Integrand size = 26, antiderivative size = 344 \[ \int f^{a+b x+c x^2} \cosh ^3\left (d+e x+f x^2\right ) \, dx=\frac {3 e^{-d+\frac {(e-b \log (f))^2}{4 (f-c \log (f))}} f^a \sqrt {\pi } \text {erf}\left (\frac {e-b \log (f)+2 x (f-c \log (f))}{2 \sqrt {f-c \log (f)}}\right )}{16 \sqrt {f-c \log (f)}}+\frac {e^{-3 d+\frac {(3 e-b \log (f))^2}{12 f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {3 e-b \log (f)+2 x (3 f-c \log (f))}{2 \sqrt {3 f-c \log (f)}}\right )}{16 \sqrt {3 f-c \log (f)}}+\frac {3 e^{d-\frac {(e+b \log (f))^2}{4 (f+c \log (f))}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e+b \log (f)+2 x (f+c \log (f))}{2 \sqrt {f+c \log (f)}}\right )}{16 \sqrt {f+c \log (f)}}+\frac {e^{3 d-\frac {(3 e+b \log (f))^2}{4 (3 f+c \log (f))}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 e+b \log (f)+2 x (3 f+c \log (f))}{2 \sqrt {3 f+c \log (f)}}\right )}{16 \sqrt {3 f+c \log (f)}} \] Output:
3/16*exp(-d+(e-b*ln(f))^2/(4*f-4*c*ln(f)))*f^a*Pi^(1/2)*erf(1/2*(e-b*ln(f) +2*x*(f-c*ln(f)))/(f-c*ln(f))^(1/2))/(f-c*ln(f))^(1/2)+1/16*exp(-3*d+(3*e- b*ln(f))^2/(12*f-4*c*ln(f)))*f^a*Pi^(1/2)*erf(1/2*(3*e-b*ln(f)+2*x*(3*f-c* ln(f)))/(3*f-c*ln(f))^(1/2))/(3*f-c*ln(f))^(1/2)+3/16*exp(d-(e+b*ln(f))^2/ (4*f+4*c*ln(f)))*f^a*Pi^(1/2)*erfi(1/2*(e+b*ln(f)+2*x*(f+c*ln(f)))/(f+c*ln (f))^(1/2))/(f+c*ln(f))^(1/2)+1/16*exp(3*d-(3*e+b*ln(f))^2/(12*f+4*c*ln(f) ))*f^a*Pi^(1/2)*erfi(1/2*(3*e+b*ln(f)+2*x*(3*f+c*ln(f)))/(3*f+c*ln(f))^(1/ 2))/(3*f+c*ln(f))^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(2991\) vs. \(2(344)=688\).
Time = 6.43 (sec) , antiderivative size = 2991, normalized size of antiderivative = 8.69 \[ \int f^{a+b x+c x^2} \cosh ^3\left (d+e x+f x^2\right ) \, dx=\text {Result too large to show} \] Input:
Integrate[f^(a + b*x + c*x^2)*Cosh[d + e*x + f*x^2]^3,x]
Output:
(f^a*Sqrt[Pi]*((27*f^3*Cosh[d]*Erf[(e + 2*f*x - b*Log[f] - 2*c*x*Log[f])/( 2*Sqrt[f - c*Log[f]])]*Sqrt[f - c*Log[f]])/E^((-e^2 + 2*b*e*Log[f] - b^2*L og[f]^2)/(4*(f - c*Log[f]))) + (27*c*f^2*Cosh[d]*Erf[(e + 2*f*x - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[f - c*Log[f]])]*Log[f]*Sqrt[f - c*Log[f]])/E^((-e ^2 + 2*b*e*Log[f] - b^2*Log[f]^2)/(4*(f - c*Log[f]))) - (3*c^2*f*Cosh[d]*E rf[(e + 2*f*x - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[f - c*Log[f]])]*Log[f]^2* Sqrt[f - c*Log[f]])/E^((-e^2 + 2*b*e*Log[f] - b^2*Log[f]^2)/(4*(f - c*Log[ f]))) - (3*c^3*Cosh[d]*Erf[(e + 2*f*x - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[f - c*Log[f]])]*Log[f]^3*Sqrt[f - c*Log[f]])/E^((-e^2 + 2*b*e*Log[f] - b^2* Log[f]^2)/(4*(f - c*Log[f]))) + (3*f^3*Cosh[3*d]*Erf[(3*e + 6*f*x - b*Log[ f] - 2*c*x*Log[f])/(2*Sqrt[3*f - c*Log[f]])]*Sqrt[3*f - c*Log[f]])/E^((-9* e^2 + 6*b*e*Log[f] - b^2*Log[f]^2)/(4*(3*f - c*Log[f]))) + (c*f^2*Cosh[3*d ]*Erf[(3*e + 6*f*x - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[3*f - c*Log[f]])]*Lo g[f]*Sqrt[3*f - c*Log[f]])/E^((-9*e^2 + 6*b*e*Log[f] - b^2*Log[f]^2)/(4*(3 *f - c*Log[f]))) - (3*c^2*f*Cosh[3*d]*Erf[(3*e + 6*f*x - b*Log[f] - 2*c*x* Log[f])/(2*Sqrt[3*f - c*Log[f]])]*Log[f]^2*Sqrt[3*f - c*Log[f]])/E^((-9*e^ 2 + 6*b*e*Log[f] - b^2*Log[f]^2)/(4*(3*f - c*Log[f]))) - (c^3*Cosh[3*d]*Er f[(3*e + 6*f*x - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[3*f - c*Log[f]])]*Log[f] ^3*Sqrt[3*f - c*Log[f]])/E^((-9*e^2 + 6*b*e*Log[f] - b^2*Log[f]^2)/(4*(3*f - c*Log[f]))) + (27*f^3*Cosh[d]*Erfi[(e + 2*f*x + b*Log[f] + 2*c*x*Log...
Time = 1.17 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, 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.
| \(\displaystyle \int f^{a+b x+c x^2} \cosh ^3\left (d+e x+f x^2\right ) \, dx\) |
| \(\Big \downarrow \) 6039 |
| \(\displaystyle \int \left (\frac {3}{8} f^{a+b x+c x^2} \exp \left (-3 \left (d+e x+f x^2\right )+2 d+2 e x+2 f x^2\right )+\frac {3}{8} f^{a+b x+c x^2} \exp \left (-3 \left (d+e x+f x^2\right )+4 d+4 e x+4 f x^2\right )+\frac {1}{8} f^{a+b x+c x^2} \exp \left (-3 \left (d+e x+f x^2\right )+6 d+6 e x+6 f x^2\right )+\frac {1}{8} e^{-3 \left (d+e x+f x^2\right )} f^{a+b x+c x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {\pi } f^a \exp \left (\frac {(3 e-b \log (f))^2}{12 f-4 c \log (f)}-3 d\right ) \text {erf}\left (\frac {-b \log (f)+2 x (3 f-c \log (f))+3 e}{2 \sqrt {3 f-c \log (f)}}\right )}{16 \sqrt {3 f-c \log (f)}}+\frac {3 \sqrt {\pi } f^a e^{\frac {(e-b \log (f))^2}{4 (f-c \log (f))}-d} \text {erf}\left (\frac {-b \log (f)+2 x (f-c \log (f))+e}{2 \sqrt {f-c \log (f)}}\right )}{16 \sqrt {f-c \log (f)}}+\frac {\sqrt {\pi } f^a \exp \left (3 d-\frac {(b \log (f)+3 e)^2}{4 (c \log (f)+3 f)}\right ) \text {erfi}\left (\frac {b \log (f)+2 x (c \log (f)+3 f)+3 e}{2 \sqrt {c \log (f)+3 f}}\right )}{16 \sqrt {c \log (f)+3 f}}+\frac {3 \sqrt {\pi } f^a e^{d-\frac {(b \log (f)+e)^2}{4 (c \log (f)+f)}} \text {erfi}\left (\frac {b \log (f)+2 x (c \log (f)+f)+e}{2 \sqrt {c \log (f)+f}}\right )}{16 \sqrt {c \log (f)+f}}\) |
Input:
Int[f^(a + b*x + c*x^2)*Cosh[d + e*x + f*x^2]^3,x]
Output:
(3*E^(-d + (e - b*Log[f])^2/(4*(f - c*Log[f])))*f^a*Sqrt[Pi]*Erf[(e - b*Lo g[f] + 2*x*(f - c*Log[f]))/(2*Sqrt[f - c*Log[f]])])/(16*Sqrt[f - c*Log[f]] ) + (E^(-3*d + (3*e - b*Log[f])^2/(12*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(3 *e - b*Log[f] + 2*x*(3*f - c*Log[f]))/(2*Sqrt[3*f - c*Log[f]])])/(16*Sqrt[ 3*f - c*Log[f]]) + (3*E^(d - (e + b*Log[f])^2/(4*(f + c*Log[f])))*f^a*Sqrt [Pi]*Erfi[(e + b*Log[f] + 2*x*(f + c*Log[f]))/(2*Sqrt[f + c*Log[f]])])/(16 *Sqrt[f + c*Log[f]]) + (E^(3*d - (3*e + b*Log[f])^2/(4*(3*f + c*Log[f])))* f^a*Sqrt[Pi]*Erfi[(3*e + b*Log[f] + 2*x*(3*f + c*Log[f]))/(2*Sqrt[3*f + c* Log[f]])])/(16*Sqrt[3*f + c*Log[f]])
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 = 5.05 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.12
| method | result | size |
| risch | \(-\frac {\operatorname {erf}\left (-x \sqrt {3 f -c \ln \left (f \right )}+\frac {b \ln \left (f \right )-3 e}{2 \sqrt {3 f -c \ln \left (f \right )}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}-6 \ln \left (f \right ) b e +12 d \ln \left (f \right ) c -36 d f +9 e^{2}}{4 \left (c \ln \left (f \right )-3 f \right )}}}{16 \sqrt {3 f -c \ln \left (f \right )}}-\frac {\operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )-3 f}\, x +\frac {3 e +b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )-3 f}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}+6 \ln \left (f \right ) b e -12 d \ln \left (f \right ) c -36 d f +9 e^{2}}{4 \left (3 f +c \ln \left (f \right )\right )}}}{16 \sqrt {-c \ln \left (f \right )-3 f}}-\frac {3 \,\operatorname {erf}\left (-x \sqrt {f -c \ln \left (f \right )}+\frac {b \ln \left (f \right )-e}{2 \sqrt {f -c \ln \left (f \right )}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}-2 \ln \left (f \right ) b e +4 d \ln \left (f \right ) c -4 d f +e^{2}}{4 \left (c \ln \left (f \right )-f \right )}}}{16 \sqrt {f -c \ln \left (f \right )}}-\frac {3 \,\operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )-f}\, x +\frac {e +b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )-f}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}+2 \ln \left (f \right ) b e -4 d \ln \left (f \right ) c -4 d f +e^{2}}{4 \left (f +c \ln \left (f \right )\right )}}}{16 \sqrt {-c \ln \left (f \right )-f}}\) | \(384\) |
Input:
int(f^(c*x^2+b*x+a)*cosh(f*x^2+e*x+d)^3,x,method=_RETURNVERBOSE)
Output:
-1/16*erf(-x*(3*f-c*ln(f))^(1/2)+1/2*(b*ln(f)-3*e)/(3*f-c*ln(f))^(1/2))/(3 *f-c*ln(f))^(1/2)*Pi^(1/2)*f^a*exp(-1/4*(b^2*ln(f)^2-6*ln(f)*b*e+12*d*ln(f )*c-36*d*f+9*e^2)/(c*ln(f)-3*f))-1/16*erf(-(-c*ln(f)-3*f)^(1/2)*x+1/2*(3*e +b*ln(f))/(-c*ln(f)-3*f)^(1/2))/(-c*ln(f)-3*f)^(1/2)*Pi^(1/2)*f^a*exp(-1/4 *(b^2*ln(f)^2+6*ln(f)*b*e-12*d*ln(f)*c-36*d*f+9*e^2)/(3*f+c*ln(f)))-3/16*e rf(-x*(f-c*ln(f))^(1/2)+1/2*(b*ln(f)-e)/(f-c*ln(f))^(1/2))/(f-c*ln(f))^(1/ 2)*Pi^(1/2)*f^a*exp(-1/4*(b^2*ln(f)^2-2*ln(f)*b*e+4*d*ln(f)*c-4*d*f+e^2)/( c*ln(f)-f))-3/16*erf(-(-c*ln(f)-f)^(1/2)*x+1/2*(e+b*ln(f))/(-c*ln(f)-f)^(1 /2))/(-c*ln(f)-f)^(1/2)*Pi^(1/2)*f^a*exp(-1/4*(b^2*ln(f)^2+2*ln(f)*b*e-4*d *ln(f)*c-4*d*f+e^2)/(f+c*ln(f)))
Leaf count of result is larger than twice the leaf count of optimal. 939 vs. \(2 (295) = 590\).
Time = 0.11 (sec) , antiderivative size = 939, normalized size of antiderivative = 2.73 \[ \int f^{a+b x+c x^2} \cosh ^3\left (d+e x+f x^2\right ) \, dx=\text {Too large to display} \] Input:
integrate(f^(c*x^2+b*x+a)*cosh(f*x^2+e*x+d)^3,x, algorithm="fricas")
Output:
-1/16*((sqrt(pi)*(c^3*log(f)^3 + 3*c^2*f*log(f)^2 - c*f^2*log(f) - 3*f^3)* cosh(-1/4*((b^2 - 4*a*c)*log(f)^2 + 9*e^2 - 36*d*f + 6*(2*c*d - b*e + 2*a* f)*log(f))/(c*log(f) - 3*f)) + sqrt(pi)*(c^3*log(f)^3 + 3*c^2*f*log(f)^2 - c*f^2*log(f) - 3*f^3)*sinh(-1/4*((b^2 - 4*a*c)*log(f)^2 + 9*e^2 - 36*d*f + 6*(2*c*d - b*e + 2*a*f)*log(f))/(c*log(f) - 3*f)))*sqrt(-c*log(f) + 3*f) *erf(-1/2*(6*f*x - (2*c*x + b)*log(f) + 3*e)*sqrt(-c*log(f) + 3*f)/(c*log( f) - 3*f)) + 3*(sqrt(pi)*(c^3*log(f)^3 + c^2*f*log(f)^2 - 9*c*f^2*log(f) - 9*f^3)*cosh(-1/4*((b^2 - 4*a*c)*log(f)^2 + e^2 - 4*d*f + 2*(2*c*d - b*e + 2*a*f)*log(f))/(c*log(f) - f)) + sqrt(pi)*(c^3*log(f)^3 + c^2*f*log(f)^2 - 9*c*f^2*log(f) - 9*f^3)*sinh(-1/4*((b^2 - 4*a*c)*log(f)^2 + e^2 - 4*d*f + 2*(2*c*d - b*e + 2*a*f)*log(f))/(c*log(f) - f)))*sqrt(-c*log(f) + f)*erf (-1/2*(2*f*x - (2*c*x + b)*log(f) + e)*sqrt(-c*log(f) + f)/(c*log(f) - f)) + 3*(sqrt(pi)*(c^3*log(f)^3 - c^2*f*log(f)^2 - 9*c*f^2*log(f) + 9*f^3)*co sh(-1/4*((b^2 - 4*a*c)*log(f)^2 + e^2 - 4*d*f - 2*(2*c*d - b*e + 2*a*f)*lo g(f))/(c*log(f) + f)) + sqrt(pi)*(c^3*log(f)^3 - c^2*f*log(f)^2 - 9*c*f^2* log(f) + 9*f^3)*sinh(-1/4*((b^2 - 4*a*c)*log(f)^2 + e^2 - 4*d*f - 2*(2*c*d - b*e + 2*a*f)*log(f))/(c*log(f) + f)))*sqrt(-c*log(f) - f)*erf(1/2*(2*f* x + (2*c*x + b)*log(f) + e)*sqrt(-c*log(f) - f)/(c*log(f) + f)) + (sqrt(pi )*(c^3*log(f)^3 - 3*c^2*f*log(f)^2 - c*f^2*log(f) + 3*f^3)*cosh(-1/4*((b^2 - 4*a*c)*log(f)^2 + 9*e^2 - 36*d*f - 6*(2*c*d - b*e + 2*a*f)*log(f))/(...
Timed out. \[ \int f^{a+b x+c x^2} \cosh ^3\left (d+e x+f x^2\right ) \, dx=\text {Timed out} \] Input:
integrate(f**(c*x**2+b*x+a)*cosh(f*x**2+e*x+d)**3,x)
Output:
Timed out
Time = 0.06 (sec) , antiderivative size = 315, normalized size of antiderivative = 0.92 \[ \int f^{a+b x+c x^2} \cosh ^3\left (d+e x+f x^2\right ) \, dx=\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - 3 \, f} x - \frac {b \log \left (f\right ) + 3 \, e}{2 \, \sqrt {-c \log \left (f\right ) - 3 \, f}}\right ) e^{\left (-\frac {{\left (b \log \left (f\right ) + 3 \, e\right )}^{2}}{4 \, {\left (c \log \left (f\right ) + 3 \, f\right )}} + 3 \, d\right )}}{16 \, \sqrt {-c \log \left (f\right ) - 3 \, f}} + \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - f} x - \frac {b \log \left (f\right ) + e}{2 \, \sqrt {-c \log \left (f\right ) - f}}\right ) e^{\left (-\frac {{\left (b \log \left (f\right ) + e\right )}^{2}}{4 \, {\left (c \log \left (f\right ) + f\right )}} + d\right )}}{16 \, \sqrt {-c \log \left (f\right ) - f}} + \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + f} x - \frac {b \log \left (f\right ) - e}{2 \, \sqrt {-c \log \left (f\right ) + f}}\right ) e^{\left (-\frac {{\left (b \log \left (f\right ) - e\right )}^{2}}{4 \, {\left (c \log \left (f\right ) - f\right )}} - d\right )}}{16 \, \sqrt {-c \log \left (f\right ) + f}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + 3 \, f} x - \frac {b \log \left (f\right ) - 3 \, e}{2 \, \sqrt {-c \log \left (f\right ) + 3 \, f}}\right ) e^{\left (-\frac {{\left (b \log \left (f\right ) - 3 \, e\right )}^{2}}{4 \, {\left (c \log \left (f\right ) - 3 \, f\right )}} - 3 \, d\right )}}{16 \, \sqrt {-c \log \left (f\right ) + 3 \, f}} \] Input:
integrate(f^(c*x^2+b*x+a)*cosh(f*x^2+e*x+d)^3,x, algorithm="maxima")
Output:
1/16*sqrt(pi)*f^a*erf(sqrt(-c*log(f) - 3*f)*x - 1/2*(b*log(f) + 3*e)/sqrt( -c*log(f) - 3*f))*e^(-1/4*(b*log(f) + 3*e)^2/(c*log(f) + 3*f) + 3*d)/sqrt( -c*log(f) - 3*f) + 3/16*sqrt(pi)*f^a*erf(sqrt(-c*log(f) - f)*x - 1/2*(b*lo g(f) + e)/sqrt(-c*log(f) - f))*e^(-1/4*(b*log(f) + e)^2/(c*log(f) + f) + d )/sqrt(-c*log(f) - f) + 3/16*sqrt(pi)*f^a*erf(sqrt(-c*log(f) + f)*x - 1/2* (b*log(f) - e)/sqrt(-c*log(f) + f))*e^(-1/4*(b*log(f) - e)^2/(c*log(f) - f ) - d)/sqrt(-c*log(f) + f) + 1/16*sqrt(pi)*f^a*erf(sqrt(-c*log(f) + 3*f)*x - 1/2*(b*log(f) - 3*e)/sqrt(-c*log(f) + 3*f))*e^(-1/4*(b*log(f) - 3*e)^2/ (c*log(f) - 3*f) - 3*d)/sqrt(-c*log(f) + 3*f)
Time = 0.15 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.24 \[ \int f^{a+b x+c x^2} \cosh ^3\left (d+e x+f x^2\right ) \, dx=-\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) - 3 \, f} {\left (2 \, x + \frac {b \log \left (f\right ) + 3 \, e}{c \log \left (f\right ) + 3 \, f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} - 12 \, c d \log \left (f\right ) + 6 \, b e \log \left (f\right ) - 12 \, a f \log \left (f\right ) + 9 \, e^{2} - 36 \, d f}{4 \, {\left (c \log \left (f\right ) + 3 \, f\right )}}\right )}}{16 \, \sqrt {-c \log \left (f\right ) - 3 \, f}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) - f} {\left (2 \, x + \frac {b \log \left (f\right ) + e}{c \log \left (f\right ) + f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) + 2 \, b e \log \left (f\right ) - 4 \, a f \log \left (f\right ) + e^{2} - 4 \, d f}{4 \, {\left (c \log \left (f\right ) + f\right )}}\right )}}{16 \, \sqrt {-c \log \left (f\right ) - f}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) + f} {\left (2 \, x + \frac {b \log \left (f\right ) - e}{c \log \left (f\right ) - f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} + 4 \, c d \log \left (f\right ) - 2 \, b e \log \left (f\right ) + 4 \, a f \log \left (f\right ) + e^{2} - 4 \, d f}{4 \, {\left (c \log \left (f\right ) - f\right )}}\right )}}{16 \, \sqrt {-c \log \left (f\right ) + f}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) + 3 \, f} {\left (2 \, x + \frac {b \log \left (f\right ) - 3 \, e}{c \log \left (f\right ) - 3 \, f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} + 12 \, c d \log \left (f\right ) - 6 \, b e \log \left (f\right ) + 12 \, a f \log \left (f\right ) + 9 \, e^{2} - 36 \, d f}{4 \, {\left (c \log \left (f\right ) - 3 \, f\right )}}\right )}}{16 \, \sqrt {-c \log \left (f\right ) + 3 \, f}} \] Input:
integrate(f^(c*x^2+b*x+a)*cosh(f*x^2+e*x+d)^3,x, algorithm="giac")
Output:
-1/16*sqrt(pi)*erf(-1/2*sqrt(-c*log(f) - 3*f)*(2*x + (b*log(f) + 3*e)/(c*l og(f) + 3*f)))*e^(-1/4*(b^2*log(f)^2 - 4*a*c*log(f)^2 - 12*c*d*log(f) + 6* b*e*log(f) - 12*a*f*log(f) + 9*e^2 - 36*d*f)/(c*log(f) + 3*f))/sqrt(-c*log (f) - 3*f) - 3/16*sqrt(pi)*erf(-1/2*sqrt(-c*log(f) - f)*(2*x + (b*log(f) + e)/(c*log(f) + f)))*e^(-1/4*(b^2*log(f)^2 - 4*a*c*log(f)^2 - 4*c*d*log(f) + 2*b*e*log(f) - 4*a*f*log(f) + e^2 - 4*d*f)/(c*log(f) + f))/sqrt(-c*log( f) - f) - 3/16*sqrt(pi)*erf(-1/2*sqrt(-c*log(f) + f)*(2*x + (b*log(f) - e) /(c*log(f) - f)))*e^(-1/4*(b^2*log(f)^2 - 4*a*c*log(f)^2 + 4*c*d*log(f) - 2*b*e*log(f) + 4*a*f*log(f) + e^2 - 4*d*f)/(c*log(f) - f))/sqrt(-c*log(f) + f) - 1/16*sqrt(pi)*erf(-1/2*sqrt(-c*log(f) + 3*f)*(2*x + (b*log(f) - 3*e )/(c*log(f) - 3*f)))*e^(-1/4*(b^2*log(f)^2 - 4*a*c*log(f)^2 + 12*c*d*log(f ) - 6*b*e*log(f) + 12*a*f*log(f) + 9*e^2 - 36*d*f)/(c*log(f) - 3*f))/sqrt( -c*log(f) + 3*f)
Timed out. \[ \int f^{a+b x+c x^2} \cosh ^3\left (d+e x+f x^2\right ) \, dx=\int f^{c\,x^2+b\,x+a}\,{\mathrm {cosh}\left (f\,x^2+e\,x+d\right )}^3 \,d x \] Input:
int(f^(a + b*x + c*x^2)*cosh(d + e*x + f*x^2)^3,x)
Output:
int(f^(a + b*x + c*x^2)*cosh(d + e*x + f*x^2)^3, x)
\[ \int f^{a+b x+c x^2} \cosh ^3\left (d+e x+f x^2\right ) \, dx=f^{a} \left (\int f^{c \,x^{2}+b x} \cosh \left (f \,x^{2}+e x +d \right )^{3}d x \right ) \] Input:
int(f^(c*x^2+b*x+a)*cosh(f*x^2+e*x+d)^3,x)
Output:
f**a*int(f**(b*x + c*x**2)*cosh(d + e*x + f*x**2)**3,x)