3.16 \(\int x^2 \cosh ^3(a+b x^2) \, dx\)

Optimal. Leaf size=160 \[ \frac {3 \sqrt {\pi } e^{-a} \text {erf}\left (\sqrt {b} x\right )}{32 b^{3/2}}+\frac {\sqrt {\frac {\pi }{3}} e^{-3 a} \text {erf}\left (\sqrt {3} \sqrt {b} x\right )}{96 b^{3/2}}-\frac {3 \sqrt {\pi } e^a \text {erfi}\left (\sqrt {b} x\right )}{32 b^{3/2}}-\frac {\sqrt {\frac {\pi }{3}} e^{3 a} \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )}{96 b^{3/2}}+\frac {3 x \sinh \left (a+b x^2\right )}{8 b}+\frac {x \sinh \left (3 a+3 b x^2\right )}{24 b} \]

[Out]

3/8*x*sinh(b*x^2+a)/b+1/24*x*sinh(3*b*x^2+3*a)/b+1/288*erf(x*3^(1/2)*b^(1/2))*3^(1/2)*Pi^(1/2)/b^(3/2)/exp(3*a
)-1/288*exp(3*a)*erfi(x*3^(1/2)*b^(1/2))*3^(1/2)*Pi^(1/2)/b^(3/2)+3/32*erf(x*b^(1/2))*Pi^(1/2)/b^(3/2)/exp(a)-
3/32*exp(a)*erfi(x*b^(1/2))*Pi^(1/2)/b^(3/2)

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Rubi [A]  time = 0.14, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ \frac {3 \sqrt {\pi } e^{-a} \text {Erf}\left (\sqrt {b} x\right )}{32 b^{3/2}}+\frac {\sqrt {\frac {\pi }{3}} e^{-3 a} \text {Erf}\left (\sqrt {3} \sqrt {b} x\right )}{96 b^{3/2}}-\frac {3 \sqrt {\pi } e^a \text {Erfi}\left (\sqrt {b} x\right )}{32 b^{3/2}}-\frac {\sqrt {\frac {\pi }{3}} e^{3 a} \text {Erfi}\left (\sqrt {3} \sqrt {b} x\right )}{96 b^{3/2}}+\frac {3 x \sinh \left (a+b x^2\right )}{8 b}+\frac {x \sinh \left (3 a+3 b x^2\right )}{24 b} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Cosh[a + b*x^2]^3,x]

[Out]

(3*Sqrt[Pi]*Erf[Sqrt[b]*x])/(32*b^(3/2)*E^a) + (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[b]*x])/(96*b^(3/2)*E^(3*a)) - (3*E
^a*Sqrt[Pi]*Erfi[Sqrt[b]*x])/(32*b^(3/2)) - (E^(3*a)*Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[b]*x])/(96*b^(3/2)) + (3*x*S
inh[a + b*x^2])/(8*b) + (x*Sinh[3*a + 3*b*x^2])/(24*b)

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 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 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 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]

Rubi steps

\begin {align*} \int x^2 \cosh ^3\left (a+b x^2\right ) \, dx &=\int \left (\frac {3}{4} x^2 \cosh \left (a+b x^2\right )+\frac {1}{4} x^2 \cosh \left (3 a+3 b x^2\right )\right ) \, dx\\ &=\frac {1}{4} \int x^2 \cosh \left (3 a+3 b x^2\right ) \, dx+\frac {3}{4} \int x^2 \cosh \left (a+b x^2\right ) \, dx\\ &=\frac {3 x \sinh \left (a+b x^2\right )}{8 b}+\frac {x \sinh \left (3 a+3 b x^2\right )}{24 b}-\frac {\int \sinh \left (3 a+3 b x^2\right ) \, dx}{24 b}-\frac {3 \int \sinh \left (a+b x^2\right ) \, dx}{8 b}\\ &=\frac {3 x \sinh \left (a+b x^2\right )}{8 b}+\frac {x \sinh \left (3 a+3 b x^2\right )}{24 b}+\frac {\int e^{-3 a-3 b x^2} \, dx}{48 b}-\frac {\int e^{3 a+3 b x^2} \, dx}{48 b}+\frac {3 \int e^{-a-b x^2} \, dx}{16 b}-\frac {3 \int e^{a+b x^2} \, dx}{16 b}\\ &=\frac {3 e^{-a} \sqrt {\pi } \text {erf}\left (\sqrt {b} x\right )}{32 b^{3/2}}+\frac {e^{-3 a} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {b} x\right )}{96 b^{3/2}}-\frac {3 e^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x\right )}{32 b^{3/2}}-\frac {e^{3 a} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )}{96 b^{3/2}}+\frac {3 x \sinh \left (a+b x^2\right )}{8 b}+\frac {x \sinh \left (3 a+3 b x^2\right )}{24 b}\\ \end {align*}

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Mathematica [A]  time = 0.33, size = 184, normalized size = 1.15 \[ \frac {27 \sqrt {\pi } (\cosh (a)-\sinh (a)) \text {erf}\left (\sqrt {b} x\right )+\sqrt {3 \pi } (\cosh (3 a)-\sinh (3 a)) \text {erf}\left (\sqrt {3} \sqrt {b} x\right )-27 \sqrt {\pi } \sinh (a) \text {erfi}\left (\sqrt {b} x\right )-\sqrt {3 \pi } \sinh (3 a) \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )-27 \sqrt {\pi } \cosh (a) \text {erfi}\left (\sqrt {b} x\right )-\sqrt {3 \pi } \cosh (3 a) \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )+108 \sqrt {b} x \sinh \left (a+b x^2\right )+12 \sqrt {b} x \sinh \left (3 \left (a+b x^2\right )\right )}{288 b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Cosh[a + b*x^2]^3,x]

[Out]

(-27*Sqrt[Pi]*Cosh[a]*Erfi[Sqrt[b]*x] - Sqrt[3*Pi]*Cosh[3*a]*Erfi[Sqrt[3]*Sqrt[b]*x] + 27*Sqrt[Pi]*Erf[Sqrt[b]
*x]*(Cosh[a] - Sinh[a]) - 27*Sqrt[Pi]*Erfi[Sqrt[b]*x]*Sinh[a] + Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[b]*x]*(Cosh[3*a] -
Sinh[3*a]) - Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[b]*x]*Sinh[3*a] + 108*Sqrt[b]*x*Sinh[a + b*x^2] + 12*Sqrt[b]*x*Sinh
[3*(a + b*x^2)])/(288*b^(3/2))

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fricas [B]  time = 0.51, size = 903, normalized size = 5.64 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cosh(b*x^2+a)^3,x, algorithm="fricas")

[Out]

1/288*(6*b*x*cosh(b*x^2 + a)^6 + 36*b*x*cosh(b*x^2 + a)*sinh(b*x^2 + a)^5 + 6*b*x*sinh(b*x^2 + a)^6 + 54*b*x*c
osh(b*x^2 + a)^4 + 18*(5*b*x*cosh(b*x^2 + a)^2 + 3*b*x)*sinh(b*x^2 + a)^4 - 54*b*x*cosh(b*x^2 + a)^2 + 24*(5*b
*x*cosh(b*x^2 + a)^3 + 9*b*x*cosh(b*x^2 + a))*sinh(b*x^2 + a)^3 + sqrt(3)*sqrt(pi)*(cosh(b*x^2 + a)^3*cosh(3*a
) + (cosh(3*a) + sinh(3*a))*sinh(b*x^2 + a)^3 + cosh(b*x^2 + a)^3*sinh(3*a) + 3*(cosh(b*x^2 + a)*cosh(3*a) + c
osh(b*x^2 + a)*sinh(3*a))*sinh(b*x^2 + a)^2 + 3*(cosh(b*x^2 + a)^2*cosh(3*a) + cosh(b*x^2 + a)^2*sinh(3*a))*si
nh(b*x^2 + a))*sqrt(-b)*erf(sqrt(3)*sqrt(-b)*x) + sqrt(3)*sqrt(pi)*(cosh(b*x^2 + a)^3*cosh(3*a) + (cosh(3*a) -
sinh(3*a))*sinh(b*x^2 + a)^3 - cosh(b*x^2 + a)^3*sinh(3*a) + 3*(cosh(b*x^2 + a)*cosh(3*a) - cosh(b*x^2 + a)*s
inh(3*a))*sinh(b*x^2 + a)^2 + 3*(cosh(b*x^2 + a)^2*cosh(3*a) - cosh(b*x^2 + a)^2*sinh(3*a))*sinh(b*x^2 + a))*s
qrt(b)*erf(sqrt(3)*sqrt(b)*x) + 27*sqrt(pi)*(cosh(b*x^2 + a)^3*cosh(a) + (cosh(a) + sinh(a))*sinh(b*x^2 + a)^3
+ cosh(b*x^2 + a)^3*sinh(a) + 3*(cosh(b*x^2 + a)*cosh(a) + cosh(b*x^2 + a)*sinh(a))*sinh(b*x^2 + a)^2 + 3*(co
sh(b*x^2 + a)^2*cosh(a) + cosh(b*x^2 + a)^2*sinh(a))*sinh(b*x^2 + a))*sqrt(-b)*erf(sqrt(-b)*x) + 27*sqrt(pi)*(
cosh(b*x^2 + a)^3*cosh(a) + (cosh(a) - sinh(a))*sinh(b*x^2 + a)^3 - cosh(b*x^2 + a)^3*sinh(a) + 3*(cosh(b*x^2
+ a)*cosh(a) - cosh(b*x^2 + a)*sinh(a))*sinh(b*x^2 + a)^2 + 3*(cosh(b*x^2 + a)^2*cosh(a) - cosh(b*x^2 + a)^2*s
inh(a))*sinh(b*x^2 + a))*sqrt(b)*erf(sqrt(b)*x) + 18*(5*b*x*cosh(b*x^2 + a)^4 + 18*b*x*cosh(b*x^2 + a)^2 - 3*b
*x)*sinh(b*x^2 + a)^2 - 6*b*x + 36*(b*x*cosh(b*x^2 + a)^5 + 6*b*x*cosh(b*x^2 + a)^3 - 3*b*x*cosh(b*x^2 + a))*s
inh(b*x^2 + a))/(b^2*cosh(b*x^2 + a)^3 + 3*b^2*cosh(b*x^2 + a)^2*sinh(b*x^2 + a) + 3*b^2*cosh(b*x^2 + a)*sinh(
b*x^2 + a)^2 + b^2*sinh(b*x^2 + a)^3)

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giac [A]  time = 0.14, size = 166, normalized size = 1.04 \[ \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {3} \sqrt {-b} x\right ) e^{\left (3 \, a\right )}}{288 \, \sqrt {-b} b} - \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {3} \sqrt {b} x\right ) e^{\left (-3 \, a\right )}}{288 \, b^{\frac {3}{2}}} + \frac {x e^{\left (3 \, b x^{2} + 3 \, a\right )}}{48 \, b} + \frac {3 \, x e^{\left (b x^{2} + a\right )}}{16 \, b} - \frac {3 \, x e^{\left (-b x^{2} - a\right )}}{16 \, b} - \frac {x e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{48 \, b} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {b} x\right ) e^{\left (-a\right )}}{32 \, b^{\frac {3}{2}}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {-b} x\right ) e^{a}}{32 \, \sqrt {-b} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cosh(b*x^2+a)^3,x, algorithm="giac")

[Out]

1/288*sqrt(3)*sqrt(pi)*erf(-sqrt(3)*sqrt(-b)*x)*e^(3*a)/(sqrt(-b)*b) - 1/288*sqrt(3)*sqrt(pi)*erf(-sqrt(3)*sqr
t(b)*x)*e^(-3*a)/b^(3/2) + 1/48*x*e^(3*b*x^2 + 3*a)/b + 3/16*x*e^(b*x^2 + a)/b - 3/16*x*e^(-b*x^2 - a)/b - 1/4
8*x*e^(-3*b*x^2 - 3*a)/b - 3/32*sqrt(pi)*erf(-sqrt(b)*x)*e^(-a)/b^(3/2) + 3/32*sqrt(pi)*erf(-sqrt(-b)*x)*e^a/(
sqrt(-b)*b)

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maple [A]  time = 0.26, size = 157, normalized size = 0.98 \[ -\frac {{\mathrm e}^{-3 a} x \,{\mathrm e}^{-3 b \,x^{2}}}{48 b}+\frac {{\mathrm e}^{-3 a} \sqrt {\pi }\, \sqrt {3}\, \erf \left (x \sqrt {3}\, \sqrt {b}\right )}{288 b^{\frac {3}{2}}}-\frac {3 \,{\mathrm e}^{-a} x \,{\mathrm e}^{-b \,x^{2}}}{16 b}+\frac {3 \,{\mathrm e}^{-a} \sqrt {\pi }\, \erf \left (x \sqrt {b}\right )}{32 b^{\frac {3}{2}}}+\frac {3 \,{\mathrm e}^{a} {\mathrm e}^{b \,x^{2}} x}{16 b}-\frac {3 \,{\mathrm e}^{a} \sqrt {\pi }\, \erf \left (\sqrt {-b}\, x \right )}{32 b \sqrt {-b}}+\frac {{\mathrm e}^{3 a} x \,{\mathrm e}^{3 b \,x^{2}}}{48 b}-\frac {{\mathrm e}^{3 a} \sqrt {\pi }\, \erf \left (\sqrt {-3 b}\, x \right )}{96 b \sqrt {-3 b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*cosh(b*x^2+a)^3,x)

[Out]

-1/48*exp(-3*a)/b*x*exp(-3*b*x^2)+1/288*exp(-3*a)/b^(3/2)*Pi^(1/2)*3^(1/2)*erf(x*3^(1/2)*b^(1/2))-3/16*exp(-a)
/b*x*exp(-b*x^2)+3/32*exp(-a)/b^(3/2)*Pi^(1/2)*erf(x*b^(1/2))+3/16*exp(a)*exp(b*x^2)*x/b-3/32*exp(a)/b*Pi^(1/2
)/(-b)^(1/2)*erf((-b)^(1/2)*x)+1/48*exp(3*a)/b*x*exp(3*b*x^2)-1/96*exp(3*a)/b*Pi^(1/2)/(-3*b)^(1/2)*erf((-3*b)
^(1/2)*x)

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maxima [A]  time = 0.41, size = 162, normalized size = 1.01 \[ -\frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\sqrt {3} \sqrt {-b} x\right ) e^{\left (3 \, a\right )}}{288 \, \sqrt {-b} b} + \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\sqrt {3} \sqrt {b} x\right ) e^{\left (-3 \, a\right )}}{288 \, b^{\frac {3}{2}}} + \frac {x e^{\left (3 \, b x^{2} + 3 \, a\right )}}{48 \, b} + \frac {3 \, x e^{\left (b x^{2} + a\right )}}{16 \, b} - \frac {3 \, x e^{\left (-b x^{2} - a\right )}}{16 \, b} - \frac {x e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{48 \, b} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {b} x\right ) e^{\left (-a\right )}}{32 \, b^{\frac {3}{2}}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {-b} x\right ) e^{a}}{32 \, \sqrt {-b} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cosh(b*x^2+a)^3,x, algorithm="maxima")

[Out]

-1/288*sqrt(3)*sqrt(pi)*erf(sqrt(3)*sqrt(-b)*x)*e^(3*a)/(sqrt(-b)*b) + 1/288*sqrt(3)*sqrt(pi)*erf(sqrt(3)*sqrt
(b)*x)*e^(-3*a)/b^(3/2) + 1/48*x*e^(3*b*x^2 + 3*a)/b + 3/16*x*e^(b*x^2 + a)/b - 3/16*x*e^(-b*x^2 - a)/b - 1/48
*x*e^(-3*b*x^2 - 3*a)/b + 3/32*sqrt(pi)*erf(sqrt(b)*x)*e^(-a)/b^(3/2) - 3/32*sqrt(pi)*erf(sqrt(-b)*x)*e^a/(sqr
t(-b)*b)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\mathrm {cosh}\left (b\,x^2+a\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*cosh(a + b*x^2)^3,x)

[Out]

int(x^2*cosh(a + b*x^2)^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \cosh ^{3}{\left (a + b x^{2} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*cosh(b*x**2+a)**3,x)

[Out]

Integral(x**2*cosh(a + b*x**2)**3, x)

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