3.8 \(\int \cosh (a+b x-c x^2) \, dx\)

Optimal. Leaf size=91 \[ -\frac {\sqrt {\pi } e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {\sqrt {\pi } e^{-a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}} \]

[Out]

-1/4*exp(a+1/4*b^2/c)*erf(1/2*(-2*c*x+b)/c^(1/2))*Pi^(1/2)/c^(1/2)-1/4*exp(-a-1/4*b^2/c)*erfi(1/2*(-2*c*x+b)/c
^(1/2))*Pi^(1/2)/c^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5375, 2234, 2205, 2204} \[ -\frac {\sqrt {\pi } e^{a+\frac {b^2}{4 c}} \text {Erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {\sqrt {\pi } e^{-a-\frac {b^2}{4 c}} \text {Erfi}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x - c*x^2],x]

[Out]

-(E^(a + b^2/(4*c))*Sqrt[Pi]*Erf[(b - 2*c*x)/(2*Sqrt[c])])/(4*Sqrt[c]) - (E^(-a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b
- 2*c*x)/(2*Sqrt[c])])/(4*Sqrt[c])

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 5375

Int[Cosh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[1/2, Int[E^(a + b*x + c*x^2), x], x] + Dist[1/2
, Int[E^(-a - b*x - c*x^2), x], x] /; FreeQ[{a, b, c}, x]

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 109, normalized size = 1.20 \[ \frac {\sqrt {\pi } \left (\text {erf}\left (\frac {2 c x-b}{2 \sqrt {c}}\right ) \left (\sinh \left (a+\frac {b^2}{4 c}\right )+\cosh \left (a+\frac {b^2}{4 c}\right )\right )+\text {erfi}\left (\frac {2 c x-b}{2 \sqrt {c}}\right ) \left (\cosh \left (a+\frac {b^2}{4 c}\right )-\sinh \left (a+\frac {b^2}{4 c}\right )\right )\right )}{4 \sqrt {c}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x - c*x^2],x]

[Out]

(Sqrt[Pi]*(Erfi[(-b + 2*c*x)/(2*Sqrt[c])]*(Cosh[a + b^2/(4*c)] - Sinh[a + b^2/(4*c)]) + Erf[(-b + 2*c*x)/(2*Sq
rt[c])]*(Cosh[a + b^2/(4*c)] + Sinh[a + b^2/(4*c)])))/(4*Sqrt[c])

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fricas [A]  time = 0.54, size = 116, normalized size = 1.27 \[ -\frac {\sqrt {\pi } \sqrt {-c} {\left (\cosh \left (\frac {b^{2} + 4 \, a c}{4 \, c}\right ) - \sinh \left (\frac {b^{2} + 4 \, a c}{4 \, c}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x - b\right )} \sqrt {-c}}{2 \, c}\right ) - \sqrt {\pi } \sqrt {c} {\left (\cosh \left (\frac {b^{2} + 4 \, a c}{4 \, c}\right ) + \sinh \left (\frac {b^{2} + 4 \, a c}{4 \, c}\right )\right )} \operatorname {erf}\left (\frac {2 \, c x - b}{2 \, \sqrt {c}}\right )}{4 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(sqrt(pi)*sqrt(-c)*(cosh(1/4*(b^2 + 4*a*c)/c) - sinh(1/4*(b^2 + 4*a*c)/c))*erf(1/2*(2*c*x - b)*sqrt(-c)/c
) - sqrt(pi)*sqrt(c)*(cosh(1/4*(b^2 + 4*a*c)/c) + sinh(1/4*(b^2 + 4*a*c)/c))*erf(1/2*(2*c*x - b)/sqrt(c)))/c

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giac [A]  time = 0.12, size = 81, normalized size = 0.89 \[ -\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {c} {\left (2 \, x - \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} + 4 \, a c}{4 \, c}\right )}}{4 \, \sqrt {c}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c} {\left (2 \, x - \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} + 4 \, a c}{4 \, c}\right )}}{4 \, \sqrt {-c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.09, size = 79, normalized size = 0.87 \[ \frac {\sqrt {\pi }\, {\mathrm e}^{-\frac {4 a c +b^{2}}{4 c}} \erf \left (\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right )}{4 \sqrt {-c}}-\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {4 a c +b^{2}}{4 c}} \erf \left (-\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right )}{4 \sqrt {c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(-c*x^2+b*x+a),x)

[Out]

1/4*Pi^(1/2)*exp(-1/4*(4*a*c+b^2)/c)/(-c)^(1/2)*erf((-c)^(1/2)*x+1/2*b/(-c)^(1/2))-1/4*Pi^(1/2)*exp(1/4*(4*a*c
+b^2)/c)/c^(1/2)*erf(-c^(1/2)*x+1/2*b/c^(1/2))

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maxima [B]  time = 0.65, size = 511, normalized size = 5.62 \[ -\frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x - b\right )} b {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {\frac {{\left (2 \, c x - b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {\frac {{\left (2 \, c x - b\right )}^{2}}{c}} \left (-c\right )^{\frac {3}{2}}} - \frac {2 \, c e^{\left (-\frac {{\left (2 \, c x - b\right )}^{2}}{4 \, c}\right )}}{\left (-c\right )^{\frac {3}{2}}}\right )} b e^{\left (a + \frac {b^{2}}{4 \, c}\right )}}{8 \, \sqrt {-c}} - \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x - b\right )} b^{2} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {\frac {{\left (2 \, c x - b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {\frac {{\left (2 \, c x - b\right )}^{2}}{c}} \left (-c\right )^{\frac {5}{2}}} - \frac {4 \, b c e^{\left (-\frac {{\left (2 \, c x - b\right )}^{2}}{4 \, c}\right )}}{\left (-c\right )^{\frac {5}{2}}} - \frac {4 \, {\left (2 \, c x - b\right )}^{3} \Gamma \left (\frac {3}{2}, \frac {{\left (2 \, c x - b\right )}^{2}}{4 \, c}\right )}{\left (\frac {{\left (2 \, c x - b\right )}^{2}}{c}\right )^{\frac {3}{2}} \left (-c\right )^{\frac {5}{2}}}\right )} c e^{\left (a + \frac {b^{2}}{4 \, c}\right )}}{8 \, \sqrt {-c}} + \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x - b\right )} b {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x - b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x - b\right )}^{2}}{c}} c^{\frac {3}{2}}} + \frac {2 \, e^{\left (\frac {{\left (2 \, c x - b\right )}^{2}}{4 \, c}\right )}}{\sqrt {c}}\right )} b e^{\left (-a - \frac {b^{2}}{4 \, c}\right )}}{8 \, \sqrt {c}} - \frac {1}{8} \, {\left (\frac {\sqrt {\pi } {\left (2 \, c x - b\right )} b^{2} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x - b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x - b\right )}^{2}}{c}} c^{\frac {5}{2}}} + \frac {4 \, b e^{\left (\frac {{\left (2 \, c x - b\right )}^{2}}{4 \, c}\right )}}{c^{\frac {3}{2}}} - \frac {4 \, {\left (2 \, c x - b\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, c x - b\right )}^{2}}{4 \, c}\right )}{\left (-\frac {{\left (2 \, c x - b\right )}^{2}}{c}\right )^{\frac {3}{2}} c^{\frac {5}{2}}}\right )} \sqrt {c} e^{\left (-a - \frac {b^{2}}{4 \, c}\right )} + x \cosh \left (c x^{2} - b x - a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(sqrt(pi)*(2*c*x - b)*b*(erf(1/2*sqrt((2*c*x - b)^2/c)) - 1)/(sqrt((2*c*x - b)^2/c)*(-c)^(3/2)) - 2*c*e^(
-1/4*(2*c*x - b)^2/c)/(-c)^(3/2))*b*e^(a + 1/4*b^2/c)/sqrt(-c) - 1/8*(sqrt(pi)*(2*c*x - b)*b^2*(erf(1/2*sqrt((
2*c*x - b)^2/c)) - 1)/(sqrt((2*c*x - b)^2/c)*(-c)^(5/2)) - 4*b*c*e^(-1/4*(2*c*x - b)^2/c)/(-c)^(5/2) - 4*(2*c*
x - b)^3*gamma(3/2, 1/4*(2*c*x - b)^2/c)/(((2*c*x - b)^2/c)^(3/2)*(-c)^(5/2)))*c*e^(a + 1/4*b^2/c)/sqrt(-c) +
1/8*(sqrt(pi)*(2*c*x - b)*b*(erf(1/2*sqrt(-(2*c*x - b)^2/c)) - 1)/(sqrt(-(2*c*x - b)^2/c)*c^(3/2)) + 2*e^(1/4*
(2*c*x - b)^2/c)/sqrt(c))*b*e^(-a - 1/4*b^2/c)/sqrt(c) - 1/8*(sqrt(pi)*(2*c*x - b)*b^2*(erf(1/2*sqrt(-(2*c*x -
b)^2/c)) - 1)/(sqrt(-(2*c*x - b)^2/c)*c^(5/2)) + 4*b*e^(1/4*(2*c*x - b)^2/c)/c^(3/2) - 4*(2*c*x - b)^3*gamma(
3/2, -1/4*(2*c*x - b)^2/c)/((-(2*c*x - b)^2/c)^(3/2)*c^(5/2)))*sqrt(c)*e^(-a - 1/4*b^2/c) + x*cosh(c*x^2 - b*x
- a)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \mathrm {cosh}\left (-c\,x^2+b\,x+a\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x - c*x^2),x)

[Out]

int(cosh(a + b*x - c*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(-c*x**2+b*x+a),x)

[Out]

Integral(cosh(a + b*x - c*x**2), x)

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