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

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

[Out]

(-3*x*Cosh[a + b*x^2])/(8*b) + (x*Cosh[3*a + 3*b*x^2])/(24*b) + (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))

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Rubi [A]  time = 0.138398, antiderivative size = 160, normalized size of antiderivative = 1., 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 = {5340, 5324, 5299, 2204, 2205} \[ \frac{3 \sqrt{\pi } e^{-a} \text{Erf}\left (\sqrt{b} x\right )}{32 b^{3/2}}-\frac{\sqrt{\frac{\pi }{3}} e^{-3 a} \text{Erf}\left (\sqrt{3} \sqrt{b} x\right )}{96 b^{3/2}}+\frac{3 \sqrt{\pi } e^a \text{Erfi}\left (\sqrt{b} x\right )}{32 b^{3/2}}-\frac{\sqrt{\frac{\pi }{3}} e^{3 a} \text{Erfi}\left (\sqrt{3} \sqrt{b} x\right )}{96 b^{3/2}}-\frac{3 x \cosh \left (a+b x^2\right )}{8 b}+\frac{x \cosh \left (3 a+3 b x^2\right )}{24 b} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Sinh[a + b*x^2]^3,x]

[Out]

(-3*x*Cosh[a + b*x^2])/(8*b) + (x*Cosh[3*a + 3*b*x^2])/(24*b) + (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))

Rule 5340

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

Rule 5324

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

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 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 \sinh ^3\left (a+b x^2\right ) \, dx &=\int \left (-\frac{3}{4} x^2 \sinh \left (a+b x^2\right )+\frac{1}{4} x^2 \sinh \left (3 a+3 b x^2\right )\right ) \, dx\\ &=\frac{1}{4} \int x^2 \sinh \left (3 a+3 b x^2\right ) \, dx-\frac{3}{4} \int x^2 \sinh \left (a+b x^2\right ) \, dx\\ &=-\frac{3 x \cosh \left (a+b x^2\right )}{8 b}+\frac{x \cosh \left (3 a+3 b x^2\right )}{24 b}-\frac{\int \cosh \left (3 a+3 b x^2\right ) \, dx}{24 b}+\frac{3 \int \cosh \left (a+b x^2\right ) \, dx}{8 b}\\ &=-\frac{3 x \cosh \left (a+b x^2\right )}{8 b}+\frac{x \cosh \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 x \cosh \left (a+b x^2\right )}{8 b}+\frac{x \cosh \left (3 a+3 b x^2\right )}{24 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}}\\ \end{align*}

Mathematica [A]  time = 0.305033, size = 184, normalized size = 1.15 \[ \frac{27 \sqrt{\pi } (\cosh (a)-\sinh (a)) \text{Erf}\left (\sqrt{b} x\right )+\sqrt{3 \pi } (\sinh (3 a)-\cosh (3 a)) \text{Erf}\left (\sqrt{3} \sqrt{b} x\right )+27 \sqrt{\pi } \sinh (a) \text{Erfi}\left (\sqrt{b} x\right )-\sqrt{3 \pi } \sinh (3 a) \text{Erfi}\left (\sqrt{3} \sqrt{b} x\right )+27 \sqrt{\pi } \cosh (a) \text{Erfi}\left (\sqrt{b} x\right )-\sqrt{3 \pi } \cosh (3 a) \text{Erfi}\left (\sqrt{3} \sqrt{b} x\right )-108 \sqrt{b} x \cosh \left (a+b x^2\right )+12 \sqrt{b} x \cosh \left (3 \left (a+b x^2\right )\right )}{288 b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sinh[a + b*x^2]^3,x]

[Out]

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

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Maple [A]  time = 0.07, size = 157, normalized size = 1. \begin{align*}{\frac{{{\rm e}^{-3\,a}}x{{\rm e}^{-3\,b{x}^{2}}}}{48\,b}}-{\frac{{{\rm e}^{-3\,a}}\sqrt{\pi }\sqrt{3}}{288}{\it Erf} \left ( x\sqrt{3}\sqrt{b} \right ){b}^{-{\frac{3}{2}}}}-{\frac{3\,{{\rm e}^{-a}}x{{\rm e}^{-b{x}^{2}}}}{16\,b}}+{\frac{3\,{{\rm e}^{-a}}\sqrt{\pi }}{32}{\it Erf} \left ( x\sqrt{b} \right ){b}^{-{\frac{3}{2}}}}+{\frac{{{\rm e}^{3\,a}}x{{\rm e}^{3\,b{x}^{2}}}}{48\,b}}-{\frac{{{\rm e}^{3\,a}}\sqrt{\pi }}{96\,b}{\it Erf} \left ( \sqrt{-3\,b}x \right ){\frac{1}{\sqrt{-3\,b}}}}-{\frac{3\,{{\rm e}^{a}}{{\rm e}^{b{x}^{2}}}x}{16\,b}}+{\frac{3\,{{\rm e}^{a}}\sqrt{\pi }}{32\,b}{\it Erf} \left ( \sqrt{-b}x \right ){\frac{1}{\sqrt{-b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*sinh(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))+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)-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)

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Maxima [A]  time = 1.82847, size = 219, normalized size = 1.37 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*sinh(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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Fricas [B]  time = 1.89479, size = 2429, normalized size = 15.18 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*sinh(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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sinh ^{3}{\left (a + b x^{2} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*sinh(a + b*x**2)**3, x)

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Giac [A]  time = 1.23862, size = 224, normalized size = 1.4 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*sinh(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)