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

Optimal. Leaf size=111 \[ -\frac {\sqrt {\pi } b e^{\frac {b^2}{4 c}-a} \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}}-\frac {\sqrt {\pi } b e^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}}+\frac {\sinh \left (a+b x+c x^2\right )}{2 c} \]

[Out]

1/2*sinh(c*x^2+b*x+a)/c-1/8*b*exp(-a+1/4*b^2/c)*erf(1/2*(2*c*x+b)/c^(1/2))*Pi^(1/2)/c^(3/2)-1/8*b*exp(a-1/4*b^
2/c)*erfi(1/2*(2*c*x+b)/c^(1/2))*Pi^(1/2)/c^(3/2)

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Rubi [A]  time = 0.05, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5383, 5375, 2234, 2204, 2205} \[ -\frac {\sqrt {\pi } b e^{\frac {b^2}{4 c}-a} \text {Erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}}-\frac {\sqrt {\pi } b e^{a-\frac {b^2}{4 c}} \text {Erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}}+\frac {\sinh \left (a+b x+c x^2\right )}{2 c} \]

Antiderivative was successfully verified.

[In]

Int[x*Cosh[a + b*x + c*x^2],x]

[Out]

-(b*E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2)) - (b*E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi
[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2)) + Sinh[a + b*x + c*x^2]/(2*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 5383

Int[Cosh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*Sinh[a + b*x + c*x^2])/
(2*c), x] - Dist[(b*e - 2*c*d)/(2*c), Int[Cosh[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*
e - 2*c*d, 0]

Rubi steps

\begin {align*} \int x \cosh \left (a+b x+c x^2\right ) \, dx &=\frac {\sinh \left (a+b x+c x^2\right )}{2 c}-\frac {b \int \cosh \left (a+b x+c x^2\right ) \, dx}{2 c}\\ &=\frac {\sinh \left (a+b x+c x^2\right )}{2 c}-\frac {b \int e^{-a-b x-c x^2} \, dx}{4 c}-\frac {b \int e^{a+b x+c x^2} \, dx}{4 c}\\ &=\frac {\sinh \left (a+b x+c x^2\right )}{2 c}-\frac {\left (b e^{a-\frac {b^2}{4 c}}\right ) \int e^{\frac {(b+2 c x)^2}{4 c}} \, dx}{4 c}-\frac {\left (b e^{-a+\frac {b^2}{4 c}}\right ) \int e^{-\frac {(-b-2 c x)^2}{4 c}} \, dx}{4 c}\\ &=-\frac {b e^{-a+\frac {b^2}{4 c}} \sqrt {\pi } \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}}-\frac {b e^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}}+\frac {\sinh \left (a+b x+c x^2\right )}{2 c}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 130, normalized size = 1.17 \[ \frac {\sqrt {\pi } b \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right ) \left (\sinh \left (a-\frac {b^2}{4 c}\right )-\cosh \left (a-\frac {b^2}{4 c}\right )\right )-\sqrt {\pi } b \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right ) \left (\sinh \left (a-\frac {b^2}{4 c}\right )+\cosh \left (a-\frac {b^2}{4 c}\right )\right )+4 \sqrt {c} \sinh (a+x (b+c x))}{8 c^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Cosh[a + b*x + c*x^2],x]

[Out]

(b*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])]*(-Cosh[a - b^2/(4*c)] + Sinh[a - b^2/(4*c)]) - b*Sqrt[Pi]*Erfi[(b + 2
*c*x)/(2*Sqrt[c])]*(Cosh[a - b^2/(4*c)] + Sinh[a - b^2/(4*c)]) + 4*Sqrt[c]*Sinh[a + x*(b + c*x)])/(8*c^(3/2))

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fricas [B]  time = 0.51, size = 344, normalized size = 3.10 \[ \frac {2 \, c \cosh \left (c x^{2} + b x + a\right )^{2} + \sqrt {\pi } {\left (b \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + b \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + {\left (b \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + b \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \sinh \left (c x^{2} + b x + a\right )\right )} \sqrt {-c} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, c}\right ) - \sqrt {\pi } {\left (b \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) - b \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + {\left (b \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) - b \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \sinh \left (c x^{2} + b x + a\right )\right )} \sqrt {c} \operatorname {erf}\left (\frac {2 \, c x + b}{2 \, \sqrt {c}}\right ) + 4 \, c \cosh \left (c x^{2} + b x + a\right ) \sinh \left (c x^{2} + b x + a\right ) + 2 \, c \sinh \left (c x^{2} + b x + a\right )^{2} - 2 \, c}{8 \, {\left (c^{2} \cosh \left (c x^{2} + b x + a\right ) + c^{2} \sinh \left (c x^{2} + b x + a\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.12, size = 121, normalized size = 1.09 \[ \frac {\frac {\sqrt {\pi } b \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} - 4 \, a c}{4 \, c}\right )}}{\sqrt {c}} - 2 \, e^{\left (-c x^{2} - b x - a\right )}}{8 \, c} + \frac {\frac {\sqrt {\pi } b \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )}}{\sqrt {-c}} + 2 \, e^{\left (c x^{2} + b x + a\right )}}{8 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.10, size = 124, normalized size = 1.12 \[ -\frac {{\mathrm e}^{-c \,x^{2}-b x -a}}{4 c}-\frac {b \sqrt {\pi }\, {\mathrm e}^{-\frac {4 a c -b^{2}}{4 c}} \erf \left (\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right )}{8 c^{\frac {3}{2}}}+\frac {{\mathrm e}^{c \,x^{2}+b x +a}}{4 c}+\frac {b \sqrt {\pi }\, {\mathrm e}^{\frac {4 a c -b^{2}}{4 c}} \erf \left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right )}{8 c \sqrt {-c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/c*exp(-c*x^2-b*x-a)-1/8*b/c^(3/2)*Pi^(1/2)*exp(-1/4*(4*a*c-b^2)/c)*erf(c^(1/2)*x+1/2*b/c^(1/2))+1/4/c*exp
(c*x^2+b*x+a)+1/8*b/c*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 = 0.61, size = 611, normalized size = 5.50 \[ \frac {1}{2} \, x^{2} \cosh \left (c x^{2} + b x + a\right ) - \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}} 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 )} b e^{\left (a - \frac {b^{2}}{4 \, c}\right )}}{32 \, \sqrt {c}} + \frac {1}{32} \, {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{3} {\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 {7}{2}}} - \frac {6 \, b^{2} e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{c^{\frac {5}{2}}} - \frac {12 \, {\left (2 \, c x + b\right )}^{3} b \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 {7}{2}}} + \frac {8 \, \Gamma \left (2, -\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}{c^{\frac {3}{2}}}\right )} \sqrt {c} e^{\left (a - \frac {b^{2}}{4 \, c}\right )} - \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 )} b e^{\left (-a + \frac {b^{2}}{4 \, c}\right )}}{32 \, \sqrt {-c}} - \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{3} {\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 {7}{2}}} + \frac {6 \, b^{2} c e^{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{\left (-c\right )^{\frac {7}{2}}} - \frac {12 \, {\left (2 \, c x + b\right )}^{3} b \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 {7}{2}}} + \frac {8 \, c^{2} \Gamma \left (2, \frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}{\left (-c\right )^{\frac {7}{2}}}\right )} c e^{\left (-a + \frac {b^{2}}{4 \, c}\right )}}{32 \, \sqrt {-c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*cosh(c*x^2 + b*x + a) - 1/32*(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)))*b*e^(a - 1/4*b^2/c)/sqrt(c) + 1/32*(sqrt(pi)*(2*c*x + b)*b^3*(erf(1/2*s
qrt(-(2*c*x + b)^2/c)) - 1)/(sqrt(-(2*c*x + b)^2/c)*c^(7/2)) - 6*b^2*e^(1/4*(2*c*x + b)^2/c)/c^(5/2) - 12*(2*c
*x + b)^3*b*gamma(3/2, -1/4*(2*c*x + b)^2/c)/((-(2*c*x + b)^2/c)^(3/2)*c^(7/2)) + 8*gamma(2, -1/4*(2*c*x + b)^
2/c)/c^(3/2))*sqrt(c)*e^(a - 1/4*b^2/c) - 1/32*(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)))*b*e^(-a + 1/4*b^2/c)/sqrt(-c) - 1/32*(sqrt(pi)*(2*c*x
+ b)*b^3*(erf(1/2*sqrt((2*c*x + b)^2/c)) - 1)/(sqrt((2*c*x + b)^2/c)*(-c)^(7/2)) + 6*b^2*c*e^(-1/4*(2*c*x + b
)^2/c)/(-c)^(7/2) - 12*(2*c*x + b)^3*b*gamma(3/2, 1/4*(2*c*x + b)^2/c)/(((2*c*x + b)^2/c)^(3/2)*(-c)^(7/2)) +
8*c^2*gamma(2, 1/4*(2*c*x + b)^2/c)/(-c)^(7/2))*c*e^(-a + 1/4*b^2/c)/sqrt(-c)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cosh(a + b*x + c*x^2),x)

[Out]

int(x*cosh(a + b*x + c*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*cosh(a + b*x + c*x**2), x)

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