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

Optimal. Leaf size=91 \[ \frac{3}{8} b \sinh (a) \text{Chi}\left (b x^2\right )+\frac{3}{8} b \sinh (3 a) \text{Chi}\left (3 b x^2\right )+\frac{3}{8} b \cosh (a) \text{Shi}\left (b x^2\right )+\frac{3}{8} b \cosh (3 a) \text{Shi}\left (3 b x^2\right )-\frac{3 \cosh \left (a+b x^2\right )}{8 x^2}-\frac{\cosh \left (3 \left (a+b x^2\right )\right )}{8 x^2} \]

[Out]

(-3*Cosh[a + b*x^2])/(8*x^2) - Cosh[3*(a + b*x^2)]/(8*x^2) + (3*b*CoshIntegral[b*x^2]*Sinh[a])/8 + (3*b*CoshIn
tegral[3*b*x^2]*Sinh[3*a])/8 + (3*b*Cosh[a]*SinhIntegral[b*x^2])/8 + (3*b*Cosh[3*a]*SinhIntegral[3*b*x^2])/8

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Rubi [A]  time = 0.218949, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5341, 5321, 3297, 3303, 3298, 3301} \[ \frac{3}{8} b \sinh (a) \text{Chi}\left (b x^2\right )+\frac{3}{8} b \sinh (3 a) \text{Chi}\left (3 b x^2\right )+\frac{3}{8} b \cosh (a) \text{Shi}\left (b x^2\right )+\frac{3}{8} b \cosh (3 a) \text{Shi}\left (3 b x^2\right )-\frac{3 \cosh \left (a+b x^2\right )}{8 x^2}-\frac{\cosh \left (3 \left (a+b x^2\right )\right )}{8 x^2} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x^2]^3/x^3,x]

[Out]

(-3*Cosh[a + b*x^2])/(8*x^2) - Cosh[3*(a + b*x^2)]/(8*x^2) + (3*b*CoshIntegral[b*x^2]*Sinh[a])/8 + (3*b*CoshIn
tegral[3*b*x^2]*Sinh[3*a])/8 + (3*b*Cosh[a]*SinhIntegral[b*x^2])/8 + (3*b*Cosh[3*a]*SinhIntegral[3*b*x^2])/8

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 5321

Int[((a_.) + Cosh[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Cosh[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Sim
plify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rule 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3298

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(I*SinhIntegral[(c*f*fz)
/d + f*fz*x])/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rubi steps

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

Mathematica [A]  time = 0.0811908, size = 92, normalized size = 1.01 \[ \frac{3 b x^2 \sinh (a) \text{Chi}\left (b x^2\right )+3 b x^2 \sinh (3 a) \text{Chi}\left (3 b x^2\right )+3 b x^2 \cosh (a) \text{Shi}\left (b x^2\right )+3 b x^2 \cosh (3 a) \text{Shi}\left (3 b x^2\right )-3 \cosh \left (a+b x^2\right )-\cosh \left (3 \left (a+b x^2\right )\right )}{8 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x^2]^3/x^3,x]

[Out]

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

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Maple [A]  time = 0.052, size = 120, normalized size = 1.3 \begin{align*} -{\frac{{{\rm e}^{-3\,a}}{{\rm e}^{-3\,b{x}^{2}}}}{16\,{x}^{2}}}+{\frac{3\,{{\rm e}^{-3\,a}}b{\it Ei} \left ( 1,3\,b{x}^{2} \right ) }{16}}-{\frac{3\,{{\rm e}^{-a}}{{\rm e}^{-b{x}^{2}}}}{16\,{x}^{2}}}+{\frac{3\,{{\rm e}^{-a}}b{\it Ei} \left ( 1,b{x}^{2} \right ) }{16}}-{\frac{{{\rm e}^{3\,a}}{{\rm e}^{3\,b{x}^{2}}}}{16\,{x}^{2}}}-{\frac{3\,{{\rm e}^{3\,a}}b{\it Ei} \left ( 1,-3\,b{x}^{2} \right ) }{16}}-{\frac{3\,{{\rm e}^{a}}{{\rm e}^{b{x}^{2}}}}{16\,{x}^{2}}}-{\frac{3\,{{\rm e}^{a}}b{\it Ei} \left ( 1,-b{x}^{2} \right ) }{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*exp(-3*a)/x^2*exp(-3*b*x^2)+3/16*exp(-3*a)*b*Ei(1,3*b*x^2)-3/16*exp(-a)/x^2*exp(-b*x^2)+3/16*exp(-a)*b*E
i(1,b*x^2)-1/16*exp(3*a)/x^2*exp(3*b*x^2)-3/16*exp(3*a)*b*Ei(1,-3*b*x^2)-3/16*exp(a)*exp(b*x^2)/x^2-3/16*exp(a
)*b*Ei(1,-b*x^2)

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Maxima [A]  time = 1.33815, size = 78, normalized size = 0.86 \begin{align*} -\frac{3}{16} \, b e^{\left (-3 \, a\right )} \Gamma \left (-1, 3 \, b x^{2}\right ) - \frac{3}{16} \, b e^{\left (-a\right )} \Gamma \left (-1, b x^{2}\right ) + \frac{3}{16} \, b e^{a} \Gamma \left (-1, -b x^{2}\right ) + \frac{3}{16} \, b e^{\left (3 \, a\right )} \Gamma \left (-1, -3 \, b x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.69295, size = 404, normalized size = 4.44 \begin{align*} -\frac{2 \, \cosh \left (b x^{2} + a\right )^{3} + 6 \, \cosh \left (b x^{2} + a\right ) \sinh \left (b x^{2} + a\right )^{2} - 3 \,{\left (b x^{2}{\rm Ei}\left (3 \, b x^{2}\right ) - b x^{2}{\rm Ei}\left (-3 \, b x^{2}\right )\right )} \cosh \left (3 \, a\right ) - 3 \,{\left (b x^{2}{\rm Ei}\left (b x^{2}\right ) - b x^{2}{\rm Ei}\left (-b x^{2}\right )\right )} \cosh \left (a\right ) - 3 \,{\left (b x^{2}{\rm Ei}\left (3 \, b x^{2}\right ) + b x^{2}{\rm Ei}\left (-3 \, b x^{2}\right )\right )} \sinh \left (3 \, a\right ) - 3 \,{\left (b x^{2}{\rm Ei}\left (b x^{2}\right ) + b x^{2}{\rm Ei}\left (-b x^{2}\right )\right )} \sinh \left (a\right ) + 6 \, \cosh \left (b x^{2} + a\right )}{16 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(2*cosh(b*x^2 + a)^3 + 6*cosh(b*x^2 + a)*sinh(b*x^2 + a)^2 - 3*(b*x^2*Ei(3*b*x^2) - b*x^2*Ei(-3*b*x^2))*
cosh(3*a) - 3*(b*x^2*Ei(b*x^2) - b*x^2*Ei(-b*x^2))*cosh(a) - 3*(b*x^2*Ei(3*b*x^2) + b*x^2*Ei(-3*b*x^2))*sinh(3
*a) - 3*(b*x^2*Ei(b*x^2) + b*x^2*Ei(-b*x^2))*sinh(a) + 6*cosh(b*x^2 + a))/x^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cosh(a + b*x**2)**3/x**3, x)

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Giac [B]  time = 1.2537, size = 302, normalized size = 3.32 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )} b^{2}{\rm Ei}\left (3 \, b x^{2}\right ) e^{\left (3 \, a\right )} - 3 \, a b^{2}{\rm Ei}\left (3 \, b x^{2}\right ) e^{\left (3 \, a\right )} - 3 \,{\left (b x^{2} + a\right )} b^{2}{\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} + 3 \, a b^{2}{\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} - 3 \,{\left (b x^{2} + a\right )} b^{2}{\rm Ei}\left (-3 \, b x^{2}\right ) e^{\left (-3 \, a\right )} + 3 \, a b^{2}{\rm Ei}\left (-3 \, b x^{2}\right ) e^{\left (-3 \, a\right )} + 3 \,{\left (b x^{2} + a\right )} b^{2}{\rm Ei}\left (b x^{2}\right ) e^{a} - 3 \, a b^{2}{\rm Ei}\left (b x^{2}\right ) e^{a} - b^{2} e^{\left (3 \, b x^{2} + 3 \, a\right )} - 3 \, b^{2} e^{\left (b x^{2} + a\right )} - 3 \, b^{2} e^{\left (-b x^{2} - a\right )} - b^{2} e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{16 \, b^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(3*(b*x^2 + a)*b^2*Ei(3*b*x^2)*e^(3*a) - 3*a*b^2*Ei(3*b*x^2)*e^(3*a) - 3*(b*x^2 + a)*b^2*Ei(-b*x^2)*e^(-a
) + 3*a*b^2*Ei(-b*x^2)*e^(-a) - 3*(b*x^2 + a)*b^2*Ei(-3*b*x^2)*e^(-3*a) + 3*a*b^2*Ei(-3*b*x^2)*e^(-3*a) + 3*(b
*x^2 + a)*b^2*Ei(b*x^2)*e^a - 3*a*b^2*Ei(b*x^2)*e^a - b^2*e^(3*b*x^2 + 3*a) - 3*b^2*e^(b*x^2 + a) - 3*b^2*e^(-
b*x^2 - a) - b^2*e^(-3*b*x^2 - 3*a))/(b^2*x^2)