\(\int \frac {\cosh (c+d x)}{x^2 (a+b x^2)^2} \, dx\) [71]

Optimal result

Integrand size = 19, antiderivative size = 500 \[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^2\right )^2} \, dx=-\frac {\cosh (c+d x)}{a^2 x}+\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 \sqrt {b} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{5/2}}+\frac {3 \sqrt {b} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{5/2}}+\frac {d \text {Chi}(d x) \sinh (c)}{a^2}+\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a^2}+\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a^2}+\frac {d \cosh (c) \text {Shi}(d x)}{a^2}-\frac {d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a^2}+\frac {3 \sqrt {b} \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{5/2}}+\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 a^2}+\frac {3 \sqrt {b} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{5/2}} \]

[Out]

-cosh(d*x+c)/a^2/x+d*cosh(c)*Shi(d*x)/a^2+1/4*d*cosh(c+d*(-a)^(1/2)/b^(1/2))*Shi(d*x-d*(-a)^(1/2)/b^(1/2))/a^2
+1/4*d*cosh(c-d*(-a)^(1/2)/b^(1/2))*Shi(d*x+d*(-a)^(1/2)/b^(1/2))/a^2+d*Chi(d*x)*sinh(c)/a^2+1/4*d*Chi(d*x+d*(
-a)^(1/2)/b^(1/2))*sinh(c-d*(-a)^(1/2)/b^(1/2))/a^2+1/4*d*Chi(-d*x+d*(-a)^(1/2)/b^(1/2))*sinh(c+d*(-a)^(1/2)/b
^(1/2))/a^2+3/4*Chi(d*x+d*(-a)^(1/2)/b^(1/2))*cosh(c-d*(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(5/2)-3/4*Chi(-d*x+d*(
-a)^(1/2)/b^(1/2))*cosh(c+d*(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(5/2)+3/4*Shi(d*x+d*(-a)^(1/2)/b^(1/2))*sinh(c-d*
(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(5/2)-3/4*Shi(d*x-d*(-a)^(1/2)/b^(1/2))*sinh(c+d*(-a)^(1/2)/b^(1/2))*b^(1/2)/
(-a)^(5/2)+1/4*cosh(d*x+c)*b^(1/2)/a^2/((-a)^(1/2)-x*b^(1/2))-1/4*cosh(d*x+c)*b^(1/2)/a^2/((-a)^(1/2)+x*b^(1/2
))

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5401, 3378, 3384, 3379, 3382, 5389} \[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^2\right )^2} \, dx=\frac {d \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a^2}+\frac {d \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a^2}-\frac {d \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a^2}+\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a^2}+\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d \sinh (c) \text {Chi}(d x)}{a^2}+\frac {d \cosh (c) \text {Shi}(d x)}{a^2}-\frac {\cosh (c+d x)}{a^2 x}-\frac {3 \sqrt {b} \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{5/2}}+\frac {3 \sqrt {b} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{5/2}}+\frac {3 \sqrt {b} \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{5/2}}+\frac {3 \sqrt {b} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{5/2}} \]

[In]

Int[Cosh[c + d*x]/(x^2*(a + b*x^2)^2),x]

[Out]

-(Cosh[c + d*x]/(a^2*x)) + (Sqrt[b]*Cosh[c + d*x])/(4*a^2*(Sqrt[-a] - Sqrt[b]*x)) - (Sqrt[b]*Cosh[c + d*x])/(4
*a^2*(Sqrt[-a] + Sqrt[b]*x)) - (3*Sqrt[b]*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d
*x])/(4*(-a)^(5/2)) + (3*Sqrt[b]*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*(
-a)^(5/2)) + (d*CoshIntegral[d*x]*Sinh[c])/a^2 + (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a
]*d)/Sqrt[b]])/(4*a^2) + (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(4*a^2) +
(d*Cosh[c]*SinhIntegral[d*x])/a^2 - (d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x
])/(4*a^2) + (3*Sqrt[b]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*(-a)^(5/2)
) + (d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*a^2) + (3*Sqrt[b]*Sinh[c -
(Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*(-a)^(5/2))

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 5389

Int[Cosh[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Cosh[c + d*x], (a
+ b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])

Rule 5401

Int[Cosh[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Cosh[c
+ d*x], x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[m] && IGtQ[n, 0] && (Eq
Q[n, 2] || EqQ[p, -1])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\cosh (c+d x)}{a^2 x^2}-\frac {b \cosh (c+d x)}{a \left (a+b x^2\right )^2}-\frac {b \cosh (c+d x)}{a^2 \left (a+b x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {\cosh (c+d x)}{x^2} \, dx}{a^2}-\frac {b \int \frac {\cosh (c+d x)}{a+b x^2} \, dx}{a^2}-\frac {b \int \frac {\cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx}{a} \\ & = -\frac {\cosh (c+d x)}{a^2 x}-\frac {b \int \left (\frac {\sqrt {-a} \cosh (c+d x)}{2 a \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {-a} \cosh (c+d x)}{2 a \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{a^2}-\frac {b \int \left (-\frac {b \cosh (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {b \cosh (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}+b x\right )^2}-\frac {b \cosh (c+d x)}{2 a \left (-a b-b^2 x^2\right )}\right ) \, dx}{a}+\frac {d \int \frac {\sinh (c+d x)}{x} \, dx}{a^2} \\ & = -\frac {\cosh (c+d x)}{a^2 x}+\frac {b \int \frac {\cosh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 (-a)^{5/2}}+\frac {b \int \frac {\cosh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 (-a)^{5/2}}+\frac {b^2 \int \frac {\cosh (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^2} \, dx}{4 a^2}+\frac {b^2 \int \frac {\cosh (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^2} \, dx}{4 a^2}+\frac {b^2 \int \frac {\cosh (c+d x)}{-a b-b^2 x^2} \, dx}{2 a^2}+\frac {(d \cosh (c)) \int \frac {\sinh (d x)}{x} \, dx}{a^2}+\frac {(d \sinh (c)) \int \frac {\cosh (d x)}{x} \, dx}{a^2} \\ & = -\frac {\cosh (c+d x)}{a^2 x}+\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d \text {Chi}(d x) \sinh (c)}{a^2}+\frac {d \cosh (c) \text {Shi}(d x)}{a^2}+\frac {b^2 \int \left (-\frac {\sqrt {-a} \cosh (c+d x)}{2 a b \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {-a} \cosh (c+d x)}{2 a b \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{2 a^2}-\frac {(b d) \int \frac {\sinh (c+d x)}{\sqrt {-a} \sqrt {b}-b x} \, dx}{4 a^2}+\frac {(b d) \int \frac {\sinh (c+d x)}{\sqrt {-a} \sqrt {b}+b x} \, dx}{4 a^2}+\frac {\left (b \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 (-a)^{5/2}}+\frac {\left (b \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 (-a)^{5/2}}+\frac {\left (b \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 (-a)^{5/2}}-\frac {\left (b \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 (-a)^{5/2}} \\ & = -\frac {\cosh (c+d x)}{a^2 x}+\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {\sqrt {b} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{5/2}}+\frac {\sqrt {b} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 (-a)^{5/2}}+\frac {d \text {Chi}(d x) \sinh (c)}{a^2}+\frac {d \cosh (c) \text {Shi}(d x)}{a^2}+\frac {\sqrt {b} \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{5/2}}+\frac {\sqrt {b} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 (-a)^{5/2}}+\frac {b \int \frac {\cosh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 (-a)^{5/2}}+\frac {b \int \frac {\cosh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 (-a)^{5/2}}+\frac {\left (b d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a} \sqrt {b}+b x} \, dx}{4 a^2}+\frac {\left (b d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a} \sqrt {b}-b x} \, dx}{4 a^2}+\frac {\left (b d \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a} \sqrt {b}+b x} \, dx}{4 a^2}-\frac {\left (b d \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a} \sqrt {b}-b x} \, dx}{4 a^2} \\ & = -\frac {\cosh (c+d x)}{a^2 x}+\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {\sqrt {b} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{5/2}}+\frac {\sqrt {b} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 (-a)^{5/2}}+\frac {d \text {Chi}(d x) \sinh (c)}{a^2}+\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a^2}+\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a^2}+\frac {d \cosh (c) \text {Shi}(d x)}{a^2}-\frac {d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a^2}+\frac {\sqrt {b} \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{5/2}}+\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 a^2}+\frac {\sqrt {b} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 (-a)^{5/2}}+\frac {\left (b \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 (-a)^{5/2}}+\frac {\left (b \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 (-a)^{5/2}}+\frac {\left (b \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 (-a)^{5/2}}-\frac {\left (b \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 (-a)^{5/2}} \\ & = -\frac {\cosh (c+d x)}{a^2 x}+\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 \sqrt {b} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{5/2}}+\frac {3 \sqrt {b} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{5/2}}+\frac {d \text {Chi}(d x) \sinh (c)}{a^2}+\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a^2}+\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a^2}+\frac {d \cosh (c) \text {Shi}(d x)}{a^2}-\frac {d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a^2}+\frac {3 \sqrt {b} \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{5/2}}+\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 a^2}+\frac {3 \sqrt {b} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{5/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.15 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.67 \[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^2\right )^2} \, dx=\frac {-\frac {4 \sqrt {a} \left (2 a+3 b x^2\right ) \cosh (c) \cosh (d x)}{x \left (a+b x^2\right )}+e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (3 i \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\left (-3 i \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )-e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (3 i \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\left (-3 i \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )-\frac {4 \sqrt {a} \left (2 a+3 b x^2\right ) \sinh (c) \sinh (d x)}{x \left (a+b x^2\right )}+8 \sqrt {a} d (\text {Chi}(d x) \sinh (c)+\cosh (c) \text {Shi}(d x))}{8 a^{5/2}} \]

[In]

Integrate[Cosh[c + d*x]/(x^2*(a + b*x^2)^2),x]

[Out]

((-4*Sqrt[a]*(2*a + 3*b*x^2)*Cosh[c]*Cosh[d*x])/(x*(a + b*x^2)) + E^(c - (I*Sqrt[a]*d)/Sqrt[b])*(((3*I)*Sqrt[b
] + Sqrt[a]*d)*E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[d*(((-I)*Sqrt[a])/Sqrt[b] + x)] + ((-3*I)*Sqrt[b] +
Sqrt[a]*d)*ExpIntegralEi[d*((I*Sqrt[a])/Sqrt[b] + x)]) - E^(-c - (I*Sqrt[a]*d)/Sqrt[b])*(((3*I)*Sqrt[b] + Sqr
t[a]*d)*E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[((-I)*Sqrt[a]*d)/Sqrt[b] - d*x] + ((-3*I)*Sqrt[b] + Sqrt[a
]*d)*ExpIntegralEi[(I*Sqrt[a]*d)/Sqrt[b] - d*x]) - (4*Sqrt[a]*(2*a + 3*b*x^2)*Sinh[c]*Sinh[d*x])/(x*(a + b*x^2
)) + 8*Sqrt[a]*d*(CoshIntegral[d*x]*Sinh[c] + Cosh[c]*SinhIntegral[d*x]))/(8*a^(5/2))

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 595, normalized size of antiderivative = 1.19

method	result	size
risch	\(-\frac {3 \,{\mathrm e}^{-d x -c} x \,d^{2} b}{4 a^{2} \left (b \,d^{2} x^{2}+a \,d^{2}\right )}-\frac {{\mathrm e}^{-d x -c} d^{2}}{2 a \left (b \,d^{2} x^{2}+a \,d^{2}\right ) x}+\frac {d \,{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{8 a^{2}}+\frac {d \,{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{8 a^{2}}+\frac {3 \,{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) b}{8 a^{2} \sqrt {-a b}}-\frac {3 \,{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) b}{8 a^{2} \sqrt {-a b}}+\frac {d \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right )}{2 a^{2}}-\frac {3 \,{\mathrm e}^{d x +c} x \,d^{2} b}{4 a^{2} \left (b \,d^{2} x^{2}+a \,d^{2}\right )}-\frac {{\mathrm e}^{d x +c} d^{2}}{2 a \left (b \,d^{2} x^{2}+a \,d^{2}\right ) x}-\frac {d \,{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{8 a^{2}}-\frac {d \,{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{8 a^{2}}+\frac {3 \,{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) b}{8 a^{2} \sqrt {-a b}}-\frac {3 \,{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) b}{8 a^{2} \sqrt {-a b}}-\frac {d \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right )}{2 a^{2}}\)	\(595\)

[In]

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

[Out]

-3/4*exp(-d*x-c)/a^2/(b*d^2*x^2+a*d^2)*x*d^2*b-1/2*exp(-d*x-c)/a/(b*d^2*x^2+a*d^2)/x*d^2+1/8*d/a^2*exp(-(d*(-a
*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)+1/8*d/a^2*exp(-(-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*
b)^(1/2)+(d*x+c)*b-c*b)/b)+3/8/a^2/(-a*b)^(1/2)*exp(-(d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)-(d*x+c)*b+c
*b)/b)*b-3/8/a^2/(-a*b)^(1/2)*exp(-(-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*b+1/2*d/a^2
*exp(-c)*Ei(1,d*x)-3/4*exp(d*x+c)/a^2/(b*d^2*x^2+a*d^2)*x*d^2*b-1/2*exp(d*x+c)/a/(b*d^2*x^2+a*d^2)/x*d^2-1/8*d
/a^2*exp((d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)-1/8*d/a^2*exp((-d*(-a*b)^(1/2)+c*b)/b)
*Ei(1,-(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)+3/8/a^2/(-a*b)^(1/2)*exp((d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)
-(d*x+c)*b+c*b)/b)*b-3/8/a^2/(-a*b)^(1/2)*exp((-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)
*b-1/2*d/a^2*exp(c)*Ei(1,-d*x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1310 vs. \(2 (389) = 778\).

Time = 0.29 (sec) , antiderivative size = 1310, normalized size of antiderivative = 2.62 \[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/8*(4*(3*a*b*d*x^2 + 2*a^2*d)*cosh(d*x + c) - (((a*b*d^2*x^3 + a^2*d^2*x)*cosh(d*x + c)^2 - (a*b*d^2*x^3 + a
^2*d^2*x)*sinh(d*x + c)^2 + 3*((b^2*x^3 + a*b*x)*cosh(d*x + c)^2 - (b^2*x^3 + a*b*x)*sinh(d*x + c)^2)*sqrt(-a*
d^2/b))*Ei(d*x - sqrt(-a*d^2/b)) - ((a*b*d^2*x^3 + a^2*d^2*x)*cosh(d*x + c)^2 - (a*b*d^2*x^3 + a^2*d^2*x)*sinh
(d*x + c)^2 - 3*((b^2*x^3 + a*b*x)*cosh(d*x + c)^2 - (b^2*x^3 + a*b*x)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(-d*
x + sqrt(-a*d^2/b)))*cosh(c + sqrt(-a*d^2/b)) - 4*((a*b*d^2*x^3 + a^2*d^2*x)*Ei(d*x) - (a*b*d^2*x^3 + a^2*d^2*
x)*Ei(-d*x))*cosh(c) - (((a*b*d^2*x^3 + a^2*d^2*x)*cosh(d*x + c)^2 - (a*b*d^2*x^3 + a^2*d^2*x)*sinh(d*x + c)^2
- 3*((b^2*x^3 + a*b*x)*cosh(d*x + c)^2 - (b^2*x^3 + a*b*x)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x + sqrt(-a*
d^2/b)) - ((a*b*d^2*x^3 + a^2*d^2*x)*cosh(d*x + c)^2 - (a*b*d^2*x^3 + a^2*d^2*x)*sinh(d*x + c)^2 + 3*((b^2*x^3
+ a*b*x)*cosh(d*x + c)^2 - (b^2*x^3 + a*b*x)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(-d*x - sqrt(-a*d^2/b)))*cosh
(-c + sqrt(-a*d^2/b)) - (((a*b*d^2*x^3 + a^2*d^2*x)*cosh(d*x + c)^2 - (a*b*d^2*x^3 + a^2*d^2*x)*sinh(d*x + c)^
2 + 3*((b^2*x^3 + a*b*x)*cosh(d*x + c)^2 - (b^2*x^3 + a*b*x)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x - sqrt(-a
*d^2/b)) + ((a*b*d^2*x^3 + a^2*d^2*x)*cosh(d*x + c)^2 - (a*b*d^2*x^3 + a^2*d^2*x)*sinh(d*x + c)^2 - 3*((b^2*x^
3 + a*b*x)*cosh(d*x + c)^2 - (b^2*x^3 + a*b*x)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(-d*x + sqrt(-a*d^2/b)))*sin
h(c + sqrt(-a*d^2/b)) - 4*((a*b*d^2*x^3 + a^2*d^2*x)*Ei(d*x) + (a*b*d^2*x^3 + a^2*d^2*x)*Ei(-d*x))*sinh(c) + (
((a*b*d^2*x^3 + a^2*d^2*x)*cosh(d*x + c)^2 - (a*b*d^2*x^3 + a^2*d^2*x)*sinh(d*x + c)^2 - 3*((b^2*x^3 + a*b*x)*
cosh(d*x + c)^2 - (b^2*x^3 + a*b*x)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x + sqrt(-a*d^2/b)) + ((a*b*d^2*x^3
+ a^2*d^2*x)*cosh(d*x + c)^2 - (a*b*d^2*x^3 + a^2*d^2*x)*sinh(d*x + c)^2 + 3*((b^2*x^3 + a*b*x)*cosh(d*x + c)^
2 - (b^2*x^3 + a*b*x)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(-d*x - sqrt(-a*d^2/b)))*sinh(-c + sqrt(-a*d^2/b)))/(
(a^3*b*d*x^3 + a^4*d*x)*cosh(d*x + c)^2 - (a^3*b*d*x^3 + a^4*d*x)*sinh(d*x + c)^2)

Sympy [F]

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

[In]

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

[Out]

Integral(cosh(c + d*x)/(x**2*(a + b*x**2)**2), x)

Maxima [F]

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

[In]

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

[Out]

integrate(cosh(d*x + c)/((b*x^2 + a)^2*x^2), x)

Giac [F]

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

[In]

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

[Out]

integrate(cosh(d*x + c)/((b*x^2 + a)^2*x^2), x)

Mupad [F(-1)]

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

[In]

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

[Out]

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