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

Optimal. Leaf size=110 \[ \frac {\sqrt {\frac {\pi }{2}} e^{\frac {b^2}{2 c}-2 a} \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{8 \sqrt {c}}+\frac {\sqrt {\frac {\pi }{2}} e^{2 a-\frac {b^2}{2 c}} \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{8 \sqrt {c}}+\frac {x}{2} \]

[Out]

1/2*x+1/16*exp(-2*a+1/2*b^2/c)*erf(1/2*(2*c*x+b)*2^(1/2)/c^(1/2))*2^(1/2)*Pi^(1/2)/c^(1/2)+1/16*exp(2*a-1/2*b^
2/c)*erfi(1/2*(2*c*x+b)*2^(1/2)/c^(1/2))*2^(1/2)*Pi^(1/2)/c^(1/2)

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Rubi [A]  time = 0.07, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5377, 5375, 2234, 2204, 2205} \[ \frac {\sqrt {\frac {\pi }{2}} e^{\frac {b^2}{2 c}-2 a} \text {Erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{8 \sqrt {c}}+\frac {\sqrt {\frac {\pi }{2}} e^{2 a-\frac {b^2}{2 c}} \text {Erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{8 \sqrt {c}}+\frac {x}{2} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x + c*x^2]^2,x]

[Out]

x/2 + (E^(-2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(8*Sqrt[c]) + (E^(2*a - b^2/(2*c))*
Sqrt[Pi/2]*Erfi[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(8*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]

Rule 5377

Int[Cosh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]^(n_), x_Symbol] :> Int[ExpandTrigReduce[Cosh[a + b*x + c*x^2]^n, x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 1]

Rubi steps

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

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Mathematica [A]  time = 0.15, size = 140, normalized size = 1.27 \[ \frac {\sqrt {\pi } \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right ) \left (\cosh \left (2 a-\frac {b^2}{2 c}\right )-\sinh \left (2 a-\frac {b^2}{2 c}\right )\right )+\sqrt {\pi } \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right ) \left (\sinh \left (2 a-\frac {b^2}{2 c}\right )+\cosh \left (2 a-\frac {b^2}{2 c}\right )\right )+4 \sqrt {2} \sqrt {c} x}{8 \sqrt {2} \sqrt {c}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x + c*x^2]^2,x]

[Out]

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

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fricas [A]  time = 0.61, size = 128, normalized size = 1.16 \[ -\frac {\sqrt {2} \sqrt {\pi } \sqrt {-c} {\left (\cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + \sinh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )\right )} \operatorname {erf}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, c}\right ) - \sqrt {2} \sqrt {\pi } \sqrt {c} {\left (\cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) - \sinh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )\right )} \operatorname {erf}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )}}{2 \, \sqrt {c}}\right ) - 8 \, c x}{16 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(sqrt(2)*sqrt(pi)*sqrt(-c)*(cosh(-1/2*(b^2 - 4*a*c)/c) + sinh(-1/2*(b^2 - 4*a*c)/c))*erf(1/2*sqrt(2)*(2*
c*x + b)*sqrt(-c)/c) - sqrt(2)*sqrt(pi)*sqrt(c)*(cosh(-1/2*(b^2 - 4*a*c)/c) - sinh(-1/2*(b^2 - 4*a*c)/c))*erf(
1/2*sqrt(2)*(2*c*x + b)/sqrt(c)) - 8*c*x)/c

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giac [A]  time = 0.13, size = 94, normalized size = 0.85 \[ -\frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} - 4 \, a c}{2 \, c}\right )}}{16 \, \sqrt {c}} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {-c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )}}{16 \, \sqrt {-c}} + \frac {1}{2} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*sqrt(2)*sqrt(pi)*erf(-1/2*sqrt(2)*sqrt(c)*(2*x + b/c))*e^(1/2*(b^2 - 4*a*c)/c)/sqrt(c) - 1/16*sqrt(2)*sq
rt(pi)*erf(-1/2*sqrt(2)*sqrt(-c)*(2*x + b/c))*e^(-1/2*(b^2 - 4*a*c)/c)/sqrt(-c) + 1/2*x

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maple [A]  time = 0.18, size = 94, normalized size = 0.85 \[ \frac {x}{2}+\frac {\sqrt {\pi }\, {\mathrm e}^{-\frac {4 a c -b^{2}}{2 c}} \sqrt {2}\, \erf \left (\sqrt {2}\, \sqrt {c}\, x +\frac {b \sqrt {2}}{2 \sqrt {c}}\right )}{16 \sqrt {c}}-\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {4 a c -b^{2}}{2 c}} \erf \left (-\sqrt {-2 c}\, x +\frac {b}{\sqrt {-2 c}}\right )}{8 \sqrt {-2 c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x+1/16*Pi^(1/2)*exp(-1/2*(4*a*c-b^2)/c)*2^(1/2)/c^(1/2)*erf(2^(1/2)*c^(1/2)*x+1/2*b*2^(1/2)/c^(1/2))-1/8*P
i^(1/2)*exp(1/2*(4*a*c-b^2)/c)/(-2*c)^(1/2)*erf(-(-2*c)^(1/2)*x+b/(-2*c)^(1/2))

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maxima [A]  time = 0.40, size = 96, normalized size = 0.87 \[ \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} \sqrt {-c} x - \frac {\sqrt {2} b}{2 \, \sqrt {-c}}\right ) e^{\left (2 \, a - \frac {b^{2}}{2 \, c}\right )}}{16 \, \sqrt {-c}} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} \sqrt {c} x + \frac {\sqrt {2} b}{2 \, \sqrt {c}}\right ) e^{\left (-2 \, a + \frac {b^{2}}{2 \, c}\right )}}{16 \, \sqrt {c}} + \frac {1}{2} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*sqrt(2)*sqrt(pi)*erf(sqrt(2)*sqrt(-c)*x - 1/2*sqrt(2)*b/sqrt(-c))*e^(2*a - 1/2*b^2/c)/sqrt(-c) + 1/16*sqr
t(2)*sqrt(pi)*erf(sqrt(2)*sqrt(c)*x + 1/2*sqrt(2)*b/sqrt(c))*e^(-2*a + 1/2*b^2/c)/sqrt(c) + 1/2*x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x + c*x^2)^2,x)

[Out]

int(cosh(a + b*x + c*x^2)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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