Integrand size = 19, antiderivative size = 115 \[ \int f^{a+b x} \cosh \left (d+e x+f x^2\right ) \, dx=\frac {1}{4} e^{-d+\frac {(e-b \log (f))^2}{4 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erf}\left (\frac {e+2 f x-b \log (f)}{2 \sqrt {f}}\right )+\frac {1}{4} e^{d-\frac {(e+b \log (f))^2}{4 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erfi}\left (\frac {e+2 f x+b \log (f)}{2 \sqrt {f}}\right ) \]
1/4*exp(-d+1/4*(e-b*ln(f))^2/f)*f^(-1/2+a)*erf(1/2*(e+2*f*x-b*ln(f))/f^(1/ 2))*Pi^(1/2)+1/4*exp(d-1/4*(e+b*ln(f))^2/f)*f^(-1/2+a)*erfi(1/2*(e+2*f*x+b *ln(f))/f^(1/2))*Pi^(1/2)
Time = 0.21 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.07 \[ \int f^{a+b x} \cosh \left (d+e x+f x^2\right ) \, dx=\frac {1}{4} e^{-\frac {e^2+b^2 \log ^2(f)}{4 f}} f^{a-\frac {b e+f}{2 f}} \sqrt {\pi } \left (e^{\frac {e^2+b^2 \log ^2(f)}{2 f}} \text {erf}\left (\frac {e+2 f x-b \log (f)}{2 \sqrt {f}}\right ) (\cosh (d)-\sinh (d))+\text {erfi}\left (\frac {e+2 f x+b \log (f)}{2 \sqrt {f}}\right ) (\cosh (d)+\sinh (d))\right ) \]
(f^(a - (b*e + f)/(2*f))*Sqrt[Pi]*(E^((e^2 + b^2*Log[f]^2)/(2*f))*Erf[(e + 2*f*x - b*Log[f])/(2*Sqrt[f])]*(Cosh[d] - Sinh[d]) + Erfi[(e + 2*f*x + b* Log[f])/(2*Sqrt[f])]*(Cosh[d] + Sinh[d])))/(4*E^((e^2 + b^2*Log[f]^2)/(4*f )))
Time = 0.43 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6039, 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.
\(\displaystyle \int f^{a+b x} \cosh \left (d+e x+f x^2\right ) \, dx\) |
\(\Big \downarrow \) 6039 |
\(\displaystyle \int \left (\frac {1}{2} f^{a+b x} e^{-d-e x-f x^2}+\frac {1}{2} f^{a+b x} e^{d+e x+f x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \sqrt {\pi } f^{a-\frac {1}{2}} e^{\frac {(e-b \log (f))^2}{4 f}-d} \text {erf}\left (\frac {-b \log (f)+e+2 f x}{2 \sqrt {f}}\right )+\frac {1}{4} \sqrt {\pi } f^{a-\frac {1}{2}} e^{d-\frac {(b \log (f)+e)^2}{4 f}} \text {erfi}\left (\frac {b \log (f)+e+2 f x}{2 \sqrt {f}}\right )\) |
(E^(-d + (e - b*Log[f])^2/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erf[(e + 2*f*x - b* Log[f])/(2*Sqrt[f])])/4 + (E^(d - (e + b*Log[f])^2/(4*f))*f^(-1/2 + a)*Sqr t[Pi]*Erfi[(e + 2*f*x + b*Log[f])/(2*Sqrt[f])])/4
3.4.8.3.1 Defintions of rubi rules used
Int[Cosh[v_]^(n_.)*(F_)^(u_), x_Symbol] :> Int[ExpandTrigToExp[F^u, Cosh[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 = 0.15 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.10
method | result | size |
risch | \(-\frac {\operatorname {erf}\left (-\sqrt {f}\, x +\frac {b \ln \left (f \right )-e}{2 \sqrt {f}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {b^{2} \ln \left (f \right )^{2}-2 \ln \left (f \right ) b e -4 d f +e^{2}}{4 f}}}{4 \sqrt {f}}-\frac {\operatorname {erf}\left (-\sqrt {-f}\, x +\frac {e +b \ln \left (f \right )}{2 \sqrt {-f}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}+2 \ln \left (f \right ) b e -4 d f +e^{2}}{4 f}}}{4 \sqrt {-f}}\) | \(126\) |
-1/4*erf(-f^(1/2)*x+1/2*(b*ln(f)-e)/f^(1/2))/f^(1/2)*Pi^(1/2)*f^a*exp(1/4* (b^2*ln(f)^2-2*ln(f)*b*e-4*d*f+e^2)/f)-1/4*erf(-(-f)^(1/2)*x+1/2*(e+b*ln(f ))/(-f)^(1/2))/(-f)^(1/2)*Pi^(1/2)*f^a*exp(-1/4*(b^2*ln(f)^2+2*ln(f)*b*e-4 *d*f+e^2)/f)
Leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (90) = 180\).
Time = 0.27 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.18 \[ \int f^{a+b x} \cosh \left (d+e x+f x^2\right ) \, dx=-\frac {\sqrt {\pi } \sqrt {-f} \cosh \left (\frac {b^{2} \log \left (f\right )^{2} + e^{2} - 4 \, d f + 2 \, {\left (b e - 2 \, a f\right )} \log \left (f\right )}{4 \, f}\right ) \operatorname {erf}\left (\frac {{\left (2 \, f x + b \log \left (f\right ) + e\right )} \sqrt {-f}}{2 \, f}\right ) + \sqrt {\pi } \sqrt {f} \cosh \left (\frac {b^{2} \log \left (f\right )^{2} + e^{2} - 4 \, d f - 2 \, {\left (b e - 2 \, a f\right )} \log \left (f\right )}{4 \, f}\right ) \operatorname {erf}\left (-\frac {2 \, f x - b \log \left (f\right ) + e}{2 \, \sqrt {f}}\right ) - \sqrt {\pi } \sqrt {-f} \operatorname {erf}\left (\frac {{\left (2 \, f x + b \log \left (f\right ) + e\right )} \sqrt {-f}}{2 \, f}\right ) \sinh \left (\frac {b^{2} \log \left (f\right )^{2} + e^{2} - 4 \, d f + 2 \, {\left (b e - 2 \, a f\right )} \log \left (f\right )}{4 \, f}\right ) + \sqrt {\pi } \sqrt {f} \operatorname {erf}\left (-\frac {2 \, f x - b \log \left (f\right ) + e}{2 \, \sqrt {f}}\right ) \sinh \left (\frac {b^{2} \log \left (f\right )^{2} + e^{2} - 4 \, d f - 2 \, {\left (b e - 2 \, a f\right )} \log \left (f\right )}{4 \, f}\right )}{4 \, f} \]
-1/4*(sqrt(pi)*sqrt(-f)*cosh(1/4*(b^2*log(f)^2 + e^2 - 4*d*f + 2*(b*e - 2* a*f)*log(f))/f)*erf(1/2*(2*f*x + b*log(f) + e)*sqrt(-f)/f) + sqrt(pi)*sqrt (f)*cosh(1/4*(b^2*log(f)^2 + e^2 - 4*d*f - 2*(b*e - 2*a*f)*log(f))/f)*erf( -1/2*(2*f*x - b*log(f) + e)/sqrt(f)) - sqrt(pi)*sqrt(-f)*erf(1/2*(2*f*x + b*log(f) + e)*sqrt(-f)/f)*sinh(1/4*(b^2*log(f)^2 + e^2 - 4*d*f + 2*(b*e - 2*a*f)*log(f))/f) + sqrt(pi)*sqrt(f)*erf(-1/2*(2*f*x - b*log(f) + e)/sqrt( f))*sinh(1/4*(b^2*log(f)^2 + e^2 - 4*d*f - 2*(b*e - 2*a*f)*log(f))/f))/f
\[ \int f^{a+b x} \cosh \left (d+e x+f x^2\right ) \, dx=\int f^{a + b x} \cosh {\left (d + e x + f x^{2} \right )}\, dx \]
Time = 0.21 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.89 \[ \int f^{a+b x} \cosh \left (d+e x+f x^2\right ) \, dx=\frac {1}{4} \, \sqrt {\pi } f^{a - \frac {1}{2}} \operatorname {erf}\left (\sqrt {f} x - \frac {b \log \left (f\right ) - e}{2 \, \sqrt {f}}\right ) e^{\left (-d + \frac {{\left (b \log \left (f\right ) - e\right )}^{2}}{4 \, f}\right )} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-f} x - \frac {b \log \left (f\right ) + e}{2 \, \sqrt {-f}}\right ) e^{\left (d - \frac {{\left (b \log \left (f\right ) + e\right )}^{2}}{4 \, f}\right )}}{4 \, \sqrt {-f}} \]
1/4*sqrt(pi)*f^(a - 1/2)*erf(sqrt(f)*x - 1/2*(b*log(f) - e)/sqrt(f))*e^(-d + 1/4*(b*log(f) - e)^2/f) + 1/4*sqrt(pi)*f^a*erf(sqrt(-f)*x - 1/2*(b*log( f) + e)/sqrt(-f))*e^(d - 1/4*(b*log(f) + e)^2/f)/sqrt(-f)
Time = 0.27 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.15 \[ \int f^{a+b x} \cosh \left (d+e x+f x^2\right ) \, dx=-\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-f} {\left (2 \, x + \frac {b \log \left (f\right ) + e}{f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} + 2 \, b e \log \left (f\right ) - 4 \, a f \log \left (f\right ) + e^{2} - 4 \, d f}{4 \, f}\right )}}{4 \, \sqrt {-f}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {f} {\left (2 \, x - \frac {b \log \left (f\right ) - e}{f}\right )}\right ) e^{\left (\frac {b^{2} \log \left (f\right )^{2} - 2 \, b e \log \left (f\right ) + 4 \, a f \log \left (f\right ) + e^{2} - 4 \, d f}{4 \, f}\right )}}{4 \, \sqrt {f}} \]
-1/4*sqrt(pi)*erf(-1/2*sqrt(-f)*(2*x + (b*log(f) + e)/f))*e^(-1/4*(b^2*log (f)^2 + 2*b*e*log(f) - 4*a*f*log(f) + e^2 - 4*d*f)/f)/sqrt(-f) - 1/4*sqrt( pi)*erf(-1/2*sqrt(f)*(2*x - (b*log(f) - e)/f))*e^(1/4*(b^2*log(f)^2 - 2*b* e*log(f) + 4*a*f*log(f) + e^2 - 4*d*f)/f)/sqrt(f)
Timed out. \[ \int f^{a+b x} \cosh \left (d+e x+f x^2\right ) \, dx=\int f^{a+b\,x}\,\mathrm {cosh}\left (f\,x^2+e\,x+d\right ) \,d x \]