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

Optimal. Leaf size=219 \[ \frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {\sqrt {\pi } f^a e^{-\frac {(2 e-b \log (f))^2}{4 c \log (f)}-2 d} \text {erfi}\left (\frac {-b \log (f)-2 c x \log (f)+2 e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{2 d-\frac {(b \log (f)+2 e)^2}{4 c \log (f)}} \text {erfi}\left (\frac {b \log (f)+2 c x \log (f)+2 e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}} \]

[Out]

1/8*exp(-2*d-1/4*(2*e-b*ln(f))^2/c/ln(f))*f^a*erfi(1/2*(-2*e+b*ln(f)+2*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-1/4*(2*e+b*ln(f))^2/c/ln(f))*f^a*erfi(1/2*(2*e+b*ln(f)+2*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-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)

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Rubi [A]  time = 0.37, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5513, 2234, 2204, 2287} \[ \frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} \text {Erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {\sqrt {\pi } f^a e^{-\frac {(2 e-b \log (f))^2}{4 c \log (f)}-2 d} \text {Erfi}\left (\frac {-b \log (f)-2 c x \log (f)+2 e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{2 d-\frac {(b \log (f)+2 e)^2}{4 c \log (f)}} \text {Erfi}\left (\frac {b \log (f)+2 c x \log (f)+2 e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x + c*x^2)*Cosh[d + e*x]^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/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(2*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])]
)/(8*Sqrt[c]*Sqrt[Log[f]]) + (E^(2*d - (2*e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(2*e + b*Log[f] + 2*
c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[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 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(d+e x) \, dx &=\int \left (\frac {1}{2} f^{a+b x+c x^2}+\frac {1}{4} e^{-2 d-2 e x} f^{a+b x+c x^2}+\frac {1}{4} e^{2 d+2 e x} f^{a+b x+c x^2}\right ) \, dx\\ &=\frac {1}{4} \int e^{-2 d-2 e x} f^{a+b x+c x^2} \, dx+\frac {1}{4} \int e^{2 d+2 e x} 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)+c x^2 \log (f)-x (2 e-b \log (f))\right ) \, dx+\frac {1}{4} \int \exp \left (2 d+a \log (f)+c x^2 \log (f)+x (2 e+b \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 (e^{-2 d-\frac {(2 e-b \log (f))^2}{4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(-2 e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}\right ) \, dx+\frac {1}{4} \left (e^{2 d-\frac {(2 e+b \log (f))^2}{4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(2 e+b \log (f)+2 c x \log (f))^2}{4 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 {e^{-2 d-\frac {(2 e-b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {2 e-b \log (f)-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{2 d-\frac {(2 e+b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {2 e+b \log (f)+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}\\ \end {align*}

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Mathematica [A]  time = 0.56, size = 183, normalized size = 0.84 \[ \frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} e^{-\frac {e (b \log (f)+e)}{c \log (f)}} \left (e^{\frac {2 b e}{c}} (\cosh (2 d)-\sinh (2 d)) \text {erfi}\left (\frac {\log (f) (b+2 c x)-2 e}{2 \sqrt {c} \sqrt {\log (f)}}\right )+(\sinh (2 d)+\cosh (2 d)) \text {erfi}\left (\frac {\log (f) (b+2 c x)+2 e}{2 \sqrt {c} \sqrt {\log (f)}}\right )+2 e^{\frac {e (b \log (f)+e)}{c \log (f)}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )\right )}{8 \sqrt {c} \sqrt {\log (f)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.51, size = 341, normalized size = 1.56 \[ -\frac {2 \, \sqrt {-c \log \relax (f)} {\left (\sqrt {\pi } \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)}{4 \, c}\right ) + \sqrt {\pi } \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)}{4 \, c}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c \log \relax (f)}}{2 \, c}\right ) + \sqrt {-c \log \relax (f)} {\left (\sqrt {\pi } \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)^{2} + 4 \, e^{2} - 4 \, {\left (2 \, c d - b e\right )} \log \relax (f)}{4 \, c \log \relax (f)}\right ) + \sqrt {\pi } \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)^{2} + 4 \, e^{2} - 4 \, {\left (2 \, c d - b e\right )} \log \relax (f)}{4 \, c \log \relax (f)}\right )\right )} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c x + b\right )} \log \relax (f) + 2 \, e\right )} \sqrt {-c \log \relax (f)}}{2 \, c \log \relax (f)}\right ) + \sqrt {-c \log \relax (f)} {\left (\sqrt {\pi } \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)^{2} + 4 \, e^{2} + 4 \, {\left (2 \, c d - b e\right )} \log \relax (f)}{4 \, c \log \relax (f)}\right ) + \sqrt {\pi } \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)^{2} + 4 \, e^{2} + 4 \, {\left (2 \, c d - b e\right )} \log \relax (f)}{4 \, c \log \relax (f)}\right )\right )} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c x + b\right )} \log \relax (f) - 2 \, e\right )} \sqrt {-c \log \relax (f)}}{2 \, c \log \relax (f)}\right )}{8 \, c \log \relax (f)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 225, normalized size = 1.03 \[ -\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f) - 4 \, a c \log \relax (f)}{4 \, c}\right )}}{4 \, \sqrt {-c \log \relax (f)}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b \log \relax (f) - 2 \, e}{c \log \relax (f)}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f)^{2} - 4 \, a c \log \relax (f)^{2} + 8 \, c d \log \relax (f) - 4 \, b e \log \relax (f) + 4 \, e^{2}}{4 \, c \log \relax (f)}\right )}}{8 \, \sqrt {-c \log \relax (f)}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b \log \relax (f) + 2 \, e}{c \log \relax (f)}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f)^{2} - 4 \, a c \log \relax (f)^{2} - 8 \, c d \log \relax (f) + 4 \, b e \log \relax (f) + 4 \, e^{2}}{4 \, c \log \relax (f)}\right )}}{8 \, \sqrt {-c \log \relax (f)}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.28, size = 211, normalized size = 0.96 \[ -\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}-4 \ln \relax (f ) b e +8 d \ln \relax (f ) c +4 e^{2}}{4 \ln \relax (f ) c}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {b \ln \relax (f )-2 e}{2 \sqrt {-c \ln \relax (f )}}\right )}{8 \sqrt {-c \ln \relax (f )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}+4 \ln \relax (f ) b e -8 d \ln \relax (f ) c +4 e^{2}}{4 \ln \relax (f ) c}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {2 e +b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}\right )}{8 \sqrt {-c \ln \relax (f )}}-\frac {\sqrt {\pi }\, f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}\right )}{4 \sqrt {-c \ln \relax (f )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c*x^2+b*x+a)*cosh(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+4*e^2)/ln(f)/c)/(-c*ln(f))^(1/2)*erf(-(-c*ln(f
))^(1/2)*x+1/2*(b*ln(f)-2*e)/(-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+
4*e^2)/ln(f)/c)/(-c*ln(f))^(1/2)*erf(-(-c*ln(f))^(1/2)*x+1/2*(2*e+b*ln(f))/(-c*ln(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))

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maxima [A]  time = 0.34, size = 185, normalized size = 0.84 \[ \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x - \frac {b \log \relax (f) + 2 \, e}{2 \, \sqrt {-c \log \relax (f)}}\right ) e^{\left (2 \, d - \frac {{\left (b \log \relax (f) + 2 \, e\right )}^{2}}{4 \, c \log \relax (f)}\right )}}{8 \, \sqrt {-c \log \relax (f)}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x - \frac {b \log \relax (f) - 2 \, e}{2 \, \sqrt {-c \log \relax (f)}}\right ) e^{\left (-2 \, d - \frac {{\left (b \log \relax (f) - 2 \, e\right )}^{2}}{4 \, c \log \relax (f)}\right )}}{8 \, \sqrt {-c \log \relax (f)}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x - \frac {b \log \relax (f)}{2 \, \sqrt {-c \log \relax (f)}}\right )}{4 \, \sqrt {-c \log \relax (f)} f^{\frac {b^{2}}{4 \, c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + b x + c x^{2}} \cosh ^{2}{\left (d + e x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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