Integrand size = 23, antiderivative size = 225 \[ \int f^{a+b x+c x^2} \cosh ^2\left (d+f x^2\right ) \, dx=\frac {f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {e^{-2 d+\frac {b^2 \log ^2(f)}{8 f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {b \log (f)-2 x (2 f-c \log (f))}{2 \sqrt {2 f-c \log (f)}}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {e^{2 d-\frac {b^2 \log ^2(f)}{8 f+4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {b \log (f)+2 x (2 f+c \log (f))}{2 \sqrt {2 f+c \log (f)}}\right )}{8 \sqrt {2 f+c \log (f)}} \]
1/4*f^(a-1/4*b^2/c)*erfi(1/2*(2*c*x+b)*ln(f)^(1/2)/c^(1/2))*Pi^(1/2)/c^(1/ 2)/ln(f)^(1/2)-1/8*exp(-2*d+b^2*ln(f)^2/(8*f-4*c*ln(f)))*f^a*erf(1/2*(b*ln (f)-2*x*(2*f-c*ln(f)))/(2*f-c*ln(f))^(1/2))*Pi^(1/2)/(2*f-c*ln(f))^(1/2)+1 /8*exp(2*d-b^2*ln(f)^2/(8*f+4*c*ln(f)))*f^a*erfi(1/2*(b*ln(f)+2*x*(2*f+c*l n(f)))/(2*f+c*ln(f))^(1/2))*Pi^(1/2)/(2*f+c*ln(f))^(1/2)
Time = 1.58 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.14 \[ \int f^{a+b x+c x^2} \cosh ^2\left (d+f x^2\right ) \, dx=\frac {1}{8} f^a \sqrt {\pi } \left (\frac {2 f^{-\frac {b^2}{4 c}} \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{\sqrt {c} \sqrt {\log (f)}}-\frac {e^{-\frac {b^2 \log ^2(f)}{8 f+4 c \log (f)}} \left (e^{\frac {b^2 f \log ^2(f)}{4 f^2-c^2 \log ^2(f)}} \text {erf}\left (\frac {4 f x-(b+2 c x) \log (f)}{2 \sqrt {2 f-c \log (f)}}\right ) \sqrt {2 f-c \log (f)} (2 f+c \log (f)) (\cosh (2 d)-\sinh (2 d))+\text {erfi}\left (\frac {4 f x+(b+2 c x) \log (f)}{2 \sqrt {2 f+c \log (f)}}\right ) (2 f-c \log (f)) \sqrt {2 f+c \log (f)} (\cosh (2 d)+\sinh (2 d))\right )}{-4 f^2+c^2 \log ^2(f)}\right ) \]
(f^a*Sqrt[Pi]*((2*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(Sqrt[c]*f ^(b^2/(4*c))*Sqrt[Log[f]]) - (E^((b^2*f*Log[f]^2)/(4*f^2 - c^2*Log[f]^2))* Erf[(4*f*x - (b + 2*c*x)*Log[f])/(2*Sqrt[2*f - c*Log[f]])]*Sqrt[2*f - c*Lo g[f]]*(2*f + c*Log[f])*(Cosh[2*d] - Sinh[2*d]) + Erfi[(4*f*x + (b + 2*c*x) *Log[f])/(2*Sqrt[2*f + c*Log[f]])]*(2*f - c*Log[f])*Sqrt[2*f + c*Log[f]]*( Cosh[2*d] + Sinh[2*d]))/(E^((b^2*Log[f]^2)/(8*f + 4*c*Log[f]))*(-4*f^2 + c ^2*Log[f]^2))))/8
Time = 0.64 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, 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.
| \(\displaystyle \int \cosh ^2\left (d+f x^2\right ) f^{a+b x+c x^2} \, dx\) |
| \(\Big \downarrow \) 6039 |
| \(\displaystyle \int \left (\frac {1}{4} e^{-2 d-2 f x^2} f^{a+b x+c x^2}+\frac {1}{4} e^{2 d+2 f x^2} f^{a+b x+c x^2}+\frac {1}{2} f^{a+b x+c x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {\pi } f^a e^{\frac {b^2 \log ^2(f)}{8 f-4 c \log (f)}-2 d} \text {erf}\left (\frac {b \log (f)-2 x (2 f-c \log (f))}{2 \sqrt {2 f-c \log (f)}}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {\sqrt {\pi } f^a e^{2 d-\frac {b^2 \log ^2(f)}{4 c \log (f)+8 f}} \text {erfi}\left (\frac {b \log (f)+2 x (c \log (f)+2 f)}{2 \sqrt {c \log (f)+2 f}}\right )}{8 \sqrt {c \log (f)+2 f}}+\frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}\) |
(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*d + (b^2*Log[f]^2)/(8*f - 4*c*Log[f]))*f ^a*Sqrt[Pi]*Erf[(b*Log[f] - 2*x*(2*f - c*Log[f]))/(2*Sqrt[2*f - c*Log[f]]) ])/(8*Sqrt[2*f - c*Log[f]]) + (E^(2*d - (b^2*Log[f]^2)/(8*f + 4*c*Log[f])) *f^a*Sqrt[Pi]*Erfi[(b*Log[f] + 2*x*(2*f + c*Log[f]))/(2*Sqrt[2*f + c*Log[f ]])])/(8*Sqrt[2*f + c*Log[f]])
3.4.24.3.1 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 = 0.40 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.96
| method | result | size |
| risch | \(-\frac {\operatorname {erf}\left (-x \sqrt {2 f -c \ln \left (f \right )}+\frac {\ln \left (f \right ) b}{2 \sqrt {2 f -c \ln \left (f \right )}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}+8 d \ln \left (f \right ) c -16 d f}{4 \left (c \ln \left (f \right )-2 f \right )}}}{8 \sqrt {2 f -c \ln \left (f \right )}}-\frac {\operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )-2 f}\, x +\frac {\ln \left (f \right ) b}{2 \sqrt {-c \ln \left (f \right )-2 f}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}-8 d \ln \left (f \right ) c -16 d f}{4 \left (2 f +c \ln \left (f \right )\right )}}}{8 \sqrt {-c \ln \left (f \right )-2 f}}-\frac {f^{a} \sqrt {\pi }\, f^{-\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {\ln \left (f \right ) b}{2 \sqrt {-c \ln \left (f \right )}}\right )}{4 \sqrt {-c \ln \left (f \right )}}\) | \(217\) |
-1/8*erf(-x*(2*f-c*ln(f))^(1/2)+1/2*ln(f)*b/(2*f-c*ln(f))^(1/2))/(2*f-c*ln (f))^(1/2)*Pi^(1/2)*f^a*exp(-1/4*(b^2*ln(f)^2+8*d*ln(f)*c-16*d*f)/(c*ln(f) -2*f))-1/8*erf(-(-c*ln(f)-2*f)^(1/2)*x+1/2*ln(f)*b/(-c*ln(f)-2*f)^(1/2))/( -c*ln(f)-2*f)^(1/2)*Pi^(1/2)*f^a*exp(-1/4*(b^2*ln(f)^2-8*d*ln(f)*c-16*d*f) /(2*f+c*ln(f)))-1/4*f^a*Pi^(1/2)*f^(-1/4*b^2/c)/(-c*ln(f))^(1/2)*erf(-(-c* ln(f))^(1/2)*x+1/2*ln(f)*b/(-c*ln(f))^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (185) = 370\).
Time = 0.27 (sec) , antiderivative size = 466, normalized size of antiderivative = 2.07 \[ \int f^{a+b x+c x^2} \cosh ^2\left (d+f x^2\right ) \, dx=-\frac {{\left (\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} + 2 \, c f \log \left (f\right )\right )} \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 16 \, d f + 8 \, {\left (c d + a f\right )} \log \left (f\right )}{4 \, {\left (c \log \left (f\right ) - 2 \, f\right )}}\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} + 2 \, c f \log \left (f\right )\right )} \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 16 \, d f + 8 \, {\left (c d + a f\right )} \log \left (f\right )}{4 \, {\left (c \log \left (f\right ) - 2 \, f\right )}}\right )\right )} \sqrt {-c \log \left (f\right ) + 2 \, f} \operatorname {erf}\left (-\frac {{\left (4 \, f x - {\left (2 \, c x + b\right )} \log \left (f\right )\right )} \sqrt {-c \log \left (f\right ) + 2 \, f}}{2 \, {\left (c \log \left (f\right ) - 2 \, f\right )}}\right ) + {\left (\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 2 \, c f \log \left (f\right )\right )} \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 16 \, d f - 8 \, {\left (c d + a f\right )} \log \left (f\right )}{4 \, {\left (c \log \left (f\right ) + 2 \, f\right )}}\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 2 \, c f \log \left (f\right )\right )} \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 16 \, d f - 8 \, {\left (c d + a f\right )} \log \left (f\right )}{4 \, {\left (c \log \left (f\right ) + 2 \, f\right )}}\right )\right )} \sqrt {-c \log \left (f\right ) - 2 \, f} \operatorname {erf}\left (\frac {{\left (4 \, f x + {\left (2 \, c x + b\right )} \log \left (f\right )\right )} \sqrt {-c \log \left (f\right ) - 2 \, f}}{2 \, {\left (c \log \left (f\right ) + 2 \, f\right )}}\right ) + 2 \, {\left (\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 4 \, f^{2}\right )} \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )}{4 \, c}\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 4 \, f^{2}\right )} \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )}{4 \, c}\right )\right )} \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 )}{8 \, {\left (c^{3} \log \left (f\right )^{3} - 4 \, c f^{2} \log \left (f\right )\right )}} \]
-1/8*((sqrt(pi)*(c^2*log(f)^2 + 2*c*f*log(f))*cosh(-1/4*((b^2 - 4*a*c)*log (f)^2 - 16*d*f + 8*(c*d + a*f)*log(f))/(c*log(f) - 2*f)) + sqrt(pi)*(c^2*l og(f)^2 + 2*c*f*log(f))*sinh(-1/4*((b^2 - 4*a*c)*log(f)^2 - 16*d*f + 8*(c* d + a*f)*log(f))/(c*log(f) - 2*f)))*sqrt(-c*log(f) + 2*f)*erf(-1/2*(4*f*x - (2*c*x + b)*log(f))*sqrt(-c*log(f) + 2*f)/(c*log(f) - 2*f)) + (sqrt(pi)* (c^2*log(f)^2 - 2*c*f*log(f))*cosh(-1/4*((b^2 - 4*a*c)*log(f)^2 - 16*d*f - 8*(c*d + a*f)*log(f))/(c*log(f) + 2*f)) + sqrt(pi)*(c^2*log(f)^2 - 2*c*f* log(f))*sinh(-1/4*((b^2 - 4*a*c)*log(f)^2 - 16*d*f - 8*(c*d + a*f)*log(f)) /(c*log(f) + 2*f)))*sqrt(-c*log(f) - 2*f)*erf(1/2*(4*f*x + (2*c*x + b)*log (f))*sqrt(-c*log(f) - 2*f)/(c*log(f) + 2*f)) + 2*(sqrt(pi)*(c^2*log(f)^2 - 4*f^2)*cosh(-1/4*(b^2 - 4*a*c)*log(f)/c) + sqrt(pi)*(c^2*log(f)^2 - 4*f^2 )*sinh(-1/4*(b^2 - 4*a*c)*log(f)/c))*sqrt(-c*log(f))*erf(1/2*(2*c*x + b)*s qrt(-c*log(f))/c))/(c^3*log(f)^3 - 4*c*f^2*log(f))
\[ \int f^{a+b x+c x^2} \cosh ^2\left (d+f x^2\right ) \, dx=\int f^{a + b x + c x^{2}} \cosh ^{2}{\left (d + f x^{2} \right )}\, dx \]
Time = 0.21 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.88 \[ \int f^{a+b x+c x^2} \cosh ^2\left (d+f x^2\right ) \, dx=\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - 2 \, f} x - \frac {b \log \left (f\right )}{2 \, \sqrt {-c \log \left (f\right ) - 2 \, f}}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2}}{4 \, {\left (c \log \left (f\right ) + 2 \, f\right )}} + 2 \, d\right )}}{8 \, \sqrt {-c \log \left (f\right ) - 2 \, f}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + 2 \, f} x - \frac {b \log \left (f\right )}{2 \, \sqrt {-c \log \left (f\right ) + 2 \, f}}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2}}{4 \, {\left (c \log \left (f\right ) - 2 \, f\right )}} - 2 \, d\right )}}{8 \, \sqrt {-c \log \left (f\right ) + 2 \, f}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x - \frac {b \log \left (f\right )}{2 \, \sqrt {-c \log \left (f\right )}}\right )}{4 \, \sqrt {-c \log \left (f\right )} f^{\frac {b^{2}}{4 \, c}}} \]
1/8*sqrt(pi)*f^a*erf(sqrt(-c*log(f) - 2*f)*x - 1/2*b*log(f)/sqrt(-c*log(f) - 2*f))*e^(-1/4*b^2*log(f)^2/(c*log(f) + 2*f) + 2*d)/sqrt(-c*log(f) - 2*f ) + 1/8*sqrt(pi)*f^a*erf(sqrt(-c*log(f) + 2*f)*x - 1/2*b*log(f)/sqrt(-c*lo g(f) + 2*f))*e^(-1/4*b^2*log(f)^2/(c*log(f) - 2*f) - 2*d)/sqrt(-c*log(f) + 2*f) + 1/4*sqrt(pi)*f^a*erf(sqrt(-c*log(f))*x - 1/2*b*log(f)/sqrt(-c*log( f)))/(sqrt(-c*log(f))*f^(1/4*b^2/c))
Time = 0.29 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.06 \[ \int f^{a+b x+c x^2} \cosh ^2\left (d+f x^2\right ) \, dx=-\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) - 2 \, f} {\left (2 \, x + \frac {b \log \left (f\right )}{c \log \left (f\right ) + 2 \, f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} - 8 \, c d \log \left (f\right ) - 8 \, a f \log \left (f\right ) - 16 \, d f}{4 \, {\left (c \log \left (f\right ) + 2 \, f\right )}}\right )}}{8 \, \sqrt {-c \log \left (f\right ) - 2 \, f}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) + 2 \, f} {\left (2 \, x + \frac {b \log \left (f\right )}{c \log \left (f\right ) - 2 \, f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} + 8 \, c d \log \left (f\right ) + 8 \, a f \log \left (f\right ) - 16 \, d f}{4 \, {\left (c \log \left (f\right ) - 2 \, f\right )}}\right )}}{8 \, \sqrt {-c \log \left (f\right ) + 2 \, f}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right ) - 4 \, a c \log \left (f\right )}{4 \, c}\right )}}{4 \, \sqrt {-c \log \left (f\right )}} \]
-1/8*sqrt(pi)*erf(-1/2*sqrt(-c*log(f) - 2*f)*(2*x + b*log(f)/(c*log(f) + 2 *f)))*e^(-1/4*(b^2*log(f)^2 - 4*a*c*log(f)^2 - 8*c*d*log(f) - 8*a*f*log(f) - 16*d*f)/(c*log(f) + 2*f))/sqrt(-c*log(f) - 2*f) - 1/8*sqrt(pi)*erf(-1/2 *sqrt(-c*log(f) + 2*f)*(2*x + b*log(f)/(c*log(f) - 2*f)))*e^(-1/4*(b^2*log (f)^2 - 4*a*c*log(f)^2 + 8*c*d*log(f) + 8*a*f*log(f) - 16*d*f)/(c*log(f) - 2*f))/sqrt(-c*log(f) + 2*f) - 1/4*sqrt(pi)*erf(-1/2*sqrt(-c*log(f))*(2*x + b/c))*e^(-1/4*(b^2*log(f) - 4*a*c*log(f))/c)/sqrt(-c*log(f))
Timed out. \[ \int f^{a+b x+c x^2} \cosh ^2\left (d+f x^2\right ) \, dx=\int f^{c\,x^2+b\,x+a}\,{\mathrm {cosh}\left (f\,x^2+d\right )}^2 \,d x \]