\(\int x^2 \cosh ((a+b x)^2) \, dx\) [54]

Optimal result

Integrand size = 12, antiderivative size = 113 \[ \int x^2 \cosh \left ((a+b x)^2\right ) \, dx=\frac {\sqrt {\pi } \text {erf}(a+b x)}{8 b^3}+\frac {a^2 \sqrt {\pi } \text {erf}(a+b x)}{4 b^3}-\frac {\sqrt {\pi } \text {erfi}(a+b x)}{8 b^3}+\frac {a^2 \sqrt {\pi } \text {erfi}(a+b x)}{4 b^3}-\frac {a \sinh \left ((a+b x)^2\right )}{b^3}+\frac {(a+b x) \sinh \left ((a+b x)^2\right )}{2 b^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5473, 6874, 5407, 2235, 2236, 5429, 2717, 5433, 5406} \[ \int x^2 \cosh \left ((a+b x)^2\right ) \, dx=\frac {\sqrt {\pi } a^2 \text {erf}(a+b x)}{4 b^3}+\frac {\sqrt {\pi } a^2 \text {erfi}(a+b x)}{4 b^3}+\frac {\sqrt {\pi } \text {erf}(a+b x)}{8 b^3}-\frac {\sqrt {\pi } \text {erfi}(a+b x)}{8 b^3}-\frac {a \sinh \left ((a+b x)^2\right )}{b^3}+\frac {(a+b x) \sinh \left ((a+b x)^2\right )}{2 b^3} \]

[In]

Int[x^2*Cosh[(a + b*x)^2],x]

[Out]

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

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 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 5406

Int[Sinh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Dist[1/2, Int[E^(c + d*x^n), x], x] - Dist[1/2, Int[E^(-c - d*
x^n), x], x] /; FreeQ[{c, d}, x] && IGtQ[n, 1]

Rule 5407

Int[Cosh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Dist[1/2, Int[E^(c + d*x^n), x], x] + Dist[1/2, Int[E^(-c - d*
x^n), x], x] /; FreeQ[{c, d}, x] && IGtQ[n, 1]

Rule 5429

Int[((a_.) + Cosh[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Cosh[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Sim
plify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rule 5433

Int[Cosh[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[e^(n - 1)*(e*x)^(m - n + 1)*(Sinh[c +
d*x^n]/(d*n)), x] - Dist[e^n*((m - n + 1)/(d*n)), Int[(e*x)^(m - n)*Sinh[c + d*x^n], x], x] /; FreeQ[{c, d, e}
, x] && IGtQ[n, 0] && LtQ[0, n, m + 1]

Rule 5473

Int[((a_.) + Cosh[(c_.) + (d_.)*(u_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/Coefficient[u, x, 1]^(
m + 1), Subst[Int[(x - Coefficient[u, x, 0])^m*(a + b*Cosh[c + d*x^n])^p, x], x, u], x] /; FreeQ[{a, b, c, d,
n, p}, x] && LinearQ[u, x] && NeQ[u, x] && IntegerQ[m]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.55 \[ \int x^2 \cosh \left ((a+b x)^2\right ) \, dx=\frac {\left (1+2 a^2\right ) \sqrt {\pi } \text {erf}(a+b x)+\left (-1+2 a^2\right ) \sqrt {\pi } \text {erfi}(a+b x)-4 (a-b x) \sinh \left ((a+b x)^2\right )}{8 b^3} \]

[In]

Integrate[x^2*Cosh[(a + b*x)^2],x]

[Out]

((1 + 2*a^2)*Sqrt[Pi]*Erf[a + b*x] + (-1 + 2*a^2)*Sqrt[Pi]*Erfi[a + b*x] - 4*(a - b*x)*Sinh[(a + b*x)^2])/(8*b
^3)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.08 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.20

method	result	size
risch	\(-\frac {x \,{\mathrm e}^{-\left (b x +a \right )^{2}}}{4 b^{2}}+\frac {a \,{\mathrm e}^{-\left (b x +a \right )^{2}}}{4 b^{3}}+\frac {a^{2} \operatorname {erf}\left (b x +a \right ) \sqrt {\pi }}{4 b^{3}}+\frac {\operatorname {erf}\left (b x +a \right ) \sqrt {\pi }}{8 b^{3}}+\frac {x \,{\mathrm e}^{\left (b x +a \right )^{2}}}{4 b^{2}}-\frac {a \,{\mathrm e}^{\left (b x +a \right )^{2}}}{4 b^{3}}-\frac {i a^{2} \sqrt {\pi }\, \operatorname {erf}\left (i x b +i a \right )}{4 b^{3}}+\frac {i \sqrt {\pi }\, \operatorname {erf}\left (i x b +i a \right )}{8 b^{3}}\)	\(136\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (95) = 190\).

Time = 0.26 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.84 \[ \int x^2 \cosh \left ((a+b x)^2\right ) \, dx=-\frac {2 \, b^{2} x - 2 \, {\left (b^{2} x - a b\right )} \cosh \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )^{2} + \sqrt {\pi } \sqrt {-b^{2}} {\left ({\left (2 \, a^{2} - 1\right )} \cosh \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right ) + {\left (2 \, a^{2} - 1\right )} \sinh \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )\right )} \operatorname {erf}\left (\frac {\sqrt {-b^{2}} {\left (b x + a\right )}}{b}\right ) - \sqrt {\pi } \sqrt {b^{2}} {\left ({\left (2 \, a^{2} + 1\right )} \cosh \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right ) + {\left (2 \, a^{2} + 1\right )} \sinh \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )\right )} \operatorname {erf}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - 4 \, {\left (b^{2} x - a b\right )} \cosh \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right ) \sinh \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right ) - 2 \, {\left (b^{2} x - a b\right )} \sinh \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )^{2} - 2 \, a b}{8 \, {\left (b^{4} \cosh \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right ) + b^{4} \sinh \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 818 vs. \(2 (95) = 190\).

Time = 0.44 (sec) , antiderivative size = 818, normalized size of antiderivative = 7.24 \[ \int x^2 \cosh \left ((a+b x)^2\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/3*x^3*cosh((b*x + a)^2) - 1/6*((sqrt(pi)*(b^2*x + a*b)*a^3*b^4*(erf(sqrt((b^2*x + a*b)^2)/b) - 1)/(sqrt((b^2
*x + a*b)^2)*(-b^2)^(7/2)) - 3*(b^2*x + a*b)^3*a*b^4*gamma(3/2, (b^2*x + a*b)^2/b^2)/(((b^2*x + a*b)^2)^(3/2)*
(-b^2)^(7/2)) + 3*a^2*b^4*e^(-(b^2*x + a*b)^2/b^2)/(-b^2)^(7/2) + b^4*gamma(2, (b^2*x + a*b)^2/b^2)/(-b^2)^(7/
2))*a/sqrt(-b^2) + (sqrt(pi)*(b^2*x + a*b)*a^4*b^5*(erf(sqrt((b^2*x + a*b)^2)/b) - 1)/(sqrt((b^2*x + a*b)^2)*(
-b^2)^(9/2)) - 6*(b^2*x + a*b)^3*a^2*b^5*gamma(3/2, (b^2*x + a*b)^2/b^2)/(((b^2*x + a*b)^2)^(3/2)*(-b^2)^(9/2)
) + 4*a^3*b^5*e^(-(b^2*x + a*b)^2/b^2)/(-b^2)^(9/2) - (b^2*x + a*b)^5*b^5*gamma(5/2, (b^2*x + a*b)^2/b^2)/(((b
^2*x + a*b)^2)^(5/2)*(-b^2)^(9/2)) + 4*a*b^5*gamma(2, (b^2*x + a*b)^2/b^2)/(-b^2)^(9/2))*b/sqrt(-b^2) - a*(sqr
t(pi)*(b^2*x + a*b)*a^3*(erf(sqrt(-(b^2*x + a*b)^2/b^2)) - 1)/(b^4*sqrt(-(b^2*x + a*b)^2/b^2)) - 3*a^2*e^((b^2
*x + a*b)^2/b^2)/b^3 + gamma(2, -(b^2*x + a*b)^2/b^2)/b^3 - 3*(b^2*x + a*b)^3*a*gamma(3/2, -(b^2*x + a*b)^2/b^
2)/(b^6*(-(b^2*x + a*b)^2/b^2)^(3/2)))/b + sqrt(pi)*(b^2*x + a*b)*a^4*(erf(sqrt(-(b^2*x + a*b)^2/b^2)) - 1)/(b
^5*sqrt(-(b^2*x + a*b)^2/b^2)) - 4*a^3*e^((b^2*x + a*b)^2/b^2)/b^4 + 4*a*gamma(2, -(b^2*x + a*b)^2/b^2)/b^4 -
6*(b^2*x + a*b)^3*a^2*gamma(3/2, -(b^2*x + a*b)^2/b^2)/(b^7*(-(b^2*x + a*b)^2/b^2)^(3/2)) - (b^2*x + a*b)^5*ga
mma(5/2, -(b^2*x + a*b)^2/b^2)/(b^9*(-(b^2*x + a*b)^2/b^2)^(5/2)))*b

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.21 \[ \int x^2 \cosh \left ((a+b x)^2\right ) \, dx=-\frac {\frac {i \, \sqrt {\pi } {\left (2 \, a^{2} - 1\right )} \operatorname {erf}\left (i \, b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {2 \, {\left (b {\left (x + \frac {a}{b}\right )} - 2 \, a\right )} e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}}{b}}{8 \, b^{2}} - \frac {\frac {\sqrt {\pi } {\left (2 \, a^{2} + 1\right )} \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} + \frac {2 \, {\left (b {\left (x + \frac {a}{b}\right )} - 2 \, a\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{8 \, b^{2}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(x^2*cosh((a + b*x)^2),x)

[Out]

int(x^2*cosh((a + b*x)^2), x)