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

Optimal. Leaf size=99 \[ \frac{\sqrt{\frac{\pi }{2}} e^{-2 a} \text{Erf}\left (\sqrt{2} \sqrt{b} x\right )}{32 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} e^{2 a} \text{Erfi}\left (\sqrt{2} \sqrt{b} x\right )}{32 b^{3/2}}+\frac{x \sinh \left (2 a+2 b x^2\right )}{8 b}+\frac{x^3}{6} \]

[Out]

x^3/6 + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[b]*x])/(32*b^(3/2)*E^(2*a)) - (E^(2*a)*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[b]*x]
)/(32*b^(3/2)) + (x*Sinh[2*a + 2*b*x^2])/(8*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0876185, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {5341, 5325, 5298, 2204, 2205} \[ \frac{\sqrt{\frac{\pi }{2}} e^{-2 a} \text{Erf}\left (\sqrt{2} \sqrt{b} x\right )}{32 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} e^{2 a} \text{Erfi}\left (\sqrt{2} \sqrt{b} x\right )}{32 b^{3/2}}+\frac{x \sinh \left (2 a+2 b x^2\right )}{8 b}+\frac{x^3}{6} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Cosh[a + b*x^2]^2,x]

[Out]

x^3/6 + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[b]*x])/(32*b^(3/2)*E^(2*a)) - (E^(2*a)*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[b]*x]
)/(32*b^(3/2)) + (x*Sinh[2*a + 2*b*x^2])/(8*b)

Rule 5341

Int[((a_.) + Cosh[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandTrigReduce[(
e*x)^m, (a + b*Cosh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 1] && IGtQ[n, 0]

Rule 5325

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 5298

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 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]

Rubi steps

\begin{align*} \int x^2 \cosh ^2\left (a+b x^2\right ) \, dx &=\int \left (\frac{x^2}{2}+\frac{1}{2} x^2 \cosh \left (2 a+2 b x^2\right )\right ) \, dx\\ &=\frac{x^3}{6}+\frac{1}{2} \int x^2 \cosh \left (2 a+2 b x^2\right ) \, dx\\ &=\frac{x^3}{6}+\frac{x \sinh \left (2 a+2 b x^2\right )}{8 b}-\frac{\int \sinh \left (2 a+2 b x^2\right ) \, dx}{8 b}\\ &=\frac{x^3}{6}+\frac{x \sinh \left (2 a+2 b x^2\right )}{8 b}+\frac{\int e^{-2 a-2 b x^2} \, dx}{16 b}-\frac{\int e^{2 a+2 b x^2} \, dx}{16 b}\\ &=\frac{x^3}{6}+\frac{e^{-2 a} \sqrt{\frac{\pi }{2}} \text{erf}\left (\sqrt{2} \sqrt{b} x\right )}{32 b^{3/2}}-\frac{e^{2 a} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\sqrt{2} \sqrt{b} x\right )}{32 b^{3/2}}+\frac{x \sinh \left (2 a+2 b x^2\right )}{8 b}\\ \end{align*}

Mathematica [A]  time = 0.218033, size = 101, normalized size = 1.02 \[ \frac{3 \sqrt{2 \pi } (\cosh (2 a)-\sinh (2 a)) \text{Erf}\left (\sqrt{2} \sqrt{b} x\right )-3 \sqrt{2 \pi } (\sinh (2 a)+\cosh (2 a)) \text{Erfi}\left (\sqrt{2} \sqrt{b} x\right )+8 \sqrt{b} x \left (3 \sinh \left (2 \left (a+b x^2\right )\right )+4 b x^2\right )}{192 b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Cosh[a + b*x^2]^2,x]

[Out]

(3*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[b]*x]*(Cosh[2*a] - Sinh[2*a]) - 3*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[b]*x]*(Cosh[2*a]
+ Sinh[2*a]) + 8*Sqrt[b]*x*(4*b*x^2 + 3*Sinh[2*(a + b*x^2)]))/(192*b^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.044, size = 90, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{6}}-{\frac{{{\rm e}^{-2\,a}}x{{\rm e}^{-2\,b{x}^{2}}}}{16\,b}}+{\frac{{{\rm e}^{-2\,a}}\sqrt{\pi }\sqrt{2}}{64}{\it Erf} \left ( x\sqrt{2}\sqrt{b} \right ){b}^{-{\frac{3}{2}}}}+{\frac{{{\rm e}^{2\,a}}x{{\rm e}^{2\,b{x}^{2}}}}{16\,b}}-{\frac{{{\rm e}^{2\,a}}\sqrt{\pi }}{32\,b}{\it Erf} \left ( \sqrt{-2\,b}x \right ){\frac{1}{\sqrt{-2\,b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^3-1/16*exp(-2*a)/b*x*exp(-2*b*x^2)+1/64*exp(-2*a)/b^(3/2)*Pi^(1/2)*2^(1/2)*erf(x*2^(1/2)*b^(1/2))+1/16*e
xp(2*a)/b*x*exp(2*b*x^2)-1/32*exp(2*a)/b*Pi^(1/2)/(-2*b)^(1/2)*erf((-2*b)^(1/2)*x)

________________________________________________________________________________________

Maxima [A]  time = 1.64222, size = 128, normalized size = 1.29 \begin{align*} \frac{1}{6} \, x^{3} - \frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\sqrt{2} \sqrt{-b} x\right ) e^{\left (2 \, a\right )}}{64 \, \sqrt{-b} b} + \frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\sqrt{2} \sqrt{b} x\right ) e^{\left (-2 \, a\right )}}{64 \, b^{\frac{3}{2}}} + \frac{x e^{\left (2 \, b x^{2} + 2 \, a\right )}}{16 \, b} - \frac{x e^{\left (-2 \, b x^{2} - 2 \, a\right )}}{16 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^3 - 1/64*sqrt(2)*sqrt(pi)*erf(sqrt(2)*sqrt(-b)*x)*e^(2*a)/(sqrt(-b)*b) + 1/64*sqrt(2)*sqrt(pi)*erf(sqrt(
2)*sqrt(b)*x)*e^(-2*a)/b^(3/2) + 1/16*x*e^(2*b*x^2 + 2*a)/b - 1/16*x*e^(-2*b*x^2 - 2*a)/b

________________________________________________________________________________________

Fricas [B]  time = 1.76446, size = 1125, normalized size = 11.36 \begin{align*} \frac{32 \, b^{2} x^{3} \cosh \left (b x^{2} + a\right )^{2} + 12 \, b x \cosh \left (b x^{2} + a\right )^{4} + 48 \, b x \cosh \left (b x^{2} + a\right ) \sinh \left (b x^{2} + a\right )^{3} + 12 \, b x \sinh \left (b x^{2} + a\right )^{4} + 3 \, \sqrt{2} \sqrt{\pi }{\left (\cosh \left (b x^{2} + a\right )^{2} \cosh \left (2 \, a\right ) +{\left (\cosh \left (2 \, a\right ) + \sinh \left (2 \, a\right )\right )} \sinh \left (b x^{2} + a\right )^{2} + \cosh \left (b x^{2} + a\right )^{2} \sinh \left (2 \, a\right ) + 2 \,{\left (\cosh \left (b x^{2} + a\right ) \cosh \left (2 \, a\right ) + \cosh \left (b x^{2} + a\right ) \sinh \left (2 \, a\right )\right )} \sinh \left (b x^{2} + a\right )\right )} \sqrt{-b} \operatorname{erf}\left (\sqrt{2} \sqrt{-b} x\right ) + 3 \, \sqrt{2} \sqrt{\pi }{\left (\cosh \left (b x^{2} + a\right )^{2} \cosh \left (2 \, a\right ) +{\left (\cosh \left (2 \, a\right ) - \sinh \left (2 \, a\right )\right )} \sinh \left (b x^{2} + a\right )^{2} - \cosh \left (b x^{2} + a\right )^{2} \sinh \left (2 \, a\right ) + 2 \,{\left (\cosh \left (b x^{2} + a\right ) \cosh \left (2 \, a\right ) - \cosh \left (b x^{2} + a\right ) \sinh \left (2 \, a\right )\right )} \sinh \left (b x^{2} + a\right )\right )} \sqrt{b} \operatorname{erf}\left (\sqrt{2} \sqrt{b} x\right ) + 8 \,{\left (4 \, b^{2} x^{3} + 9 \, b x \cosh \left (b x^{2} + a\right )^{2}\right )} \sinh \left (b x^{2} + a\right )^{2} - 12 \, b x + 16 \,{\left (4 \, b^{2} x^{3} \cosh \left (b x^{2} + a\right ) + 3 \, b x \cosh \left (b x^{2} + a\right )^{3}\right )} \sinh \left (b x^{2} + a\right )}{192 \,{\left (b^{2} \cosh \left (b x^{2} + a\right )^{2} + 2 \, b^{2} \cosh \left (b x^{2} + a\right ) \sinh \left (b x^{2} + a\right ) + b^{2} \sinh \left (b x^{2} + a\right )^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(32*b^2*x^3*cosh(b*x^2 + a)^2 + 12*b*x*cosh(b*x^2 + a)^4 + 48*b*x*cosh(b*x^2 + a)*sinh(b*x^2 + a)^3 + 12
*b*x*sinh(b*x^2 + a)^4 + 3*sqrt(2)*sqrt(pi)*(cosh(b*x^2 + a)^2*cosh(2*a) + (cosh(2*a) + sinh(2*a))*sinh(b*x^2
+ a)^2 + cosh(b*x^2 + a)^2*sinh(2*a) + 2*(cosh(b*x^2 + a)*cosh(2*a) + cosh(b*x^2 + a)*sinh(2*a))*sinh(b*x^2 +
a))*sqrt(-b)*erf(sqrt(2)*sqrt(-b)*x) + 3*sqrt(2)*sqrt(pi)*(cosh(b*x^2 + a)^2*cosh(2*a) + (cosh(2*a) - sinh(2*a
))*sinh(b*x^2 + a)^2 - cosh(b*x^2 + a)^2*sinh(2*a) + 2*(cosh(b*x^2 + a)*cosh(2*a) - cosh(b*x^2 + a)*sinh(2*a))
*sinh(b*x^2 + a))*sqrt(b)*erf(sqrt(2)*sqrt(b)*x) + 8*(4*b^2*x^3 + 9*b*x*cosh(b*x^2 + a)^2)*sinh(b*x^2 + a)^2 -
12*b*x + 16*(4*b^2*x^3*cosh(b*x^2 + a) + 3*b*x*cosh(b*x^2 + a)^3)*sinh(b*x^2 + a))/(b^2*cosh(b*x^2 + a)^2 + 2
*b^2*cosh(b*x^2 + a)*sinh(b*x^2 + a) + b^2*sinh(b*x^2 + a)^2)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \cosh ^{2}{\left (a + b x^{2} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*cosh(a + b*x**2)**2, x)

________________________________________________________________________________________

Giac [A]  time = 1.39855, size = 131, normalized size = 1.32 \begin{align*} \frac{1}{6} \, x^{3} + \frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\sqrt{2} \sqrt{-b} x\right ) e^{\left (2 \, a\right )}}{64 \, \sqrt{-b} b} - \frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\sqrt{2} \sqrt{b} x\right ) e^{\left (-2 \, a\right )}}{64 \, b^{\frac{3}{2}}} + \frac{x e^{\left (2 \, b x^{2} + 2 \, a\right )}}{16 \, b} - \frac{x e^{\left (-2 \, b x^{2} - 2 \, a\right )}}{16 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^3 + 1/64*sqrt(2)*sqrt(pi)*erf(-sqrt(2)*sqrt(-b)*x)*e^(2*a)/(sqrt(-b)*b) - 1/64*sqrt(2)*sqrt(pi)*erf(-sqr
t(2)*sqrt(b)*x)*e^(-2*a)/b^(3/2) + 1/16*x*e^(2*b*x^2 + 2*a)/b - 1/16*x*e^(-2*b*x^2 - 2*a)/b