Integrand size = 18, antiderivative size = 161 \[ \int f^{a+c x^2} \cosh ^2(d+e x) \, dx=\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {e^{-2 d-\frac {e^2}{c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e-c x \log (f)}{\sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{2 d-\frac {e^2}{c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e+c x \log (f)}{\sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}} \]
1/8*exp(-2*d-e^2/c/ln(f))*f^a*erfi((-e+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*d-e^2/c/ln(f))*f^a*erfi((e+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*erfi(x*c^(1/2)*l n(f)^(1/2))*Pi^(1/2)/c^(1/2)/ln(f)^(1/2)
Time = 0.17 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.81 \[ \int f^{a+c x^2} \cosh ^2(d+e x) \, dx=\frac {e^{-\frac {e^2}{c \log (f)}} f^a \sqrt {\pi } \left (2 e^{\frac {e^2}{c \log (f)}} \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )+\text {erfi}\left (\frac {-e+c x \log (f)}{\sqrt {c} \sqrt {\log (f)}}\right ) (\cosh (2 d)-\sinh (2 d))+\text {erfi}\left (\frac {e+c x \log (f)}{\sqrt {c} \sqrt {\log (f)}}\right ) (\cosh (2 d)+\sinh (2 d))\right )}{8 \sqrt {c} \sqrt {\log (f)}} \]
(f^a*Sqrt[Pi]*(2*E^(e^2/(c*Log[f]))*Erfi[Sqrt[c]*x*Sqrt[Log[f]]] + Erfi[(- e + c*x*Log[f])/(Sqrt[c]*Sqrt[Log[f]])]*(Cosh[2*d] - Sinh[2*d]) + Erfi[(e + c*x*Log[f])/(Sqrt[c]*Sqrt[Log[f]])]*(Cosh[2*d] + Sinh[2*d])))/(8*Sqrt[c] *E^(e^2/(c*Log[f]))*Sqrt[Log[f]])
Time = 0.43 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.
\(\displaystyle \int f^{a+c x^2} \cosh ^2(d+e x) \, dx\) |
\(\Big \downarrow \) 6039 |
\(\displaystyle \int \left (\frac {1}{4} e^{-2 d-2 e x} f^{a+c x^2}+\frac {1}{4} e^{2 d+2 e x} f^{a+c x^2}+\frac {1}{2} f^{a+c x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\sqrt {\pi } f^a e^{-\frac {e^2}{c \log (f)}-2 d} \text {erfi}\left (\frac {e-c x \log (f)}{\sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{2 d-\frac {e^2}{c \log (f)}} \text {erfi}\left (\frac {c x \log (f)+e}{\sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}}\) |
(f^a*Sqrt[Pi]*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(4*Sqrt[c]*Sqrt[Log[f]]) - (E^ (-2*d - e^2/(c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e - c*x*Log[f])/(Sqrt[c]*Sqrt[L og[f]])])/(8*Sqrt[c]*Sqrt[Log[f]]) + (E^(2*d - e^2/(c*Log[f]))*f^a*Sqrt[Pi ]*Erfi[(e + c*x*Log[f])/(Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]])
3.4.12.3.1 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 = 0.33 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.86
method | result | size |
risch | \(\frac {\operatorname {erf}\left (\sqrt {-c \ln \left (f \right )}\, x +\frac {e}{\sqrt {-c \ln \left (f \right )}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {2 d \ln \left (f \right ) c +e^{2}}{\ln \left (f \right ) c}}}{8 \sqrt {-c \ln \left (f \right )}}-\frac {\operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {e}{\sqrt {-c \ln \left (f \right )}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {2 d \ln \left (f \right ) c -e^{2}}{\ln \left (f \right ) c}}}{8 \sqrt {-c \ln \left (f \right )}}+\frac {f^{a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-c \ln \left (f \right )}\, x \right )}{4 \sqrt {-c \ln \left (f \right )}}\) | \(139\) |
1/8*erf((-c*ln(f))^(1/2)*x+e/(-c*ln(f))^(1/2))/(-c*ln(f))^(1/2)*Pi^(1/2)*f ^a*exp(-(2*d*ln(f)*c+e^2)/ln(f)/c)-1/8*erf(-(-c*ln(f))^(1/2)*x+e/(-c*ln(f) )^(1/2))/(-c*ln(f))^(1/2)*Pi^(1/2)*f^a*exp((2*d*ln(f)*c-e^2)/ln(f)/c)+1/4* f^a*Pi^(1/2)/(-c*ln(f))^(1/2)*erf((-c*ln(f))^(1/2)*x)
Time = 0.26 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.50 \[ \int f^{a+c x^2} \cosh ^2(d+e x) \, dx=-\frac {2 \, \sqrt {-c \log \left (f\right )} {\left (\sqrt {\pi } \cosh \left (a \log \left (f\right )\right ) + \sqrt {\pi } \sinh \left (a \log \left (f\right )\right )\right )} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x\right ) + \sqrt {-c \log \left (f\right )} {\left (\sqrt {\pi } \cosh \left (\frac {a c \log \left (f\right )^{2} + 2 \, c d \log \left (f\right ) - e^{2}}{c \log \left (f\right )}\right ) + \sqrt {\pi } \sinh \left (\frac {a c \log \left (f\right )^{2} + 2 \, c d \log \left (f\right ) - e^{2}}{c \log \left (f\right )}\right )\right )} \operatorname {erf}\left (\frac {{\left (c x \log \left (f\right ) + e\right )} \sqrt {-c \log \left (f\right )}}{c \log \left (f\right )}\right ) + \sqrt {-c \log \left (f\right )} {\left (\sqrt {\pi } \cosh \left (\frac {a c \log \left (f\right )^{2} - 2 \, c d \log \left (f\right ) - e^{2}}{c \log \left (f\right )}\right ) + \sqrt {\pi } \sinh \left (\frac {a c \log \left (f\right )^{2} - 2 \, c d \log \left (f\right ) - e^{2}}{c \log \left (f\right )}\right )\right )} \operatorname {erf}\left (\frac {{\left (c x \log \left (f\right ) - e\right )} \sqrt {-c \log \left (f\right )}}{c \log \left (f\right )}\right )}{8 \, c \log \left (f\right )} \]
-1/8*(2*sqrt(-c*log(f))*(sqrt(pi)*cosh(a*log(f)) + sqrt(pi)*sinh(a*log(f)) )*erf(sqrt(-c*log(f))*x) + sqrt(-c*log(f))*(sqrt(pi)*cosh((a*c*log(f)^2 + 2*c*d*log(f) - e^2)/(c*log(f))) + sqrt(pi)*sinh((a*c*log(f)^2 + 2*c*d*log( f) - e^2)/(c*log(f))))*erf((c*x*log(f) + e)*sqrt(-c*log(f))/(c*log(f))) + sqrt(-c*log(f))*(sqrt(pi)*cosh((a*c*log(f)^2 - 2*c*d*log(f) - e^2)/(c*log( f))) + sqrt(pi)*sinh((a*c*log(f)^2 - 2*c*d*log(f) - e^2)/(c*log(f))))*erf( (c*x*log(f) - e)*sqrt(-c*log(f))/(c*log(f))))/(c*log(f))
\[ \int f^{a+c x^2} \cosh ^2(d+e x) \, dx=\int f^{a + c x^{2}} \cosh ^{2}{\left (d + e x \right )}\, dx \]
Time = 0.21 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.81 \[ \int f^{a+c x^2} \cosh ^2(d+e x) \, dx=\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x - \frac {e}{\sqrt {-c \log \left (f\right )}}\right ) e^{\left (2 \, d - \frac {e^{2}}{c \log \left (f\right )}\right )}}{8 \, \sqrt {-c \log \left (f\right )}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x + \frac {e}{\sqrt {-c \log \left (f\right )}}\right ) e^{\left (-2 \, d - \frac {e^{2}}{c \log \left (f\right )}\right )}}{8 \, \sqrt {-c \log \left (f\right )}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x\right )}{4 \, \sqrt {-c \log \left (f\right )}} \]
1/8*sqrt(pi)*f^a*erf(sqrt(-c*log(f))*x - e/sqrt(-c*log(f)))*e^(2*d - e^2/( c*log(f)))/sqrt(-c*log(f)) + 1/8*sqrt(pi)*f^a*erf(sqrt(-c*log(f))*x + e/sq rt(-c*log(f)))*e^(-2*d - e^2/(c*log(f)))/sqrt(-c*log(f)) + 1/4*sqrt(pi)*f^ a*erf(sqrt(-c*log(f))*x)/sqrt(-c*log(f))
Time = 0.27 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.93 \[ \int f^{a+c x^2} \cosh ^2(d+e x) \, dx=-\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (-\sqrt {-c \log \left (f\right )} x\right )}{4 \, \sqrt {-c \log \left (f\right )}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (f\right )} {\left (x + \frac {e}{c \log \left (f\right )}\right )}\right ) e^{\left (\frac {a c \log \left (f\right )^{2} + 2 \, c d \log \left (f\right ) - e^{2}}{c \log \left (f\right )}\right )}}{8 \, \sqrt {-c \log \left (f\right )}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (f\right )} {\left (x - \frac {e}{c \log \left (f\right )}\right )}\right ) e^{\left (\frac {a c \log \left (f\right )^{2} - 2 \, c d \log \left (f\right ) - e^{2}}{c \log \left (f\right )}\right )}}{8 \, \sqrt {-c \log \left (f\right )}} \]
-1/4*sqrt(pi)*f^a*erf(-sqrt(-c*log(f))*x)/sqrt(-c*log(f)) - 1/8*sqrt(pi)*e rf(-sqrt(-c*log(f))*(x + e/(c*log(f))))*e^((a*c*log(f)^2 + 2*c*d*log(f) - e^2)/(c*log(f)))/sqrt(-c*log(f)) - 1/8*sqrt(pi)*erf(-sqrt(-c*log(f))*(x - e/(c*log(f))))*e^((a*c*log(f)^2 - 2*c*d*log(f) - e^2)/(c*log(f)))/sqrt(-c* log(f))
Timed out. \[ \int f^{a+c x^2} \cosh ^2(d+e x) \, dx=\int f^{c\,x^2+a}\,{\mathrm {cosh}\left (d+e\,x\right )}^2 \,d x \]