3.20 \(\int \frac{\sinh ^3(a+b x^2)}{x^2} \, dx\)

Optimal. Leaf size=136 \[ -\frac{3}{8} \sqrt{\pi } e^{-a} \sqrt{b} \text{Erf}\left (\sqrt{b} x\right )+\frac{1}{8} \sqrt{3 \pi } e^{-3 a} \sqrt{b} \text{Erf}\left (\sqrt{3} \sqrt{b} x\right )-\frac{3}{8} \sqrt{\pi } e^a \sqrt{b} \text{Erfi}\left (\sqrt{b} x\right )+\frac{1}{8} \sqrt{3 \pi } e^{3 a} \sqrt{b} \text{Erfi}\left (\sqrt{3} \sqrt{b} x\right )-\frac{\sinh ^3\left (a+b x^2\right )}{x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.109961, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {5330, 5618, 5299, 2204, 2205} \[ -\frac{3}{8} \sqrt{\pi } e^{-a} \sqrt{b} \text{Erf}\left (\sqrt{b} x\right )+\frac{1}{8} \sqrt{3 \pi } e^{-3 a} \sqrt{b} \text{Erf}\left (\sqrt{3} \sqrt{b} x\right )-\frac{3}{8} \sqrt{\pi } e^a \sqrt{b} \text{Erfi}\left (\sqrt{b} x\right )+\frac{1}{8} \sqrt{3 \pi } e^{3 a} \sqrt{b} \text{Erfi}\left (\sqrt{3} \sqrt{b} x\right )-\frac{\sinh ^3\left (a+b x^2\right )}{x} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*x^2]^3/x^2,x]

[Out]

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

Rule 5330

Int[(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_)]^(p_), x_Symbol] :> -Simp[Sinh[a + b*x^n]^p/((n - 1)*x^(n - 1)), x
] + Dist[(b*n*p)/(n - 1), Int[Sinh[a + b*x^n]^(p - 1)*Cosh[a + b*x^n], x], x] /; FreeQ[{a, b}, x] && IntegersQ
[n, p] && EqQ[m + n, 0] && GtQ[p, 1] && NeQ[n, 1]

Rule 5618

Int[Cosh[w_]^(q_.)*Sinh[v_]^(p_.), x_Symbol] :> Int[ExpandTrigReduce[Sinh[v]^p*Cosh[w]^q, x], x] /; IGtQ[p, 0]
&& IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/
w], 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 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 \frac{\sinh ^3\left (a+b x^2\right )}{x^2} \, dx &=-\frac{\sinh ^3\left (a+b x^2\right )}{x}+(6 b) \int \cosh \left (a+b x^2\right ) \sinh ^2\left (a+b x^2\right ) \, dx\\ &=-\frac{\sinh ^3\left (a+b x^2\right )}{x}+(6 b) \int \left (-\frac{1}{4} \cosh \left (a+b x^2\right )+\frac{1}{4} \cosh \left (3 a+3 b x^2\right )\right ) \, dx\\ &=-\frac{\sinh ^3\left (a+b x^2\right )}{x}-\frac{1}{2} (3 b) \int \cosh \left (a+b x^2\right ) \, dx+\frac{1}{2} (3 b) \int \cosh \left (3 a+3 b x^2\right ) \, dx\\ &=-\frac{\sinh ^3\left (a+b x^2\right )}{x}+\frac{1}{4} (3 b) \int e^{-3 a-3 b x^2} \, dx-\frac{1}{4} (3 b) \int e^{-a-b x^2} \, dx-\frac{1}{4} (3 b) \int e^{a+b x^2} \, dx+\frac{1}{4} (3 b) \int e^{3 a+3 b x^2} \, dx\\ &=-\frac{3}{8} \sqrt{b} e^{-a} \sqrt{\pi } \text{erf}\left (\sqrt{b} x\right )+\frac{1}{8} \sqrt{b} e^{-3 a} \sqrt{3 \pi } \text{erf}\left (\sqrt{3} \sqrt{b} x\right )-\frac{3}{8} \sqrt{b} e^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} x\right )+\frac{1}{8} \sqrt{b} e^{3 a} \sqrt{3 \pi } \text{erfi}\left (\sqrt{3} \sqrt{b} x\right )-\frac{\sinh ^3\left (a+b x^2\right )}{x}\\ \end{align*}

Mathematica [A]  time = 0.329041, size = 204, normalized size = 1.5 \[ \frac{3 \sqrt{\pi } \sqrt{b} x (\sinh (a)-\cosh (a)) \text{Erf}\left (\sqrt{b} x\right )+\sqrt{3 \pi } \sqrt{b} x (\cosh (3 a)-\sinh (3 a)) \text{Erf}\left (\sqrt{3} \sqrt{b} x\right )-3 \sqrt{\pi } \sqrt{b} x \sinh (a) \text{Erfi}\left (\sqrt{b} x\right )+\sqrt{3 \pi } \sqrt{b} x \sinh (3 a) \text{Erfi}\left (\sqrt{3} \sqrt{b} x\right )-3 \sqrt{\pi } \sqrt{b} x \cosh (a) \text{Erfi}\left (\sqrt{b} x\right )+\sqrt{3 \pi } \sqrt{b} x \cosh (3 a) \text{Erfi}\left (\sqrt{3} \sqrt{b} x\right )+6 \sinh \left (a+b x^2\right )-2 \sinh \left (3 \left (a+b x^2\right )\right )}{8 x} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b*x^2]^3/x^2,x]

[Out]

(-3*Sqrt[b]*Sqrt[Pi]*x*Cosh[a]*Erfi[Sqrt[b]*x] + Sqrt[b]*Sqrt[3*Pi]*x*Cosh[3*a]*Erfi[Sqrt[3]*Sqrt[b]*x] - 3*Sq
rt[b]*Sqrt[Pi]*x*Erfi[Sqrt[b]*x]*Sinh[a] + 3*Sqrt[b]*Sqrt[Pi]*x*Erf[Sqrt[b]*x]*(-Cosh[a] + Sinh[a]) + Sqrt[b]*
Sqrt[3*Pi]*x*Erf[Sqrt[3]*Sqrt[b]*x]*(Cosh[3*a] - Sinh[3*a]) + Sqrt[b]*Sqrt[3*Pi]*x*Erfi[Sqrt[3]*Sqrt[b]*x]*Sin
h[3*a] + 6*Sinh[a + b*x^2] - 2*Sinh[3*(a + b*x^2)])/(8*x)

________________________________________________________________________________________

Maple [A]  time = 0.052, size = 149, normalized size = 1.1 \begin{align*}{\frac{{{\rm e}^{-3\,a}}{{\rm e}^{-3\,b{x}^{2}}}}{8\,x}}+{\frac{{{\rm e}^{-3\,a}}\sqrt{\pi }\sqrt{3}}{8}\sqrt{b}{\it Erf} \left ( x\sqrt{3}\sqrt{b} \right ) }-{\frac{3\,{{\rm e}^{-a}}{{\rm e}^{-b{x}^{2}}}}{8\,x}}-{\frac{3\,{{\rm e}^{-a}}\sqrt{\pi }}{8}\sqrt{b}{\it Erf} \left ( x\sqrt{b} \right ) }-{\frac{{{\rm e}^{3\,a}}{{\rm e}^{3\,b{x}^{2}}}}{8\,x}}+{\frac{3\,{{\rm e}^{3\,a}}b\sqrt{\pi }}{8}{\it Erf} \left ( \sqrt{-3\,b}x \right ){\frac{1}{\sqrt{-3\,b}}}}+{\frac{3\,{{\rm e}^{a}}{{\rm e}^{b{x}^{2}}}}{8\,x}}-{\frac{3\,{{\rm e}^{a}}b\sqrt{\pi }}{8}{\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^2,x)

[Out]

1/8*exp(-3*a)/x*exp(-3*b*x^2)+1/8*exp(-3*a)*b^(1/2)*Pi^(1/2)*3^(1/2)*erf(x*3^(1/2)*b^(1/2))-3/8*exp(-a)/x*exp(
-b*x^2)-3/8*exp(-a)*b^(1/2)*Pi^(1/2)*erf(x*b^(1/2))-1/8*exp(3*a)/x*exp(3*b*x^2)+3/8*exp(3*a)*b*Pi^(1/2)/(-3*b)
^(1/2)*erf((-3*b)^(1/2)*x)+3/8*exp(a)*exp(b*x^2)/x-3/8*exp(a)*b*Pi^(1/2)/(-b)^(1/2)*erf((-b)^(1/2)*x)

________________________________________________________________________________________

Maxima [A]  time = 1.37234, size = 138, normalized size = 1.01 \begin{align*} \frac{\sqrt{3} \sqrt{b x^{2}} e^{\left (-3 \, a\right )} \Gamma \left (-\frac{1}{2}, 3 \, b x^{2}\right )}{16 \, x} - \frac{\sqrt{3} \sqrt{-b x^{2}} e^{\left (3 \, a\right )} \Gamma \left (-\frac{1}{2}, -3 \, b x^{2}\right )}{16 \, x} - \frac{3 \, \sqrt{b x^{2}} e^{\left (-a\right )} \Gamma \left (-\frac{1}{2}, b x^{2}\right )}{16 \, x} + \frac{3 \, \sqrt{-b x^{2}} e^{a} \Gamma \left (-\frac{1}{2}, -b x^{2}\right )}{16 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*sqrt(3)*sqrt(b*x^2)*e^(-3*a)*gamma(-1/2, 3*b*x^2)/x - 1/16*sqrt(3)*sqrt(-b*x^2)*e^(3*a)*gamma(-1/2, -3*b*
x^2)/x - 3/16*sqrt(b*x^2)*e^(-a)*gamma(-1/2, b*x^2)/x + 3/16*sqrt(-b*x^2)*e^a*gamma(-1/2, -b*x^2)/x

________________________________________________________________________________________

Fricas [B]  time = 1.86949, size = 2395, normalized size = 17.61 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(cosh(b*x^2 + a)^6 + 6*cosh(b*x^2 + a)*sinh(b*x^2 + a)^5 + sinh(b*x^2 + a)^6 + 3*(5*cosh(b*x^2 + a)^2 - 1
)*sinh(b*x^2 + a)^4 - 3*cosh(b*x^2 + a)^4 + 4*(5*cosh(b*x^2 + a)^3 - 3*cosh(b*x^2 + a))*sinh(b*x^2 + a)^3 + sq
rt(3)*sqrt(pi)*(x*cosh(b*x^2 + a)^3*cosh(3*a) + x*cosh(b*x^2 + a)^3*sinh(3*a) + (x*cosh(3*a) + x*sinh(3*a))*si
nh(b*x^2 + a)^3 + 3*(x*cosh(b*x^2 + a)*cosh(3*a) + x*cosh(b*x^2 + a)*sinh(3*a))*sinh(b*x^2 + a)^2 + 3*(x*cosh(
b*x^2 + a)^2*cosh(3*a) + x*cosh(b*x^2 + a)^2*sinh(3*a))*sinh(b*x^2 + a))*sqrt(-b)*erf(sqrt(3)*sqrt(-b)*x) - sq
rt(3)*sqrt(pi)*(x*cosh(b*x^2 + a)^3*cosh(3*a) - x*cosh(b*x^2 + a)^3*sinh(3*a) + (x*cosh(3*a) - x*sinh(3*a))*si
nh(b*x^2 + a)^3 + 3*(x*cosh(b*x^2 + a)*cosh(3*a) - x*cosh(b*x^2 + a)*sinh(3*a))*sinh(b*x^2 + a)^2 + 3*(x*cosh(
b*x^2 + a)^2*cosh(3*a) - x*cosh(b*x^2 + a)^2*sinh(3*a))*sinh(b*x^2 + a))*sqrt(b)*erf(sqrt(3)*sqrt(b)*x) - 3*sq
rt(pi)*(x*cosh(b*x^2 + a)^3*cosh(a) + x*cosh(b*x^2 + a)^3*sinh(a) + (x*cosh(a) + x*sinh(a))*sinh(b*x^2 + a)^3
+ 3*(x*cosh(b*x^2 + a)*cosh(a) + x*cosh(b*x^2 + a)*sinh(a))*sinh(b*x^2 + a)^2 + 3*(x*cosh(b*x^2 + a)^2*cosh(a)
+ x*cosh(b*x^2 + a)^2*sinh(a))*sinh(b*x^2 + a))*sqrt(-b)*erf(sqrt(-b)*x) + 3*sqrt(pi)*(x*cosh(b*x^2 + a)^3*co
sh(a) - x*cosh(b*x^2 + a)^3*sinh(a) + (x*cosh(a) - x*sinh(a))*sinh(b*x^2 + a)^3 + 3*(x*cosh(b*x^2 + a)*cosh(a)
- x*cosh(b*x^2 + a)*sinh(a))*sinh(b*x^2 + a)^2 + 3*(x*cosh(b*x^2 + a)^2*cosh(a) - x*cosh(b*x^2 + a)^2*sinh(a)
)*sinh(b*x^2 + a))*sqrt(b)*erf(sqrt(b)*x) + 3*(5*cosh(b*x^2 + a)^4 - 6*cosh(b*x^2 + a)^2 + 1)*sinh(b*x^2 + a)^
2 + 3*cosh(b*x^2 + a)^2 + 6*(cosh(b*x^2 + a)^5 - 2*cosh(b*x^2 + a)^3 + cosh(b*x^2 + a))*sinh(b*x^2 + a) - 1)/(
x*cosh(b*x^2 + a)^3 + 3*x*cosh(b*x^2 + a)^2*sinh(b*x^2 + a) + 3*x*cosh(b*x^2 + a)*sinh(b*x^2 + a)^2 + x*sinh(b
*x^2 + a)^3)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sinh(a + b*x**2)**3/x**2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sinh(b*x^2 + a)^3/x^2, x)