\(\int \frac {\cosh (c+d x)}{x^3 (a+b x^2)^3} \, dx\) [78]

Optimal result

Integrand size = 19, antiderivative size = 791 \[ \int \frac {\cosh (c+d x)}{x^3 \left (a+b x^2\right )^3} \, dx=-\frac {\cosh (c+d x)}{2 a^3 x^2}-\frac {b \cosh (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac {b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}-\frac {3 b \cosh (c) \text {Chi}(d x)}{a^4}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a^3}+\frac {3 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^4}-\frac {d^2 \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^3}+\frac {3 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^4}-\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a^3}+\frac {9 \sqrt {b} d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{7/2}}-\frac {9 \sqrt {b} d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{7/2}}-\frac {d \sinh (c+d x)}{2 a^3 x}-\frac {\sqrt {b} d \sinh (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} d \sinh (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 b \sinh (c) \text {Shi}(d x)}{a^4}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a^3}+\frac {9 \sqrt {b} d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{7/2}}-\frac {3 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^4}+\frac {d^2 \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^3}+\frac {9 \sqrt {b} d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{7/2}}+\frac {3 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^4}-\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a^3} \]

[Out]

-3*b*Chi(d*x)*cosh(c)/a^4+1/2*d^2*Chi(d*x)*cosh(c)/a^3-1/2*cosh(d*x+c)/a^3/x^2-1/4*b*cosh(d*x+c)/a^2/(b*x^2+a)
^2-b*cosh(d*x+c)/a^3/(b*x^2+a)+3/2*b*Chi(d*x+d*(-a)^(1/2)/b^(1/2))*cosh(c-d*(-a)^(1/2)/b^(1/2))/a^4-1/16*d^2*C
hi(d*x+d*(-a)^(1/2)/b^(1/2))*cosh(c-d*(-a)^(1/2)/b^(1/2))/a^3+3/2*b*Chi(-d*x+d*(-a)^(1/2)/b^(1/2))*cosh(c+d*(-
a)^(1/2)/b^(1/2))/a^4-1/16*d^2*Chi(-d*x+d*(-a)^(1/2)/b^(1/2))*cosh(c+d*(-a)^(1/2)/b^(1/2))/a^3-3*b*Shi(d*x)*si
nh(c)/a^4+1/2*d^2*Shi(d*x)*sinh(c)/a^3-1/2*d*sinh(d*x+c)/a^3/x+3/2*b*Shi(d*x+d*(-a)^(1/2)/b^(1/2))*sinh(c-d*(-
a)^(1/2)/b^(1/2))/a^4-1/16*d^2*Shi(d*x+d*(-a)^(1/2)/b^(1/2))*sinh(c-d*(-a)^(1/2)/b^(1/2))/a^3+3/2*b*Shi(d*x-d*
(-a)^(1/2)/b^(1/2))*sinh(c+d*(-a)^(1/2)/b^(1/2))/a^4-1/16*d^2*Shi(d*x-d*(-a)^(1/2)/b^(1/2))*sinh(c+d*(-a)^(1/2
)/b^(1/2))/a^3-9/16*d*cosh(c+d*(-a)^(1/2)/b^(1/2))*Shi(d*x-d*(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(7/2)+9/16*d*cos
h(c-d*(-a)^(1/2)/b^(1/2))*Shi(d*x+d*(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(7/2)+9/16*d*Chi(d*x+d*(-a)^(1/2)/b^(1/2)
)*sinh(c-d*(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(7/2)-9/16*d*Chi(-d*x+d*(-a)^(1/2)/b^(1/2))*sinh(c+d*(-a)^(1/2)/b^
(1/2))*b^(1/2)/(-a)^(7/2)-1/16*d*sinh(d*x+c)*b^(1/2)/a^3/((-a)^(1/2)-x*b^(1/2))+1/16*d*sinh(d*x+c)*b^(1/2)/a^3
/((-a)^(1/2)+x*b^(1/2))

Rubi [A] (verified)

Time = 1.34 (sec) , antiderivative size = 791, normalized size of antiderivative = 1.00, number of steps used = 46, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5401, 3378, 3384, 3379, 3382, 5397, 5388} \[ \int \frac {\cosh (c+d x)}{x^3 \left (a+b x^2\right )^3} \, dx=-\frac {3 b \cosh (c) \text {Chi}(d x)}{a^4}+\frac {3 b \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^4}+\frac {3 b \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^4}-\frac {3 b \sinh (c) \text {Shi}(d x)}{a^4}-\frac {3 b \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^4}+\frac {3 b \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^4}-\frac {d^2 \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^3}-\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a^3}+\frac {d^2 \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^3}-\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a^3}-\frac {b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}-\frac {\sqrt {b} d \sinh (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} d \sinh (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a^3}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a^3}-\frac {\cosh (c+d x)}{2 a^3 x^2}-\frac {d \sinh (c+d x)}{2 a^3 x}-\frac {b \cosh (c+d x)}{4 a^2 \left (a+b x^2\right )^2}+\frac {9 \sqrt {b} d \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{7/2}}-\frac {9 \sqrt {b} d \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{7/2}}+\frac {9 \sqrt {b} d \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{7/2}}+\frac {9 \sqrt {b} d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{7/2}} \]

[In]

Int[Cosh[c + d*x]/(x^3*(a + b*x^2)^3),x]

[Out]

-1/2*Cosh[c + d*x]/(a^3*x^2) - (b*Cosh[c + d*x])/(4*a^2*(a + b*x^2)^2) - (b*Cosh[c + d*x])/(a^3*(a + b*x^2)) -
(3*b*Cosh[c]*CoshIntegral[d*x])/a^4 + (d^2*Cosh[c]*CoshIntegral[d*x])/(2*a^3) + (3*b*Cosh[c + (Sqrt[-a]*d)/Sq
rt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^4) - (d^2*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(S
qrt[-a]*d)/Sqrt[b] - d*x])/(16*a^3) + (3*b*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] +
d*x])/(2*a^4) - (d^2*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^3) + (9*Sq
rt[b]*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(7/2)) - (9*Sqrt[b]*
d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(7/2)) - (d*Sinh[c + d*x])
/(2*a^3*x) - (Sqrt[b]*d*Sinh[c + d*x])/(16*a^3*(Sqrt[-a] - Sqrt[b]*x)) + (Sqrt[b]*d*Sinh[c + d*x])/(16*a^3*(Sq
rt[-a] + Sqrt[b]*x)) - (3*b*Sinh[c]*SinhIntegral[d*x])/a^4 + (d^2*Sinh[c]*SinhIntegral[d*x])/(2*a^3) + (9*Sqrt
[b]*d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(7/2)) - (3*b*Sinh[c +
(Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^4) + (d^2*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]
*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^3) + (9*Sqrt[b]*d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral
[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(7/2)) + (3*b*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/
Sqrt[b] + d*x])/(2*a^4) - (d^2*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^
3)

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 5388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*Sinh[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegrand[Sinh[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 5397

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

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^3 x^3}-\frac {3 b \cosh (c+d x)}{a^4 x}+\frac {b^2 x \cosh (c+d x)}{a^2 \left (a+b x^2\right )^3}+\frac {2 b^2 x \cosh (c+d x)}{a^3 \left (a+b x^2\right )^2}+\frac {3 b^2 x \cosh (c+d x)}{a^4 \left (a+b x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {\cosh (c+d x)}{x^3} \, dx}{a^3}-\frac {(3 b) \int \frac {\cosh (c+d x)}{x} \, dx}{a^4}+\frac {\left (3 b^2\right ) \int \frac {x \cosh (c+d x)}{a+b x^2} \, dx}{a^4}+\frac {\left (2 b^2\right ) \int \frac {x \cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx}{a^3}+\frac {b^2 \int \frac {x \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx}{a^2} \\ & = -\frac {\cosh (c+d x)}{2 a^3 x^2}-\frac {b \cosh (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac {b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}+\frac {\left (3 b^2\right ) \int \left (-\frac {\cosh (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\cosh (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{a^4}+\frac {d \int \frac {\sinh (c+d x)}{x^2} \, dx}{2 a^3}+\frac {(b d) \int \frac {\sinh (c+d x)}{a+b x^2} \, dx}{a^3}+\frac {(b d) \int \frac {\sinh (c+d x)}{\left (a+b x^2\right )^2} \, dx}{4 a^2}-\frac {(3 b \cosh (c)) \int \frac {\cosh (d x)}{x} \, dx}{a^4}-\frac {(3 b \sinh (c)) \int \frac {\sinh (d x)}{x} \, dx}{a^4} \\ & = -\frac {\cosh (c+d x)}{2 a^3 x^2}-\frac {b \cosh (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac {b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}-\frac {3 b \cosh (c) \text {Chi}(d x)}{a^4}-\frac {d \sinh (c+d x)}{2 a^3 x}-\frac {3 b \sinh (c) \text {Shi}(d x)}{a^4}-\frac {\left (3 b^{3/2}\right ) \int \frac {\cosh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 a^4}+\frac {\left (3 b^{3/2}\right ) \int \frac {\cosh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 a^4}+\frac {(b d) \int \left (\frac {\sqrt {-a} \sinh (c+d x)}{2 a \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {-a} \sinh (c+d x)}{2 a \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{a^3}+\frac {(b d) \int \left (-\frac {b \sinh (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {b \sinh (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}+b x\right )^2}-\frac {b \sinh (c+d x)}{2 a \left (-a b-b^2 x^2\right )}\right ) \, dx}{4 a^2}+\frac {d^2 \int \frac {\cosh (c+d x)}{x} \, dx}{2 a^3} \\ & = -\frac {\cosh (c+d x)}{2 a^3 x^2}-\frac {b \cosh (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac {b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}-\frac {3 b \cosh (c) \text {Chi}(d x)}{a^4}-\frac {d \sinh (c+d x)}{2 a^3 x}-\frac {3 b \sinh (c) \text {Shi}(d x)}{a^4}+\frac {(b d) \int \frac {\sinh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 (-a)^{7/2}}+\frac {(b d) \int \frac {\sinh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 (-a)^{7/2}}-\frac {\left (b^2 d\right ) \int \frac {\sinh (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^2} \, dx}{16 a^3}-\frac {\left (b^2 d\right ) \int \frac {\sinh (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^2} \, dx}{16 a^3}-\frac {\left (b^2 d\right ) \int \frac {\sinh (c+d x)}{-a b-b^2 x^2} \, dx}{8 a^3}+\frac {\left (d^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx}{2 a^3}+\frac {\left (3 b^{3/2} \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^4}-\frac {\left (3 b^{3/2} \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^4}+\frac {\left (d^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx}{2 a^3}+\frac {\left (3 b^{3/2} \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^4}+\frac {\left (3 b^{3/2} \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^4} \\ & = -\frac {\cosh (c+d x)}{2 a^3 x^2}-\frac {b \cosh (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac {b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}-\frac {3 b \cosh (c) \text {Chi}(d x)}{a^4}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a^3}+\frac {3 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^4}+\frac {3 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^4}-\frac {d \sinh (c+d x)}{2 a^3 x}-\frac {\sqrt {b} d \sinh (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} d \sinh (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 b \sinh (c) \text {Shi}(d x)}{a^4}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a^3}-\frac {3 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^4}+\frac {3 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^4}-\frac {\left (b^2 d\right ) \int \left (-\frac {\sqrt {-a} \sinh (c+d x)}{2 a b \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {-a} \sinh (c+d x)}{2 a b \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{8 a^3}+\frac {\left (b d^2\right ) \int \frac {\cosh (c+d x)}{\sqrt {-a} \sqrt {b}-b x} \, dx}{16 a^3}-\frac {\left (b d^2\right ) \int \frac {\cosh (c+d x)}{\sqrt {-a} \sqrt {b}+b x} \, dx}{16 a^3}+\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} x} \, dx}{2 (-a)^{7/2}}-\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} x} \, dx}{2 (-a)^{7/2}}+\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} x} \, dx}{2 (-a)^{7/2}}+\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} x} \, dx}{2 (-a)^{7/2}} \\ & = -\frac {\cosh (c+d x)}{2 a^3 x^2}-\frac {b \cosh (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac {b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}-\frac {3 b \cosh (c) \text {Chi}(d x)}{a^4}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a^3}+\frac {3 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^4}+\frac {3 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^4}+\frac {\sqrt {b} d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 (-a)^{7/2}}-\frac {\sqrt {b} d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 (-a)^{7/2}}-\frac {d \sinh (c+d x)}{2 a^3 x}-\frac {\sqrt {b} d \sinh (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} d \sinh (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 b \sinh (c) \text {Shi}(d x)}{a^4}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a^3}+\frac {\sqrt {b} d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{7/2}}-\frac {3 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^4}+\frac {\sqrt {b} d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 (-a)^{7/2}}+\frac {3 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^4}+\frac {(b d) \int \frac {\sinh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{16 (-a)^{7/2}}+\frac {(b d) \int \frac {\sinh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{16 (-a)^{7/2}}-\frac {\left (b d^2 \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}+b x} \, dx}{16 a^3}+\frac {\left (b d^2 \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}-b x} \, dx}{16 a^3}-\frac {\left (b d^2 \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}+b x} \, dx}{16 a^3}-\frac {\left (b d^2 \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}-b x} \, dx}{16 a^3} \\ & = -\frac {\cosh (c+d x)}{2 a^3 x^2}-\frac {b \cosh (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac {b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}-\frac {3 b \cosh (c) \text {Chi}(d x)}{a^4}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a^3}+\frac {3 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^4}-\frac {d^2 \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^3}+\frac {3 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^4}-\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a^3}+\frac {\sqrt {b} d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 (-a)^{7/2}}-\frac {\sqrt {b} d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 (-a)^{7/2}}-\frac {d \sinh (c+d x)}{2 a^3 x}-\frac {\sqrt {b} d \sinh (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} d \sinh (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 b \sinh (c) \text {Shi}(d x)}{a^4}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a^3}+\frac {\sqrt {b} d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{7/2}}-\frac {3 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^4}+\frac {d^2 \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^3}+\frac {\sqrt {b} d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 (-a)^{7/2}}+\frac {3 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^4}-\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a^3}+\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} x} \, dx}{16 (-a)^{7/2}}-\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} x} \, dx}{16 (-a)^{7/2}}+\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} x} \, dx}{16 (-a)^{7/2}}+\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} x} \, dx}{16 (-a)^{7/2}} \\ & = -\frac {\cosh (c+d x)}{2 a^3 x^2}-\frac {b \cosh (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac {b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}-\frac {3 b \cosh (c) \text {Chi}(d x)}{a^4}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a^3}+\frac {3 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^4}-\frac {d^2 \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^3}+\frac {3 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^4}-\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a^3}+\frac {9 \sqrt {b} d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{7/2}}-\frac {9 \sqrt {b} d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{7/2}}-\frac {d \sinh (c+d x)}{2 a^3 x}-\frac {\sqrt {b} d \sinh (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} d \sinh (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 b \sinh (c) \text {Shi}(d x)}{a^4}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a^3}+\frac {9 \sqrt {b} d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{7/2}}-\frac {3 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^4}+\frac {d^2 \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^3}+\frac {9 \sqrt {b} d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{7/2}}+\frac {3 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^4}-\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a^3} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.85 (sec) , antiderivative size = 438, normalized size of antiderivative = 0.55 \[ \int \frac {\cosh (c+d x)}{x^3 \left (a+b x^2\right )^3} \, dx=\frac {e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (24 b-9 i \sqrt {a} \sqrt {b} d-a d^2\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\left (24 b+9 i \sqrt {a} \sqrt {b} d-a d^2\right ) \operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )+e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (24 b-9 i \sqrt {a} \sqrt {b} d-a d^2\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\left (24 b+9 i \sqrt {a} \sqrt {b} d-a d^2\right ) \operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )-\frac {4 a \cosh (d x) \left (2 \left (2 a^2+9 a b x^2+6 b^2 x^4\right ) \cosh (c)+d x \left (4 a^2+7 a b x^2+3 b^2 x^4\right ) \sinh (c)\right )}{x^2 \left (a+b x^2\right )^2}-\frac {4 a \left (d x \left (4 a^2+7 a b x^2+3 b^2 x^4\right ) \cosh (c)+2 \left (2 a^2+9 a b x^2+6 b^2 x^4\right ) \sinh (c)\right ) \sinh (d x)}{x^2 \left (a+b x^2\right )^2}+16 \left (-6 b+a d^2\right ) (\cosh (c) \text {Chi}(d x)+\sinh (c) \text {Shi}(d x))}{32 a^4} \]

[In]

Integrate[Cosh[c + d*x]/(x^3*(a + b*x^2)^3),x]

[Out]

(E^(c - (I*Sqrt[a]*d)/Sqrt[b])*((24*b - (9*I)*Sqrt[a]*Sqrt[b]*d - a*d^2)*E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpInte
gralEi[d*(((-I)*Sqrt[a])/Sqrt[b] + x)] + (24*b + (9*I)*Sqrt[a]*Sqrt[b]*d - a*d^2)*ExpIntegralEi[d*((I*Sqrt[a])
/Sqrt[b] + x)]) + E^(-c - (I*Sqrt[a]*d)/Sqrt[b])*((24*b - (9*I)*Sqrt[a]*Sqrt[b]*d - a*d^2)*E^(((2*I)*Sqrt[a]*d
)/Sqrt[b])*ExpIntegralEi[((-I)*Sqrt[a]*d)/Sqrt[b] - d*x] + (24*b + (9*I)*Sqrt[a]*Sqrt[b]*d - a*d^2)*ExpIntegra
lEi[(I*Sqrt[a]*d)/Sqrt[b] - d*x]) - (4*a*Cosh[d*x]*(2*(2*a^2 + 9*a*b*x^2 + 6*b^2*x^4)*Cosh[c] + d*x*(4*a^2 + 7
*a*b*x^2 + 3*b^2*x^4)*Sinh[c]))/(x^2*(a + b*x^2)^2) - (4*a*(d*x*(4*a^2 + 7*a*b*x^2 + 3*b^2*x^4)*Cosh[c] + 2*(2
*a^2 + 9*a*b*x^2 + 6*b^2*x^4)*Sinh[c])*Sinh[d*x])/(x^2*(a + b*x^2)^2) + 16*(-6*b + a*d^2)*(Cosh[c]*CoshIntegra
l[d*x] + Sinh[c]*SinhIntegral[d*x]))/(32*a^4)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1293\) vs. \(2(629)=1258\).

Time = 0.49 (sec) , antiderivative size = 1294, normalized size of antiderivative = 1.64

method	result	size
risch	\(\text {Expression too large to display}\)	\(1294\)

[In]

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

[Out]

9/32*d/a^3/(-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-9/32*d/a^3/(-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-3/4/a^4*exp((-d*(-a*b)^(1/2)+c
*b)/b)*Ei(1,-(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*b-9/8*exp(d*x+c)*d^4/a^2/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)*b-
1/4*exp(d*x+c)/a/x^2*d^4/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)-9/8*exp(-d*x-c)*d^4/a^2/(b^2*d^4*x^4+2*a*b*d^4*x^
2+a^2*d^4)*b-1/4*exp(-d*x-c)/a/x^2*d^4/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)-7/16*d^5*exp(d*x+c)/a^2/(b^2*d^4*x^
4+2*a*b*d^4*x^2+a^2*d^4)*b*x-3/4*exp(d*x+c)/a^3*x^2*d^4/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)*b^2-3/16*d^5*exp(d
*x+c)/a^3*x^3/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)*b^2+3/16*d^5*exp(-d*x-c)/a^3*x^3/(b^2*d^4*x^4+2*a*b*d^4*x^2+
a^2*d^4)*b^2+7/16*d^5*exp(-d*x-c)/a^2/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)*b*x-3/4*exp(-d*x-c)/a^3*x^2*d^4/(b^2
*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)*b^2-1/4*d^5*exp(d*x+c)/a/x/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)+1/4*d^5*exp(-d*
x-c)/a/x/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)+1/32*d^2/a^3*exp(-(d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)-(
d*x+c)*b+c*b)/b)+1/32*d^2/a^3*exp(-(-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)-3/4/a^4*exp
(-(d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*b-3/4/a^4*exp(-(-d*(-a*b)^(1/2)+c*b)/b)*Ei(1
,(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*b-9/32*d/a^3/(-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+9/32*d/a^3/(-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+1/32*d^2/a^3*exp((d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)+1/32*d^2/a^3*exp((-d*(-a*
b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)-3/4/a^4*exp((d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1
/2)-(d*x+c)*b+c*b)/b)*b+3/2/a^4*b*exp(c)*Ei(1,-d*x)+3/2/a^4*exp(-c)*Ei(1,d*x)*b-1/4*d^2/a^3*exp(-c)*Ei(1,d*x)-
1/4*d^2/a^3*exp(c)*Ei(1,-d*x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2363 vs. \(2 (630) = 1260\).

Time = 0.30 (sec) , antiderivative size = 2363, normalized size of antiderivative = 2.99 \[ \int \frac {\cosh (c+d x)}{x^3 \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/32*(8*(6*a*b^2*x^4 + 9*a^2*b*x^2 + 2*a^3)*cosh(d*x + c) + ((((a*b^2*d^2 - 24*b^3)*x^6 + 2*(a^2*b*d^2 - 24*a
*b^2)*x^4 + (a^3*d^2 - 24*a^2*b)*x^2)*cosh(d*x + c)^2 - ((a*b^2*d^2 - 24*b^3)*x^6 + 2*(a^2*b*d^2 - 24*a*b^2)*x
^4 + (a^3*d^2 - 24*a^2*b)*x^2)*sinh(d*x + c)^2 + 9*((b^3*x^6 + 2*a*b^2*x^4 + a^2*b*x^2)*cosh(d*x + c)^2 - (b^3
*x^6 + 2*a*b^2*x^4 + a^2*b*x^2)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x - sqrt(-a*d^2/b)) + (((a*b^2*d^2 - 24*
b^3)*x^6 + 2*(a^2*b*d^2 - 24*a*b^2)*x^4 + (a^3*d^2 - 24*a^2*b)*x^2)*cosh(d*x + c)^2 - ((a*b^2*d^2 - 24*b^3)*x^
6 + 2*(a^2*b*d^2 - 24*a*b^2)*x^4 + (a^3*d^2 - 24*a^2*b)*x^2)*sinh(d*x + c)^2 - 9*((b^3*x^6 + 2*a*b^2*x^4 + a^2
*b*x^2)*cosh(d*x + c)^2 - (b^3*x^6 + 2*a*b^2*x^4 + a^2*b*x^2)*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)) - 8*(((a*b^2*d^2 - 6*b^3)*x^6 + 2*(a^2*b*d^2 - 6*a*b^2)*x^4 + (a^3*d^2 -
6*a^2*b)*x^2)*Ei(d*x) + ((a*b^2*d^2 - 6*b^3)*x^6 + 2*(a^2*b*d^2 - 6*a*b^2)*x^4 + (a^3*d^2 - 6*a^2*b)*x^2)*Ei(-
d*x))*cosh(c) + ((((a*b^2*d^2 - 24*b^3)*x^6 + 2*(a^2*b*d^2 - 24*a*b^2)*x^4 + (a^3*d^2 - 24*a^2*b)*x^2)*cosh(d*
x + c)^2 - ((a*b^2*d^2 - 24*b^3)*x^6 + 2*(a^2*b*d^2 - 24*a*b^2)*x^4 + (a^3*d^2 - 24*a^2*b)*x^2)*sinh(d*x + c)^
2 - 9*((b^3*x^6 + 2*a*b^2*x^4 + a^2*b*x^2)*cosh(d*x + c)^2 - (b^3*x^6 + 2*a*b^2*x^4 + a^2*b*x^2)*sinh(d*x + c)
^2)*sqrt(-a*d^2/b))*Ei(d*x + sqrt(-a*d^2/b)) + (((a*b^2*d^2 - 24*b^3)*x^6 + 2*(a^2*b*d^2 - 24*a*b^2)*x^4 + (a^
3*d^2 - 24*a^2*b)*x^2)*cosh(d*x + c)^2 - ((a*b^2*d^2 - 24*b^3)*x^6 + 2*(a^2*b*d^2 - 24*a*b^2)*x^4 + (a^3*d^2 -
24*a^2*b)*x^2)*sinh(d*x + c)^2 + 9*((b^3*x^6 + 2*a*b^2*x^4 + a^2*b*x^2)*cosh(d*x + c)^2 - (b^3*x^6 + 2*a*b^2*
x^4 + a^2*b*x^2)*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*(3*
a*b^2*d*x^5 + 7*a^2*b*d*x^3 + 4*a^3*d*x)*sinh(d*x + c) + ((((a*b^2*d^2 - 24*b^3)*x^6 + 2*(a^2*b*d^2 - 24*a*b^2
)*x^4 + (a^3*d^2 - 24*a^2*b)*x^2)*cosh(d*x + c)^2 - ((a*b^2*d^2 - 24*b^3)*x^6 + 2*(a^2*b*d^2 - 24*a*b^2)*x^4 +
(a^3*d^2 - 24*a^2*b)*x^2)*sinh(d*x + c)^2 + 9*((b^3*x^6 + 2*a*b^2*x^4 + a^2*b*x^2)*cosh(d*x + c)^2 - (b^3*x^6
+ 2*a*b^2*x^4 + a^2*b*x^2)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x - sqrt(-a*d^2/b)) - (((a*b^2*d^2 - 24*b^3)
*x^6 + 2*(a^2*b*d^2 - 24*a*b^2)*x^4 + (a^3*d^2 - 24*a^2*b)*x^2)*cosh(d*x + c)^2 - ((a*b^2*d^2 - 24*b^3)*x^6 +
2*(a^2*b*d^2 - 24*a*b^2)*x^4 + (a^3*d^2 - 24*a^2*b)*x^2)*sinh(d*x + c)^2 - 9*((b^3*x^6 + 2*a*b^2*x^4 + a^2*b*x
^2)*cosh(d*x + c)^2 - (b^3*x^6 + 2*a*b^2*x^4 + a^2*b*x^2)*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)) - 8*(((a*b^2*d^2 - 6*b^3)*x^6 + 2*(a^2*b*d^2 - 6*a*b^2)*x^4 + (a^3*d^2 - 6*a^
2*b)*x^2)*Ei(d*x) - ((a*b^2*d^2 - 6*b^3)*x^6 + 2*(a^2*b*d^2 - 6*a*b^2)*x^4 + (a^3*d^2 - 6*a^2*b)*x^2)*Ei(-d*x)
)*sinh(c) - ((((a*b^2*d^2 - 24*b^3)*x^6 + 2*(a^2*b*d^2 - 24*a*b^2)*x^4 + (a^3*d^2 - 24*a^2*b)*x^2)*cosh(d*x +
c)^2 - ((a*b^2*d^2 - 24*b^3)*x^6 + 2*(a^2*b*d^2 - 24*a*b^2)*x^4 + (a^3*d^2 - 24*a^2*b)*x^2)*sinh(d*x + c)^2 -
9*((b^3*x^6 + 2*a*b^2*x^4 + a^2*b*x^2)*cosh(d*x + c)^2 - (b^3*x^6 + 2*a*b^2*x^4 + a^2*b*x^2)*sinh(d*x + c)^2)*
sqrt(-a*d^2/b))*Ei(d*x + sqrt(-a*d^2/b)) - (((a*b^2*d^2 - 24*b^3)*x^6 + 2*(a^2*b*d^2 - 24*a*b^2)*x^4 + (a^3*d^
2 - 24*a^2*b)*x^2)*cosh(d*x + c)^2 - ((a*b^2*d^2 - 24*b^3)*x^6 + 2*(a^2*b*d^2 - 24*a*b^2)*x^4 + (a^3*d^2 - 24*
a^2*b)*x^2)*sinh(d*x + c)^2 + 9*((b^3*x^6 + 2*a*b^2*x^4 + a^2*b*x^2)*cosh(d*x + c)^2 - (b^3*x^6 + 2*a*b^2*x^4
+ a^2*b*x^2)*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^4*b^2*
x^6 + 2*a^5*b*x^4 + a^6*x^2)*cosh(d*x + c)^2 - (a^4*b^2*x^6 + 2*a^5*b*x^4 + a^6*x^2)*sinh(d*x + c)^2)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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