\(\int \frac {\cosh ^3(a+b x^2)}{x^3} \, dx\) [21]

Optimal result

Integrand size = 14, antiderivative size = 91 \[ \int \frac {\cosh ^3\left (a+b x^2\right )}{x^3} \, dx=-\frac {3 \cosh \left (a+b x^2\right )}{8 x^2}-\frac {\cosh \left (3 \left (a+b x^2\right )\right )}{8 x^2}+\frac {3}{8} b \text {Chi}\left (b x^2\right ) \sinh (a)+\frac {3}{8} b \text {Chi}\left (3 b x^2\right ) \sinh (3 a)+\frac {3}{8} b \cosh (a) \text {Shi}\left (b x^2\right )+\frac {3}{8} b \cosh (3 a) \text {Shi}\left (3 b x^2\right ) \]

[Out]

-3/8*cosh(b*x^2+a)/x^2-1/8*cosh(3*b*x^2+3*a)/x^2+3/8*b*cosh(a)*Shi(b*x^2)+3/8*b*cosh(3*a)*Shi(3*b*x^2)+3/8*b*C
hi(b*x^2)*sinh(a)+3/8*b*Chi(3*b*x^2)*sinh(3*a)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5449, 5429, 3378, 3384, 3379, 3382} \[ \int \frac {\cosh ^3\left (a+b x^2\right )}{x^3} \, dx=\frac {3}{8} b \sinh (a) \text {Chi}\left (b x^2\right )+\frac {3}{8} b \sinh (3 a) \text {Chi}\left (3 b x^2\right )+\frac {3}{8} b \cosh (a) \text {Shi}\left (b x^2\right )+\frac {3}{8} b \cosh (3 a) \text {Shi}\left (3 b x^2\right )-\frac {3 \cosh \left (a+b x^2\right )}{8 x^2}-\frac {\cosh \left (3 \left (a+b x^2\right )\right )}{8 x^2} \]

[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 3378

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 3379

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 3382

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]

Rule 3384

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 5429

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 5449

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*} \text {integral}& = \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} \text {Subst}\left (\int \frac {\cosh (3 a+3 b x)}{x^2} \, dx,x,x^2\right )+\frac {3}{8} \text {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) \text {Subst}\left (\int \frac {\sinh (a+b x)}{x} \, dx,x,x^2\right )+\frac {1}{8} (3 b) \text {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)) \text {Subst}\left (\int \frac {\sinh (b x)}{x} \, dx,x,x^2\right )+\frac {1}{8} (3 b \cosh (3 a)) \text {Subst}\left (\int \frac {\sinh (3 b x)}{x} \, dx,x,x^2\right )+\frac {1}{8} (3 b \sinh (a)) \text {Subst}\left (\int \frac {\cosh (b x)}{x} \, dx,x,x^2\right )+\frac {1}{8} (3 b \sinh (3 a)) \text {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] (verified)

Time = 0.06 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.01 \[ \int \frac {\cosh ^3\left (a+b x^2\right )}{x^3} \, dx=\frac {-3 \cosh \left (a+b x^2\right )-\cosh \left (3 \left (a+b x^2\right )\right )+3 b x^2 \text {Chi}\left (b x^2\right ) \sinh (a)+3 b x^2 \text {Chi}\left (3 b x^2\right ) \sinh (3 a)+3 b x^2 \cosh (a) \text {Shi}\left (b x^2\right )+3 b x^2 \cosh (3 a) \text {Shi}\left (3 b x^2\right )}{8 x^2} \]

[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)

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.33

method	result	size
risch	\(-\frac {-3 \,{\mathrm e}^{-3 a} \operatorname {Ei}_{1}\left (3 b \,x^{2}\right ) b \,x^{2}-3 \,{\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b \,x^{2}\right ) b \,x^{2}+3 \,{\mathrm e}^{3 a} \operatorname {Ei}_{1}\left (-3 b \,x^{2}\right ) b \,x^{2}+3 \,\operatorname {Ei}_{1}\left (-b \,x^{2}\right ) {\mathrm e}^{a} b \,x^{2}+{\mathrm e}^{-3 b \,x^{2}-3 a}+3 \,{\mathrm e}^{-b \,x^{2}-a}+{\mathrm e}^{3 b \,x^{2}+3 a}+3 \,{\mathrm e}^{b \,x^{2}+a}}{16 x^{2}}\)	\(121\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (80) = 160\).

Time = 0.25 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.85 \[ \int \frac {\cosh ^3\left (a+b x^2\right )}{x^3} \, dx=-\frac {2 \, \cosh \left (b x^{2} + a\right )^{3} + 6 \, \cosh \left (b x^{2} + a\right ) \sinh \left (b x^{2} + a\right )^{2} - 3 \, {\left (b x^{2} {\rm Ei}\left (3 \, b x^{2}\right ) - b x^{2} {\rm Ei}\left (-3 \, b x^{2}\right )\right )} \cosh \left (3 \, a\right ) - 3 \, {\left (b x^{2} {\rm Ei}\left (b x^{2}\right ) - b x^{2} {\rm Ei}\left (-b x^{2}\right )\right )} \cosh \left (a\right ) - 3 \, {\left (b x^{2} {\rm Ei}\left (3 \, b x^{2}\right ) + b x^{2} {\rm Ei}\left (-3 \, b x^{2}\right )\right )} \sinh \left (3 \, a\right ) - 3 \, {\left (b x^{2} {\rm Ei}\left (b x^{2}\right ) + b x^{2} {\rm Ei}\left (-b x^{2}\right )\right )} \sinh \left (a\right ) + 6 \, \cosh \left (b x^{2} + a\right )}{16 \, x^{2}} \]

[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

Sympy [F]

\[ \int \frac {\cosh ^3\left (a+b x^2\right )}{x^3} \, dx=\int \frac {\cosh ^{3}{\left (a + b x^{2} \right )}}{x^{3}}\, dx \]

[In]

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

[Out]

Integral(cosh(a + b*x**2)**3/x**3, x)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.64 \[ \int \frac {\cosh ^3\left (a+b x^2\right )}{x^3} \, dx=-\frac {3}{16} \, b e^{\left (-3 \, a\right )} \Gamma \left (-1, 3 \, b x^{2}\right ) - \frac {3}{16} \, b e^{\left (-a\right )} \Gamma \left (-1, b x^{2}\right ) + \frac {3}{16} \, b e^{a} \Gamma \left (-1, -b x^{2}\right ) + \frac {3}{16} \, b e^{\left (3 \, a\right )} \Gamma \left (-1, -3 \, b x^{2}\right ) \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (80) = 160\).

Time = 0.26 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.46 \[ \int \frac {\cosh ^3\left (a+b x^2\right )}{x^3} \, dx=\frac {3 \, {\left (b x^{2} + a\right )} b^{2} {\rm Ei}\left (3 \, b x^{2}\right ) e^{\left (3 \, a\right )} - 3 \, a b^{2} {\rm Ei}\left (3 \, b x^{2}\right ) e^{\left (3 \, a\right )} - 3 \, {\left (b x^{2} + a\right )} b^{2} {\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} + 3 \, a b^{2} {\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} - 3 \, {\left (b x^{2} + a\right )} b^{2} {\rm Ei}\left (-3 \, b x^{2}\right ) e^{\left (-3 \, a\right )} + 3 \, a b^{2} {\rm Ei}\left (-3 \, b x^{2}\right ) e^{\left (-3 \, a\right )} + 3 \, {\left (b x^{2} + a\right )} b^{2} {\rm Ei}\left (b x^{2}\right ) e^{a} - 3 \, a b^{2} {\rm Ei}\left (b x^{2}\right ) e^{a} - b^{2} e^{\left (3 \, b x^{2} + 3 \, a\right )} - 3 \, b^{2} e^{\left (b x^{2} + a\right )} - 3 \, b^{2} e^{\left (-b x^{2} - a\right )} - b^{2} e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{16 \, b^{2} x^{2}} \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cosh ^3\left (a+b x^2\right )}{x^3} \, dx=\int \frac {{\mathrm {cosh}\left (b\,x^2+a\right )}^3}{x^3} \,d x \]

[In]

int(cosh(a + b*x^2)^3/x^3,x)

[Out]

int(cosh(a + b*x^2)^3/x^3, x)