\(\int x \cosh ^2(a+b x-c x^2) \, dx\) [21]

Optimal result

Integrand size = 16, antiderivative size = 136 \[ \int x \cosh ^2\left (a+b x-c x^2\right ) \, dx=\frac {x^2}{4}-\frac {b e^{2 a+\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {b-2 c x}{\sqrt {2} \sqrt {c}}\right )}{16 c^{3/2}}-\frac {b e^{-2 a-\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {b-2 c x}{\sqrt {2} \sqrt {c}}\right )}{16 c^{3/2}}-\frac {\sinh \left (2 a+2 b x-2 c x^2\right )}{8 c} \]

[Out]

1/4*x^2-1/8*sinh(-2*c*x^2+2*b*x+2*a)/c-1/32*b*exp(2*a+1/2*b^2/c)*erf(1/2*(-2*c*x+b)*2^(1/2)/c^(1/2))*2^(1/2)*P
i^(1/2)/c^(3/2)-1/32*b*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^(3/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5503, 5491, 5483, 2266, 2236, 2235} \[ \int x \cosh ^2\left (a+b x-c x^2\right ) \, dx=-\frac {\sqrt {\frac {\pi }{2}} b e^{2 a+\frac {b^2}{2 c}} \text {erf}\left (\frac {b-2 c x}{\sqrt {2} \sqrt {c}}\right )}{16 c^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} b e^{-2 a-\frac {b^2}{2 c}} \text {erfi}\left (\frac {b-2 c x}{\sqrt {2} \sqrt {c}}\right )}{16 c^{3/2}}-\frac {\sinh \left (2 a+2 b x-2 c x^2\right )}{8 c}+\frac {x^2}{4} \]

[In]

Int[x*Cosh[a + b*x - c*x^2]^2,x]

[Out]

x^2/4 - (b*E^(2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b - 2*c*x)/(Sqrt[2]*Sqrt[c])])/(16*c^(3/2)) - (b*E^(-2*a - b^2/
(2*c))*Sqrt[Pi/2]*Erfi[(b - 2*c*x)/(Sqrt[2]*Sqrt[c])])/(16*c^(3/2)) - Sinh[2*a + 2*b*x - 2*c*x^2]/(8*c)

Rule 2235

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 2236

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 2266

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 5483

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 5491

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]

Rule 5503

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {x}{2}+\frac {1}{2} x \cosh \left (2 a+2 b x-2 c x^2\right )\right ) \, dx \\ & = \frac {x^2}{4}+\frac {1}{2} \int x \cosh \left (2 a+2 b x-2 c x^2\right ) \, dx \\ & = \frac {x^2}{4}-\frac {\sinh \left (2 a+2 b x-2 c x^2\right )}{8 c}+\frac {b \int \cosh \left (2 a+2 b x-2 c x^2\right ) \, dx}{4 c} \\ & = \frac {x^2}{4}-\frac {\sinh \left (2 a+2 b x-2 c x^2\right )}{8 c}+\frac {b \int e^{2 a+2 b x-2 c x^2} \, dx}{8 c}+\frac {b \int e^{-2 a-2 b x+2 c x^2} \, dx}{8 c} \\ & = \frac {x^2}{4}-\frac {\sinh \left (2 a+2 b x-2 c x^2\right )}{8 c}+\frac {\left (b e^{-2 a-\frac {b^2}{2 c}}\right ) \int e^{\frac {(-2 b+4 c x)^2}{8 c}} \, dx}{8 c}+\frac {\left (b e^{2 a+\frac {b^2}{2 c}}\right ) \int e^{-\frac {(2 b-4 c x)^2}{8 c}} \, dx}{8 c} \\ & = \frac {x^2}{4}-\frac {b 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 )}{16 c^{3/2}}-\frac {b 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 )}{16 c^{3/2}}-\frac {\sinh \left (2 a+2 b x-2 c x^2\right )}{8 c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.18 \[ \int x \cosh ^2\left (a+b x-c x^2\right ) \, dx=\frac {b \sqrt {2 \pi } \text {erfi}\left (\frac {-b+2 c x}{\sqrt {2} \sqrt {c}}\right ) \left (\cosh \left (2 a+\frac {b^2}{2 c}\right )-\sinh \left (2 a+\frac {b^2}{2 c}\right )\right )+b \sqrt {2 \pi } \text {erf}\left (\frac {-b+2 c x}{\sqrt {2} \sqrt {c}}\right ) \left (\cosh \left (2 a+\frac {b^2}{2 c}\right )+\sinh \left (2 a+\frac {b^2}{2 c}\right )\right )+4 \sqrt {c} \left (2 c x^2-\sinh (2 (a+x (b-c x)))\right )}{32 c^{3/2}} \]

[In]

Integrate[x*Cosh[a + b*x - c*x^2]^2,x]

[Out]

(b*Sqrt[2*Pi]*Erfi[(-b + 2*c*x)/(Sqrt[2]*Sqrt[c])]*(Cosh[2*a + b^2/(2*c)] - Sinh[2*a + b^2/(2*c)]) + b*Sqrt[2*
Pi]*Erf[(-b + 2*c*x)/(Sqrt[2]*Sqrt[c])]*(Cosh[2*a + b^2/(2*c)] + Sinh[2*a + b^2/(2*c)]) + 4*Sqrt[c]*(2*c*x^2 -
Sinh[2*(a + x*(b - c*x))]))/(32*c^(3/2))

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.01

method	result	size
risch	\(\frac {x^{2}}{4}+\frac {{\mathrm e}^{2 c \,x^{2}-2 b x -2 a}}{16 c}+\frac {b \sqrt {\pi }\, {\mathrm e}^{-\frac {4 a c +b^{2}}{2 c}} \operatorname {erf}\left (\sqrt {-2 c}\, x +\frac {b}{\sqrt {-2 c}}\right )}{16 c \sqrt {-2 c}}-\frac {{\mathrm e}^{-2 c \,x^{2}+2 b x +2 a}}{16 c}-\frac {b \sqrt {\pi }\, {\mathrm e}^{\frac {4 a c +b^{2}}{2 c}} \sqrt {2}\, \operatorname {erf}\left (-\sqrt {2}\, \sqrt {c}\, x +\frac {b \sqrt {2}}{2 \sqrt {c}}\right )}{32 c^{\frac {3}{2}}}\)	\(137\)

[In]

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

[Out]

1/4*x^2+1/16/c*exp(2*c*x^2-2*b*x-2*a)+1/16*b/c*Pi^(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))-1/16/c*exp(-2*c*x^2+2*b*x+2*a)-1/32*b/c^(3/2)*Pi^(1/2)*exp(1/2*(4*a*c+b^2)/c)*2^(1/2)*erf(-2
^(1/2)*c^(1/2)*x+1/2*b*2^(1/2)/c^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 729 vs. \(2 (110) = 220\).

Time = 0.28 (sec) , antiderivative size = 729, normalized size of antiderivative = 5.36 \[ \int x \cosh ^2\left (a+b x-c x^2\right ) \, dx=\frac {8 \, c^{2} x^{2} \cosh \left (c x^{2} - b x - a\right )^{2} + 2 \, c \cosh \left (c x^{2} - b x - a\right )^{4} + 8 \, c \cosh \left (c x^{2} - b x - a\right ) \sinh \left (c x^{2} - b x - a\right )^{3} + 2 \, c \sinh \left (c x^{2} - b x - a\right )^{4} - \sqrt {2} \sqrt {\pi } {\left (b \cosh \left (c x^{2} - b x - a\right )^{2} \cosh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) - b \cosh \left (c x^{2} - b x - a\right )^{2} \sinh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) + {\left (b \cosh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) - b \sinh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right )\right )} \sinh \left (c x^{2} - b x - a\right )^{2} + 2 \, {\left (b \cosh \left (c x^{2} - b x - a\right ) \cosh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) - b \cosh \left (c x^{2} - b x - a\right ) \sinh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right )\right )} \sinh \left (c x^{2} - b x - a\right )\right )} \sqrt {-c} \operatorname {erf}\left (\frac {\sqrt {2} {\left (2 \, c x - b\right )} \sqrt {-c}}{2 \, c}\right ) + \sqrt {2} \sqrt {\pi } {\left (b \cosh \left (c x^{2} - b x - a\right )^{2} \cosh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) + b \cosh \left (c x^{2} - b x - a\right )^{2} \sinh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) + {\left (b \cosh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) + b \sinh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right )\right )} \sinh \left (c x^{2} - b x - a\right )^{2} + 2 \, {\left (b \cosh \left (c x^{2} - b x - a\right ) \cosh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) + b \cosh \left (c x^{2} - b x - a\right ) \sinh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right )\right )} \sinh \left (c x^{2} - b x - a\right )\right )} \sqrt {c} \operatorname {erf}\left (\frac {\sqrt {2} {\left (2 \, c x - b\right )}}{2 \, \sqrt {c}}\right ) + 4 \, {\left (2 \, c^{2} x^{2} + 3 \, c \cosh \left (c x^{2} - b x - a\right )^{2}\right )} \sinh \left (c x^{2} - b x - a\right )^{2} + 8 \, {\left (2 \, c^{2} x^{2} \cosh \left (c x^{2} - b x - a\right ) + c \cosh \left (c x^{2} - b x - a\right )^{3}\right )} \sinh \left (c x^{2} - b x - a\right ) - 2 \, c}{32 \, {\left (c^{2} \cosh \left (c x^{2} - b x - a\right )^{2} + 2 \, c^{2} \cosh \left (c x^{2} - b x - a\right ) \sinh \left (c x^{2} - b x - a\right ) + c^{2} \sinh \left (c x^{2} - b x - a\right )^{2}\right )}} \]

[In]

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

[Out]

1/32*(8*c^2*x^2*cosh(c*x^2 - b*x - a)^2 + 2*c*cosh(c*x^2 - b*x - a)^4 + 8*c*cosh(c*x^2 - b*x - a)*sinh(c*x^2 -
b*x - a)^3 + 2*c*sinh(c*x^2 - b*x - a)^4 - sqrt(2)*sqrt(pi)*(b*cosh(c*x^2 - b*x - a)^2*cosh(1/2*(b^2 + 4*a*c)
/c) - b*cosh(c*x^2 - b*x - a)^2*sinh(1/2*(b^2 + 4*a*c)/c) + (b*cosh(1/2*(b^2 + 4*a*c)/c) - b*sinh(1/2*(b^2 + 4
*a*c)/c))*sinh(c*x^2 - b*x - a)^2 + 2*(b*cosh(c*x^2 - b*x - a)*cosh(1/2*(b^2 + 4*a*c)/c) - b*cosh(c*x^2 - b*x
- a)*sinh(1/2*(b^2 + 4*a*c)/c))*sinh(c*x^2 - b*x - a))*sqrt(-c)*erf(1/2*sqrt(2)*(2*c*x - b)*sqrt(-c)/c) + sqrt
(2)*sqrt(pi)*(b*cosh(c*x^2 - b*x - a)^2*cosh(1/2*(b^2 + 4*a*c)/c) + b*cosh(c*x^2 - b*x - a)^2*sinh(1/2*(b^2 +
4*a*c)/c) + (b*cosh(1/2*(b^2 + 4*a*c)/c) + b*sinh(1/2*(b^2 + 4*a*c)/c))*sinh(c*x^2 - b*x - a)^2 + 2*(b*cosh(c*
x^2 - b*x - a)*cosh(1/2*(b^2 + 4*a*c)/c) + b*cosh(c*x^2 - b*x - a)*sinh(1/2*(b^2 + 4*a*c)/c))*sinh(c*x^2 - b*x
- a))*sqrt(c)*erf(1/2*sqrt(2)*(2*c*x - b)/sqrt(c)) + 4*(2*c^2*x^2 + 3*c*cosh(c*x^2 - b*x - a)^2)*sinh(c*x^2 -
b*x - a)^2 + 8*(2*c^2*x^2*cosh(c*x^2 - b*x - a) + c*cosh(c*x^2 - b*x - a)^3)*sinh(c*x^2 - b*x - a) - 2*c)/(c^
2*cosh(c*x^2 - b*x - a)^2 + 2*c^2*cosh(c*x^2 - b*x - a)*sinh(c*x^2 - b*x - a) + c^2*sinh(c*x^2 - b*x - a)^2)

Sympy [F]

\[ \int x \cosh ^2\left (a+b x-c x^2\right ) \, dx=\int x \cosh ^{2}{\left (a + b x - c x^{2} \right )}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.59 \[ \int x \cosh ^2\left (a+b x-c x^2\right ) \, dx=\frac {1}{4} \, x^{2} + \frac {\sqrt {2} {\left (\frac {\sqrt {\pi } {\left (2 \, c x - b\right )} b {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {\frac {{\left (2 \, c x - b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {\frac {{\left (2 \, c x - b\right )}^{2}}{c}} \left (-c\right )^{\frac {3}{2}}} - \frac {\sqrt {2} c e^{\left (-\frac {{\left (2 \, c x - b\right )}^{2}}{2 \, c}\right )}}{\left (-c\right )^{\frac {3}{2}}}\right )} e^{\left (2 \, a + \frac {b^{2}}{2 \, c}\right )}}{32 \, \sqrt {-c}} + \frac {\sqrt {2} {\left (\frac {\sqrt {\pi } {\left (2 \, c x - b\right )} b {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (2 \, c x - b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x - b\right )}^{2}}{c}} c^{\frac {3}{2}}} + \frac {\sqrt {2} e^{\left (\frac {{\left (2 \, c x - b\right )}^{2}}{2 \, c}\right )}}{\sqrt {c}}\right )} e^{\left (-2 \, a - \frac {b^{2}}{2 \, c}\right )}}{32 \, \sqrt {c}} \]

[In]

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

[Out]

1/4*x^2 + 1/32*sqrt(2)*(sqrt(pi)*(2*c*x - b)*b*(erf(sqrt(1/2)*sqrt((2*c*x - b)^2/c)) - 1)/(sqrt((2*c*x - b)^2/
c)*(-c)^(3/2)) - sqrt(2)*c*e^(-1/2*(2*c*x - b)^2/c)/(-c)^(3/2))*e^(2*a + 1/2*b^2/c)/sqrt(-c) + 1/32*sqrt(2)*(s
qrt(pi)*(2*c*x - b)*b*(erf(sqrt(1/2)*sqrt(-(2*c*x - b)^2/c)) - 1)/(sqrt(-(2*c*x - b)^2/c)*c^(3/2)) + sqrt(2)*e
^(1/2*(2*c*x - b)^2/c)/sqrt(c))*e^(-2*a - 1/2*b^2/c)/sqrt(c)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.06 \[ \int x \cosh ^2\left (a+b x-c x^2\right ) \, dx=\frac {1}{4} \, x^{2} - \frac {\frac {\sqrt {2} \sqrt {\pi } b \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {c} {\left (2 \, x - \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} + 4 \, a c}{2 \, c}\right )}}{\sqrt {c}} + 2 \, e^{\left (-2 \, c x^{2} + 2 \, b x + 2 \, a\right )}}{32 \, c} - \frac {\frac {\sqrt {2} \sqrt {\pi } b \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {-c} {\left (2 \, x - \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} + 4 \, a c}{2 \, c}\right )}}{\sqrt {-c}} - 2 \, e^{\left (2 \, c x^{2} - 2 \, b x - 2 \, a\right )}}{32 \, c} \]

[In]

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

[Out]

1/4*x^2 - 1/32*(sqrt(2)*sqrt(pi)*b*erf(-1/2*sqrt(2)*sqrt(c)*(2*x - b/c))*e^(1/2*(b^2 + 4*a*c)/c)/sqrt(c) + 2*e
^(-2*c*x^2 + 2*b*x + 2*a))/c - 1/32*(sqrt(2)*sqrt(pi)*b*erf(-1/2*sqrt(2)*sqrt(-c)*(2*x - b/c))*e^(-1/2*(b^2 +
4*a*c)/c)/sqrt(-c) - 2*e^(2*c*x^2 - 2*b*x - 2*a))/c

Mupad [F(-1)]

Timed out. \[ \int x \cosh ^2\left (a+b x-c x^2\right ) \, dx=\int x\,{\mathrm {cosh}\left (-c\,x^2+b\,x+a\right )}^2 \,d x \]

[In]

int(x*cosh(a + b*x - c*x^2)^2,x)

[Out]

int(x*cosh(a + b*x - c*x^2)^2, x)