\, dx \]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.26 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.73 \[ \int f^{a+b x+c x^2} \cos ^2(d+e x) \, dx=\frac {\sqrt {\pi } {\left (f^{a} {\left (\cos \left (-\frac {2 \, c d - b e}{c}\right ) - i \, \sin \left (-\frac {2 \, c d - b e}{c}\right )\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \left (f\right )}} - \frac {1}{2} \, {\left (b \log \left (f\right ) + 2 i \, e\right )} \overline {\frac {1}{\sqrt {-c \log \left (f\right )}}}\right ) e^{\left (\frac {e^{2}}{c \log \left (f\right )}\right )} + f^{a} {\left (\cos \left (-\frac {2 \, c d - b e}{c}\right ) + i \, \sin \left (-\frac {2 \, c d - b e}{c}\right )\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \left (f\right )}} - \frac {1}{2} \, {\left (b \log \left (f\right ) - 2 i \, e\right )} \overline {\frac {1}{\sqrt {-c \log \left (f\right )}}}\right ) e^{\left (\frac {e^{2}}{c \log \left (f\right )}\right )} + f^{a} {\left (\cos \left (-\frac {2 \, c d - b e}{c}\right ) - i \, \sin \left (-\frac {2 \, c d - b e}{c}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) + b \log \left (f\right ) + 2 i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (\frac {e^{2}}{c \log \left (f\right )}\right )} + f^{a} {\left (\cos \left (-\frac {2 \, c d - b e}{c}\right ) + i \, \sin \left (-\frac {2 \, c d - b e}{c}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) + b \log \left (f\right ) - 2 i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (\frac {e^{2}}{c \log \left (f\right )}\right )} + 2 \, f^{a} \operatorname {erf}\left (-\frac {1}{2} \, b \overline {\frac {1}{\sqrt {-c \log \left (f\right )}}} \log \left (f\right ) + x \overline {\sqrt {-c \log \left (f\right )}}\right ) - 2 \, f^{a} \operatorname {erf}\left (\frac {2 \, c x \log \left (f\right ) + b \log \left (f\right )}{2 \, \sqrt {-c \log \left (f\right )}}\right )\right )}}{16 \, \sqrt {-c \log \left (f\right )} f^{\frac {b^{2}}{4 \, c}}} \]
1/16*sqrt(pi)*(f^a*(cos(-(2*c*d - b*e)/c) - I*sin(-(2*c*d - b*e)/c))*erf(x *conjugate(sqrt(-c*log(f))) - 1/2*(b*log(f) + 2*I*e)*conjugate(1/sqrt(-c*l og(f))))*e^(e^2/(c*log(f))) + f^a*(cos(-(2*c*d - b*e)/c) + I*sin(-(2*c*d - b*e)/c))*erf(x*conjugate(sqrt(-c*log(f))) - 1/2*(b*log(f) - 2*I*e)*conjug ate(1/sqrt(-c*log(f))))*e^(e^2/(c*log(f))) + f^a*(cos(-(2*c*d - b*e)/c) - I*sin(-(2*c*d - b*e)/c))*erf(1/2*(2*c*x*log(f) + b*log(f) + 2*I*e)*sqrt(-c *log(f))/(c*log(f)))*e^(e^2/(c*log(f))) + f^a*(cos(-(2*c*d - b*e)/c) + I*s in(-(2*c*d - b*e)/c))*erf(1/2*(2*c*x*log(f) + b*log(f) - 2*I*e)*sqrt(-c*lo g(f))/(c*log(f)))*e^(e^2/(c*log(f))) + 2*f^a*erf(-1/2*b*conjugate(1/sqrt(- c*log(f)))*log(f) + x*conjugate(sqrt(-c*log(f)))) - 2*f^a*erf(1/2*(2*c*x*l og(f) + b*log(f))/sqrt(-c*log(f))))/(sqrt(-c*log(f))*f^(1/4*b^2/c))
\[ \int f^{a+b x+c x^2} \cos ^2(d+e x) \, dx=\int { f^{c x^{2} + b x + a} \cos \left (e x + d\right )^{2} \,d x } \]
Timed out. \[ \int f^{a+b x+c x^2} \cos ^2(d+e x) \, dx=\int f^{c\,x^2+b\,x+a}\,{\cos \left (d+e\,x\right )}^2 \,d x \]