3.327 \(\int f^{a+b x+c x^2} \cosh ^2(d+e x+f x^2) \, dx\)

Optimal. Leaf size=239 \[ \frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a \exp \left (\frac {(2 e-b \log (f))^2}{8 f-4 c \log (f)}-2 d\right ) \text {erf}\left (\frac {-b \log (f)+2 x (2 f-c \log (f))+2 e}{2 \sqrt {2 f-c \log (f)}}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {\sqrt {\pi } f^a \exp \left (2 d-\frac {(b \log (f)+2 e)^2}{4 c \log (f)+8 f}\right ) \text {erfi}\left (\frac {b \log (f)+2 x (c \log (f)+2 f)+2 e}{2 \sqrt {c \log (f)+2 f}}\right )}{8 \sqrt {c \log (f)+2 f}} \]

[Out]

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+(2*e-b*l
n(f))^2/(8*f-4*c*ln(f)))*f^a*erf(1/2*(2*e-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-(2*e+b*ln(f))^2/(8*f+4*c*ln(f)))*f^a*erfi(1/2*(2*e+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)

________________________________________________________________________________________

Rubi [A]  time = 0.51, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {5513, 2234, 2204, 2287, 2205} \[ \frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} \text {Erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a \exp \left (\frac {(2 e-b \log (f))^2}{8 f-4 c \log (f)}-2 d\right ) \text {Erf}\left (\frac {-b \log (f)+2 x (2 f-c \log (f))+2 e}{2 \sqrt {2 f-c \log (f)}}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {\sqrt {\pi } f^a \exp \left (2 d-\frac {(b \log (f)+2 e)^2}{4 c \log (f)+8 f}\right ) \text {Erfi}\left (\frac {b \log (f)+2 x (c \log (f)+2 f)+2 e}{2 \sqrt {c \log (f)+2 f}}\right )}{8 \sqrt {c \log (f)+2 f}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x + c*x^2)*Cosh[d + e*x + f*x^2]^2,x]

[Out]

(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
+ (2*e - b*Log[f])^2/(8*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(2*e - 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 - (2*e + b*Log[f])^2/(8*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(
2*e + b*Log[f] + 2*x*(2*f + c*Log[f]))/(2*Sqrt[2*f + c*Log[f]])])/(8*Sqrt[2*f + c*Log[f]])

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 2234

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2287

Int[(u_.)*(F_)^(v_)*(G_)^(w_), x_Symbol] :> With[{z = v*Log[F] + w*Log[G]}, Int[u*NormalizeIntegrand[E^z, x],
x] /; BinomialQ[z, x] || (PolynomialQ[z, x] && LeQ[Exponent[z, x], 2])] /; FreeQ[{F, G}, x]

Rule 5513

Int[Cosh[v_]^(n_.)*(F_)^(u_), x_Symbol] :> Int[ExpandTrigToExp[F^u, Cosh[v]^n, x], x] /; FreeQ[F, x] && (Linea
rQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[n, 0]

Rubi steps

\begin {align*} \int f^{a+b x+c x^2} \cosh ^2\left (d+e x+f x^2\right ) \, dx &=\int \left (\frac {1}{2} f^{a+b x+c x^2}+\frac {1}{4} e^{-2 d-2 e x-2 f x^2} f^{a+b x+c x^2}+\frac {1}{4} e^{2 d+2 e x+2 f x^2} f^{a+b x+c x^2}\right ) \, dx\\ &=\frac {1}{4} \int e^{-2 d-2 e x-2 f x^2} f^{a+b x+c x^2} \, dx+\frac {1}{4} \int e^{2 d+2 e x+2 f x^2} f^{a+b x+c x^2} \, dx+\frac {1}{2} \int f^{a+b x+c x^2} \, dx\\ &=\frac {1}{4} \int \exp \left (-2 d+a \log (f)-x (2 e-b \log (f))-x^2 (2 f-c \log (f))\right ) \, dx+\frac {1}{4} \int \exp \left (2 d+a \log (f)+x (2 e+b \log (f))+x^2 (2 f+c \log (f))\right ) \, dx+\frac {1}{2} f^{a-\frac {b^2}{4 c}} \int f^{\frac {(b+2 c x)^2}{4 c}} \, 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 {1}{4} \left (\exp \left (-2 d+\frac {(2 e-b \log (f))^2}{8 f-4 c \log (f)}\right ) f^a\right ) \int \exp \left (\frac {(-2 e+b \log (f)+2 x (-2 f+c \log (f)))^2}{4 (-2 f+c \log (f))}\right ) \, dx+\frac {1}{4} \left (\exp \left (2 d-\frac {(2 e+b \log (f))^2}{8 f+4 c \log (f)}\right ) f^a\right ) \int \exp \left (\frac {(2 e+b \log (f)+2 x (2 f+c \log (f)))^2}{4 (2 f+c \log (f))}\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 {\exp \left (-2 d+\frac {(2 e-b \log (f))^2}{8 f-4 c \log (f)}\right ) f^a \sqrt {\pi } \text {erf}\left (\frac {2 e-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 {\exp \left (2 d-\frac {(2 e+b \log (f))^2}{8 f+4 c \log (f)}\right ) f^a \sqrt {\pi } \text {erfi}\left (\frac {2 e+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)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 6.17, size = 339, normalized size = 1.42 \[ \frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {\sqrt {\pi } e^{-\frac {b^2 \log ^2(f)+4 e^2}{4 c \log (f)+8 f}} f^{a+\frac {4 b e f}{c^2 \log ^2(f)-4 f^2}} \left (\sqrt {2 f-c \log (f)} (c \log (f)+2 f) (\cosh (2 d)-\sinh (2 d)) f^{\frac {b e}{c \log (f)+2 f}} \exp \left (\frac {f \left (b^2 \log ^2(f)+4 e^2\right )}{4 f^2-c^2 \log ^2(f)}\right ) \text {erf}\left (\frac {2 (e+2 f x)-\log (f) (b+2 c x)}{2 \sqrt {2 f-c \log (f)}}\right )+(2 f-c \log (f)) \sqrt {c \log (f)+2 f} (\sinh (2 d)+\cosh (2 d)) f^{\frac {b e}{2 f-c \log (f)}} \text {erfi}\left (\frac {\log (f) (b+2 c x)+2 (e+2 f x)}{2 \sqrt {c \log (f)+2 f}}\right )\right )}{8 \left (c^2 \log ^2(f)-4 f^2\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[f^(a + b*x + c*x^2)*Cosh[d + e*x + f*x^2]^2,x]

[Out]

(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]]) - (f^(a + (
4*b*e*f)/(-4*f^2 + c^2*Log[f]^2))*Sqrt[Pi]*(E^((f*(4*e^2 + b^2*Log[f]^2))/(4*f^2 - c^2*Log[f]^2))*f^((b*e)/(2*
f + c*Log[f]))*Erf[(2*(e + 2*f*x) - (b + 2*c*x)*Log[f])/(2*Sqrt[2*f - c*Log[f]])]*Sqrt[2*f - c*Log[f]]*(2*f +
c*Log[f])*(Cosh[2*d] - Sinh[2*d]) + f^((b*e)/(2*f - c*Log[f]))*Erfi[(2*(e + 2*f*x) + (b + 2*c*x)*Log[f])/(2*Sq
rt[2*f + c*Log[f]])]*(2*f - c*Log[f])*Sqrt[2*f + c*Log[f]]*(Cosh[2*d] + Sinh[2*d])))/(8*E^((4*e^2 + b^2*Log[f]
^2)/(8*f + 4*c*Log[f]))*(-4*f^2 + c^2*Log[f]^2))

________________________________________________________________________________________

fricas [B]  time = 0.47, size = 516, normalized size = 2.16 \[ -\frac {{\left (\sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} + 2 \, c f \log \relax (f)\right )} \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)^{2} + 4 \, e^{2} - 16 \, d f + 4 \, {\left (2 \, c d - b e + 2 \, a f\right )} \log \relax (f)}{4 \, {\left (c \log \relax (f) - 2 \, f\right )}}\right ) + \sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} + 2 \, c f \log \relax (f)\right )} \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)^{2} + 4 \, e^{2} - 16 \, d f + 4 \, {\left (2 \, c d - b e + 2 \, a f\right )} \log \relax (f)}{4 \, {\left (c \log \relax (f) - 2 \, f\right )}}\right )\right )} \sqrt {-c \log \relax (f) + 2 \, f} \operatorname {erf}\left (-\frac {{\left (4 \, f x - {\left (2 \, c x + b\right )} \log \relax (f) + 2 \, e\right )} \sqrt {-c \log \relax (f) + 2 \, f}}{2 \, {\left (c \log \relax (f) - 2 \, f\right )}}\right ) + {\left (\sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} - 2 \, c f \log \relax (f)\right )} \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)^{2} + 4 \, e^{2} - 16 \, d f - 4 \, {\left (2 \, c d - b e + 2 \, a f\right )} \log \relax (f)}{4 \, {\left (c \log \relax (f) + 2 \, f\right )}}\right ) + \sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} - 2 \, c f \log \relax (f)\right )} \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)^{2} + 4 \, e^{2} - 16 \, d f - 4 \, {\left (2 \, c d - b e + 2 \, a f\right )} \log \relax (f)}{4 \, {\left (c \log \relax (f) + 2 \, f\right )}}\right )\right )} \sqrt {-c \log \relax (f) - 2 \, f} \operatorname {erf}\left (\frac {{\left (4 \, f x + {\left (2 \, c x + b\right )} \log \relax (f) + 2 \, e\right )} \sqrt {-c \log \relax (f) - 2 \, f}}{2 \, {\left (c \log \relax (f) + 2 \, f\right )}}\right ) + 2 \, {\left (\sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} - 4 \, f^{2}\right )} \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)}{4 \, c}\right ) + \sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} - 4 \, f^{2}\right )} \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)}{4 \, c}\right )\right )} \sqrt {-c \log \relax (f)} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c \log \relax (f)}}{2 \, c}\right )}{8 \, {\left (c^{3} \log \relax (f)^{3} - 4 \, c f^{2} \log \relax (f)\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*x^2+b*x+a)*cosh(f*x^2+e*x+d)^2,x, algorithm="fricas")

[Out]

-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 + 4*e^2 - 16*d*f + 4*(2*c*d -
b*e + 2*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 + 4*e^2 - 16*d*f + 4*(2*c*d - b*e + 2*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) + 2*e)*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 + 4*e^2 - 16*d*f - 4*(2*c*d - b*e + 2*a*f)*log(f))/(c*log(f) + 2*f)) + sqr
t(pi)*(c^2*log(f)^2 - 2*c*f*log(f))*sinh(-1/4*((b^2 - 4*a*c)*log(f)^2 + 4*e^2 - 16*d*f - 4*(2*c*d - b*e + 2*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) + 2*e)*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)*sqrt(-c*log(f))/c))
/(c^3*log(f)^3 - 4*c*f^2*log(f))

________________________________________________________________________________________

giac [A]  time = 0.19, size = 273, normalized size = 1.14 \[ -\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f) - 2 \, f} {\left (2 \, x + \frac {b \log \relax (f) + 2 \, e}{c \log \relax (f) + 2 \, f}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f)^{2} - 4 \, a c \log \relax (f)^{2} - 8 \, c d \log \relax (f) - 8 \, a f \log \relax (f) + 4 \, b e \log \relax (f) - 16 \, d f + 4 \, e^{2}}{4 \, {\left (c \log \relax (f) + 2 \, f\right )}}\right )}}{8 \, \sqrt {-c \log \relax (f) - 2 \, f}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f) + 2 \, f} {\left (2 \, x + \frac {b \log \relax (f) - 2 \, e}{c \log \relax (f) - 2 \, f}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f)^{2} - 4 \, a c \log \relax (f)^{2} + 8 \, c d \log \relax (f) + 8 \, a f \log \relax (f) - 4 \, b e \log \relax (f) - 16 \, d f + 4 \, e^{2}}{4 \, {\left (c \log \relax (f) - 2 \, f\right )}}\right )}}{8 \, \sqrt {-c \log \relax (f) + 2 \, f}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f) - 4 \, a c \log \relax (f)}{4 \, c}\right )}}{4 \, \sqrt {-c \log \relax (f)}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*x^2+b*x+a)*cosh(f*x^2+e*x+d)^2,x, algorithm="giac")

[Out]

-1/8*sqrt(pi)*erf(-1/2*sqrt(-c*log(f) - 2*f)*(2*x + (b*log(f) + 2*e)/(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) + 4*b*e*log(f) - 16*d*f + 4*e^2)/(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) - 2*e)/(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) - 4*b*e*log(f) - 16*d*f + 4*e^2)/(c*log(f) - 2*f))/s
qrt(-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))

________________________________________________________________________________________

maple [A]  time = 0.33, size = 249, normalized size = 1.04 \[ -\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}-4 \ln \relax (f ) b e +8 d \ln \relax (f ) c -16 d f +4 e^{2}}{4 \left (-2 f +c \ln \relax (f )\right )}} \erf \left (-x \sqrt {2 f -c \ln \relax (f )}+\frac {b \ln \relax (f )-2 e}{2 \sqrt {2 f -c \ln \relax (f )}}\right )}{8 \sqrt {2 f -c \ln \relax (f )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}+4 \ln \relax (f ) b e -8 d \ln \relax (f ) c -16 d f +4 e^{2}}{4 \left (2 f +c \ln \relax (f )\right )}} \erf \left (-\sqrt {-c \ln \relax (f )-2 f}\, x +\frac {2 e +b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )-2 f}}\right )}{8 \sqrt {-c \ln \relax (f )-2 f}}-\frac {\sqrt {\pi }\, f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}\right )}{4 \sqrt {-c \ln \relax (f )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c*x^2+b*x+a)*cosh(f*x^2+e*x+d)^2,x)

[Out]

-1/8*Pi^(1/2)*f^a*exp(-1/4*(ln(f)^2*b^2-4*ln(f)*b*e+8*d*ln(f)*c-16*d*f+4*e^2)/(-2*f+c*ln(f)))/(2*f-c*ln(f))^(1
/2)*erf(-x*(2*f-c*ln(f))^(1/2)+1/2*(b*ln(f)-2*e)/(2*f-c*ln(f))^(1/2))-1/8*Pi^(1/2)*f^a*exp(-1/4*(ln(f)^2*b^2+4
*ln(f)*b*e-8*d*ln(f)*c-16*d*f+4*e^2)/(2*f+c*ln(f)))/(-c*ln(f)-2*f)^(1/2)*erf(-(-c*ln(f)-2*f)^(1/2)*x+1/2*(2*e+
b*ln(f))/(-c*ln(f)-2*f)^(1/2))-1/4*Pi^(1/2)*f^a*f^(-1/4*b^2/c)/(-c*ln(f))^(1/2)*erf(-(-c*ln(f))^(1/2)*x+1/2*b*
ln(f)/(-c*ln(f))^(1/2))

________________________________________________________________________________________

maxima [A]  time = 0.33, size = 215, normalized size = 0.90 \[ \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f) - 2 \, f} x - \frac {b \log \relax (f) + 2 \, e}{2 \, \sqrt {-c \log \relax (f) - 2 \, f}}\right ) e^{\left (-\frac {{\left (b \log \relax (f) + 2 \, e\right )}^{2}}{4 \, {\left (c \log \relax (f) + 2 \, f\right )}} + 2 \, d\right )}}{8 \, \sqrt {-c \log \relax (f) - 2 \, f}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f) + 2 \, f} x - \frac {b \log \relax (f) - 2 \, e}{2 \, \sqrt {-c \log \relax (f) + 2 \, f}}\right ) e^{\left (-\frac {{\left (b \log \relax (f) - 2 \, e\right )}^{2}}{4 \, {\left (c \log \relax (f) - 2 \, f\right )}} - 2 \, d\right )}}{8 \, \sqrt {-c \log \relax (f) + 2 \, f}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x - \frac {b \log \relax (f)}{2 \, \sqrt {-c \log \relax (f)}}\right )}{4 \, \sqrt {-c \log \relax (f)} f^{\frac {b^{2}}{4 \, c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*x^2+b*x+a)*cosh(f*x^2+e*x+d)^2,x, algorithm="maxima")

[Out]

1/8*sqrt(pi)*f^a*erf(sqrt(-c*log(f) - 2*f)*x - 1/2*(b*log(f) + 2*e)/sqrt(-c*log(f) - 2*f))*e^(-1/4*(b*log(f) +
2*e)^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) - 2*e)/sqrt(-c*log(f) + 2*f))*e^(-1/4*(b*log(f) - 2*e)^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))

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int f^{c\,x^2+b\,x+a}\,{\mathrm {cosh}\left (f\,x^2+e\,x+d\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b*x + c*x^2)*cosh(d + e*x + f*x^2)^2,x)

[Out]

int(f^(a + b*x + c*x^2)*cosh(d + e*x + f*x^2)^2, x)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(c*x**2+b*x+a)*cosh(f*x**2+e*x+d)**2,x)

[Out]

Timed out

________________________________________________________________________________________