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

Optimal. Leaf size=54 \[ -\frac {\sqrt {\pi } a \text {erf}(a+b x)}{4 b^2}-\frac {\sqrt {\pi } a \text {erfi}(a+b x)}{4 b^2}+\frac {\sinh \left ((a+b x)^2\right )}{2 b^2} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5365, 6742, 5299, 2204, 2205, 5321, 2637} \[ -\frac {\sqrt {\pi } a \text {Erf}(a+b x)}{4 b^2}-\frac {\sqrt {\pi } a \text {Erfi}(a+b x)}{4 b^2}+\frac {\sinh \left ((a+b x)^2\right )}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x*Cosh[(a + b*x)^2],x]

[Out]

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

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 2637

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

Rule 5299

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 5321

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 5365

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 6742

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

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 39, normalized size = 0.72 \[ \frac {2 \sinh \left ((a+b x)^2\right )-\sqrt {\pi } a (\text {erf}(a+b x)+\text {erfi}(a+b x))}{4 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Cosh[(a + b*x)^2],x]

[Out]

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

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fricas [B]  time = 0.66, size = 134, normalized size = 2.48 \[ -\frac {{\left (\sqrt {\pi } a \sqrt {b^{2}} \operatorname {erf}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} + \sqrt {\pi } a \sqrt {b^{2}} \operatorname {erfi}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} - b e^{\left (2 \, b^{2} x^{2} + 4 \, a b x + 2 \, a^{2}\right )} + b\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{4 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(sqrt(pi)*a*sqrt(b^2)*erf(sqrt(b^2)*(b*x + a)/b)*e^(b^2*x^2 + 2*a*b*x + a^2) + sqrt(pi)*a*sqrt(b^2)*erfi(
sqrt(b^2)*(b*x + a)/b)*e^(b^2*x^2 + 2*a*b*x + a^2) - b*e^(2*b^2*x^2 + 4*a*b*x + 2*a^2) + b)*e^(-b^2*x^2 - 2*a*
b*x - a^2)/b^3

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giac [C]  time = 0.17, size = 99, normalized size = 1.83 \[ -\frac {-\frac {i \, \sqrt {\pi } a \operatorname {erf}\left (i \, b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}}{b}}{4 \, b} + \frac {\frac {\sqrt {\pi } a \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.12, size = 66, normalized size = 1.22 \[ -\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{4 b^{2}}-\frac {a \erf \left (b x +a \right ) \sqrt {\pi }}{4 b^{2}}+\frac {{\mathrm e}^{\left (b x +a \right )^{2}}}{4 b^{2}}+\frac {i a \sqrt {\pi }\, \erf \left (i b x +i a \right )}{4 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.65, size = 650, normalized size = 12.04 \[ \frac {1}{2} \, x^{2} \cosh \left ({\left (b x + a\right )}^{2}\right ) - \frac {1}{4} \, {\left (\frac {{\left (\frac {\sqrt {\pi } {\left (b^{2} x + a b\right )} a^{2} b^{3} {\left (\operatorname {erf}\left (\frac {\sqrt {{\left (b^{2} x + a b\right )}^{2}}}{b}\right ) - 1\right )}}{\sqrt {{\left (b^{2} x + a b\right )}^{2}} \left (-b^{2}\right )^{\frac {5}{2}}} - \frac {{\left (b^{2} x + a b\right )}^{3} b^{3} \Gamma \left (\frac {3}{2}, \frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}{{\left ({\left (b^{2} x + a b\right )}^{2}\right )}^{\frac {3}{2}} \left (-b^{2}\right )^{\frac {5}{2}}} + \frac {2 \, a b^{3} e^{\left (-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}}{\left (-b^{2}\right )^{\frac {5}{2}}}\right )} a}{\sqrt {-b^{2}}} + \frac {{\left (\frac {\sqrt {\pi } {\left (b^{2} x + a b\right )} a^{3} b^{4} {\left (\operatorname {erf}\left (\frac {\sqrt {{\left (b^{2} x + a b\right )}^{2}}}{b}\right ) - 1\right )}}{\sqrt {{\left (b^{2} x + a b\right )}^{2}} \left (-b^{2}\right )^{\frac {7}{2}}} - \frac {3 \, {\left (b^{2} x + a b\right )}^{3} a b^{4} \Gamma \left (\frac {3}{2}, \frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}{{\left ({\left (b^{2} x + a b\right )}^{2}\right )}^{\frac {3}{2}} \left (-b^{2}\right )^{\frac {7}{2}}} + \frac {3 \, a^{2} b^{4} e^{\left (-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}}{\left (-b^{2}\right )^{\frac {7}{2}}} + \frac {b^{4} \Gamma \left (2, \frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}{\left (-b^{2}\right )^{\frac {7}{2}}}\right )} b}{\sqrt {-b^{2}}} + \frac {a {\left (\frac {\sqrt {\pi } {\left (b^{2} x + a b\right )} a^{2} {\left (\operatorname {erf}\left (\sqrt {-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}}\right ) - 1\right )}}{b^{3} \sqrt {-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}}} - \frac {2 \, a e^{\left (\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}}{b^{2}} - \frac {{\left (b^{2} x + a b\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}{b^{5} \left (-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )^{\frac {3}{2}}}\right )}}{b} - \frac {\sqrt {\pi } {\left (b^{2} x + a b\right )} a^{3} {\left (\operatorname {erf}\left (\sqrt {-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}}\right ) - 1\right )}}{b^{4} \sqrt {-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}}} + \frac {3 \, a^{2} e^{\left (\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}}{b^{3}} - \frac {\Gamma \left (2, -\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}{b^{3}} + \frac {3 \, {\left (b^{2} x + a b\right )}^{3} a \Gamma \left (\frac {3}{2}, -\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}{b^{6} \left (-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )^{\frac {3}{2}}}\right )} b \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,\mathrm {cosh}\left ({\left (a+b\,x\right )}^2\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cosh((a + b*x)^2),x)

[Out]

int(x*cosh((a + b*x)^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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