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

Optimal. Leaf size=125 \[ \frac{3 \sqrt{\pi } e^{-a} \text{Erf}\left (\sqrt{b} x\right )}{16 \sqrt{b}}-\frac{\sqrt{\frac{\pi }{3}} e^{-3 a} \text{Erf}\left (\sqrt{3} \sqrt{b} x\right )}{16 \sqrt{b}}-\frac{3 \sqrt{\pi } e^a \text{Erfi}\left (\sqrt{b} x\right )}{16 \sqrt{b}}+\frac{\sqrt{\frac{\pi }{3}} e^{3 a} \text{Erfi}\left (\sqrt{3} \sqrt{b} x\right )}{16 \sqrt{b}} \]

[Out]

(3*Sqrt[Pi]*Erf[Sqrt[b]*x])/(16*Sqrt[b]*E^a) - (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[b]*x])/(16*Sqrt[b]*E^(3*a)) - (3*E
^a*Sqrt[Pi]*Erfi[Sqrt[b]*x])/(16*Sqrt[b]) + (E^(3*a)*Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[b]*x])/(16*Sqrt[b])

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Rubi [A]  time = 0.0718229, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5300, 5298, 2204, 2205} \[ \frac{3 \sqrt{\pi } e^{-a} \text{Erf}\left (\sqrt{b} x\right )}{16 \sqrt{b}}-\frac{\sqrt{\frac{\pi }{3}} e^{-3 a} \text{Erf}\left (\sqrt{3} \sqrt{b} x\right )}{16 \sqrt{b}}-\frac{3 \sqrt{\pi } e^a \text{Erfi}\left (\sqrt{b} x\right )}{16 \sqrt{b}}+\frac{\sqrt{\frac{\pi }{3}} e^{3 a} \text{Erfi}\left (\sqrt{3} \sqrt{b} x\right )}{16 \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*x^2]^3,x]

[Out]

(3*Sqrt[Pi]*Erf[Sqrt[b]*x])/(16*Sqrt[b]*E^a) - (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[b]*x])/(16*Sqrt[b]*E^(3*a)) - (3*E
^a*Sqrt[Pi]*Erfi[Sqrt[b]*x])/(16*Sqrt[b]) + (E^(3*a)*Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[b]*x])/(16*Sqrt[b])

Rule 5300

Int[((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_), x_Symbol] :> Int[ExpandTrigReduce[(a + b*Sinh[c + d*x^
n])^p, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 1] && IGtQ[p, 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 \sinh ^3\left (a+b x^2\right ) \, dx &=\int \left (-\frac{3}{4} \sinh \left (a+b x^2\right )+\frac{1}{4} \sinh \left (3 a+3 b x^2\right )\right ) \, dx\\ &=\frac{1}{4} \int \sinh \left (3 a+3 b x^2\right ) \, dx-\frac{3}{4} \int \sinh \left (a+b x^2\right ) \, dx\\ &=-\left (\frac{1}{8} \int e^{-3 a-3 b x^2} \, dx\right )+\frac{1}{8} \int e^{3 a+3 b x^2} \, dx+\frac{3}{8} \int e^{-a-b x^2} \, dx-\frac{3}{8} \int e^{a+b x^2} \, dx\\ &=\frac{3 e^{-a} \sqrt{\pi } \text{erf}\left (\sqrt{b} x\right )}{16 \sqrt{b}}-\frac{e^{-3 a} \sqrt{\frac{\pi }{3}} \text{erf}\left (\sqrt{3} \sqrt{b} x\right )}{16 \sqrt{b}}-\frac{3 e^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} x\right )}{16 \sqrt{b}}+\frac{e^{3 a} \sqrt{\frac{\pi }{3}} \text{erfi}\left (\sqrt{3} \sqrt{b} x\right )}{16 \sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.136555, size = 136, normalized size = 1.09 \[ \frac{\sqrt{\frac{\pi }{3}} \left (3 \sqrt{3} (\cosh (a)-\sinh (a)) \text{Erf}\left (\sqrt{b} x\right )+(\sinh (3 a)-\cosh (3 a)) \text{Erf}\left (\sqrt{3} \sqrt{b} x\right )-3 \sqrt{3} \sinh (a) \text{Erfi}\left (\sqrt{b} x\right )+\sinh (3 a) \text{Erfi}\left (\sqrt{3} \sqrt{b} x\right )-3 \sqrt{3} \cosh (a) \text{Erfi}\left (\sqrt{b} x\right )+\cosh (3 a) \text{Erfi}\left (\sqrt{3} \sqrt{b} x\right )\right )}{16 \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b*x^2]^3,x]

[Out]

(Sqrt[Pi/3]*(-3*Sqrt[3]*Cosh[a]*Erfi[Sqrt[b]*x] + Cosh[3*a]*Erfi[Sqrt[3]*Sqrt[b]*x] + 3*Sqrt[3]*Erf[Sqrt[b]*x]
*(Cosh[a] - Sinh[a]) - 3*Sqrt[3]*Erfi[Sqrt[b]*x]*Sinh[a] + Erfi[Sqrt[3]*Sqrt[b]*x]*Sinh[3*a] + Erf[Sqrt[3]*Sqr
t[b]*x]*(-Cosh[3*a] + Sinh[3*a])))/(16*Sqrt[b])

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Maple [A]  time = 0.036, size = 86, normalized size = 0.7 \begin{align*} -{\frac{{{\rm e}^{-3\,a}}\sqrt{\pi }\sqrt{3}}{48}{\it Erf} \left ( x\sqrt{3}\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}+{\frac{3\,\sqrt{\pi }{{\rm e}^{-a}}}{16}{\it Erf} \left ( x\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}+{\frac{{{\rm e}^{3\,a}}\sqrt{\pi }}{16}{\it Erf} \left ( \sqrt{-3\,b}x \right ){\frac{1}{\sqrt{-3\,b}}}}-{\frac{3\,{{\rm e}^{a}}\sqrt{\pi }}{16}{\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(sinh(b*x^2+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.6408, size = 123, normalized size = 0.98 \begin{align*} \frac{\sqrt{3} \sqrt{\pi } \operatorname{erf}\left (\sqrt{3} \sqrt{-b} x\right ) e^{\left (3 \, a\right )}}{48 \, \sqrt{-b}} - \frac{\sqrt{3} \sqrt{\pi } \operatorname{erf}\left (\sqrt{3} \sqrt{b} x\right ) e^{\left (-3 \, a\right )}}{48 \, \sqrt{b}} + \frac{3 \, \sqrt{\pi } \operatorname{erf}\left (\sqrt{b} x\right ) e^{\left (-a\right )}}{16 \, \sqrt{b}} - \frac{3 \, \sqrt{\pi } \operatorname{erf}\left (\sqrt{-b} x\right ) e^{a}}{16 \, \sqrt{-b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*sqrt(3)*sqrt(pi)*erf(sqrt(3)*sqrt(-b)*x)*e^(3*a)/sqrt(-b) - 1/48*sqrt(3)*sqrt(pi)*erf(sqrt(3)*sqrt(b)*x)*
e^(-3*a)/sqrt(b) + 3/16*sqrt(pi)*erf(sqrt(b)*x)*e^(-a)/sqrt(b) - 3/16*sqrt(pi)*erf(sqrt(-b)*x)*e^a/sqrt(-b)

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Fricas [A]  time = 1.79963, size = 369, normalized size = 2.95 \begin{align*} -\frac{\sqrt{3} \sqrt{\pi } \sqrt{-b}{\left (\cosh \left (3 \, a\right ) + \sinh \left (3 \, a\right )\right )} \operatorname{erf}\left (\sqrt{3} \sqrt{-b} x\right ) + \sqrt{3} \sqrt{\pi } \sqrt{b}{\left (\cosh \left (3 \, a\right ) - \sinh \left (3 \, a\right )\right )} \operatorname{erf}\left (\sqrt{3} \sqrt{b} x\right ) - 9 \, \sqrt{\pi } \sqrt{-b}{\left (\cosh \left (a\right ) + \sinh \left (a\right )\right )} \operatorname{erf}\left (\sqrt{-b} x\right ) - 9 \, \sqrt{\pi } \sqrt{b}{\left (\cosh \left (a\right ) - \sinh \left (a\right )\right )} \operatorname{erf}\left (\sqrt{b} x\right )}{48 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(sqrt(3)*sqrt(pi)*sqrt(-b)*(cosh(3*a) + sinh(3*a))*erf(sqrt(3)*sqrt(-b)*x) + sqrt(3)*sqrt(pi)*sqrt(b)*(c
osh(3*a) - sinh(3*a))*erf(sqrt(3)*sqrt(b)*x) - 9*sqrt(pi)*sqrt(-b)*(cosh(a) + sinh(a))*erf(sqrt(-b)*x) - 9*sqr
t(pi)*sqrt(b)*(cosh(a) - sinh(a))*erf(sqrt(b)*x))/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sinh(a + b*x**2)**3, x)

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Giac [A]  time = 1.26431, size = 128, normalized size = 1.02 \begin{align*} -\frac{\sqrt{3} \sqrt{\pi } \operatorname{erf}\left (-\sqrt{3} \sqrt{-b} x\right ) e^{\left (3 \, a\right )}}{48 \, \sqrt{-b}} + \frac{\sqrt{3} \sqrt{\pi } \operatorname{erf}\left (-\sqrt{3} \sqrt{b} x\right ) e^{\left (-3 \, a\right )}}{48 \, \sqrt{b}} - \frac{3 \, \sqrt{\pi } \operatorname{erf}\left (-\sqrt{b} x\right ) e^{\left (-a\right )}}{16 \, \sqrt{b}} + \frac{3 \, \sqrt{\pi } \operatorname{erf}\left (-\sqrt{-b} x\right ) e^{a}}{16 \, \sqrt{-b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*sqrt(3)*sqrt(pi)*erf(-sqrt(3)*sqrt(-b)*x)*e^(3*a)/sqrt(-b) + 1/48*sqrt(3)*sqrt(pi)*erf(-sqrt(3)*sqrt(b)*
x)*e^(-3*a)/sqrt(b) - 3/16*sqrt(pi)*erf(-sqrt(b)*x)*e^(-a)/sqrt(b) + 3/16*sqrt(pi)*erf(-sqrt(-b)*x)*e^a/sqrt(-
b)