\(\int \frac {\cosh (c+d x)}{x^3 (a+b x)^3} \, dx\) [39]

Optimal result

Integrand size = 17, antiderivative size = 377 \[ \int \frac {\cosh (c+d x)}{x^3 (a+b x)^3} \, dx=-\frac {\cosh (c+d x)}{2 a^3 x^2}+\frac {3 b \cosh (c+d x)}{a^4 x}+\frac {b^2 \cosh (c+d x)}{2 a^3 (a+b x)^2}+\frac {3 b^2 \cosh (c+d x)}{a^4 (a+b x)}+\frac {6 b^2 \cosh (c) \text {Chi}(d x)}{a^5}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a^3}-\frac {6 b^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{a^5}-\frac {d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{2 a^3}-\frac {3 b d \text {Chi}(d x) \sinh (c)}{a^4}-\frac {3 b d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{a^4}-\frac {d \sinh (c+d x)}{2 a^3 x}+\frac {b d \sinh (c+d x)}{2 a^3 (a+b x)}-\frac {3 b d \cosh (c) \text {Shi}(d x)}{a^4}+\frac {6 b^2 \sinh (c) \text {Shi}(d x)}{a^5}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a^3}-\frac {3 b d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^4}-\frac {6 b^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^5}-\frac {d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{2 a^3} \]

[Out]

6*b^2*Chi(d*x)*cosh(c)/a^5+1/2*d^2*Chi(d*x)*cosh(c)/a^3-6*b^2*Chi(a*d/b+d*x)*cosh(-c+a*d/b)/a^5-1/2*d^2*Chi(a*
d/b+d*x)*cosh(-c+a*d/b)/a^3-1/2*cosh(d*x+c)/a^3/x^2+3*b*cosh(d*x+c)/a^4/x+1/2*b^2*cosh(d*x+c)/a^3/(b*x+a)^2+3*
b^2*cosh(d*x+c)/a^4/(b*x+a)-3*b*d*cosh(c)*Shi(d*x)/a^4-3*b*d*cosh(-c+a*d/b)*Shi(a*d/b+d*x)/a^4-3*b*d*Chi(d*x)*
sinh(c)/a^4+6*b^2*Shi(d*x)*sinh(c)/a^5+1/2*d^2*Shi(d*x)*sinh(c)/a^3+3*b*d*Chi(a*d/b+d*x)*sinh(-c+a*d/b)/a^4+6*
b^2*Shi(a*d/b+d*x)*sinh(-c+a*d/b)/a^5+1/2*d^2*Shi(a*d/b+d*x)*sinh(-c+a*d/b)/a^3-1/2*d*sinh(d*x+c)/a^3/x+1/2*b*
d*sinh(d*x+c)/a^3/(b*x+a)

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6874, 3378, 3384, 3379, 3382} \[ \int \frac {\cosh (c+d x)}{x^3 (a+b x)^3} \, dx=\frac {6 b^2 \cosh (c) \text {Chi}(d x)}{a^5}-\frac {6 b^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{a^5}+\frac {6 b^2 \sinh (c) \text {Shi}(d x)}{a^5}-\frac {6 b^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{a^5}+\frac {3 b^2 \cosh (c+d x)}{a^4 (a+b x)}-\frac {3 b d \sinh (c) \text {Chi}(d x)}{a^4}-\frac {3 b d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{a^4}-\frac {3 b d \cosh (c) \text {Shi}(d x)}{a^4}-\frac {3 b d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{a^4}+\frac {3 b \cosh (c+d x)}{a^4 x}+\frac {b^2 \cosh (c+d x)}{2 a^3 (a+b x)^2}-\frac {d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{2 a^3}-\frac {d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{2 a^3}+\frac {b d \sinh (c+d x)}{2 a^3 (a+b x)}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a^3}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a^3}-\frac {\cosh (c+d x)}{2 a^3 x^2}-\frac {d \sinh (c+d x)}{2 a^3 x} \]

[In]

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

[Out]

-1/2*Cosh[c + d*x]/(a^3*x^2) + (3*b*Cosh[c + d*x])/(a^4*x) + (b^2*Cosh[c + d*x])/(2*a^3*(a + b*x)^2) + (3*b^2*
Cosh[c + d*x])/(a^4*(a + b*x)) + (6*b^2*Cosh[c]*CoshIntegral[d*x])/a^5 + (d^2*Cosh[c]*CoshIntegral[d*x])/(2*a^
3) - (6*b^2*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/a^5 - (d^2*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b +
d*x])/(2*a^3) - (3*b*d*CoshIntegral[d*x]*Sinh[c])/a^4 - (3*b*d*CoshIntegral[(a*d)/b + d*x]*Sinh[c - (a*d)/b])
/a^4 - (d*Sinh[c + d*x])/(2*a^3*x) + (b*d*Sinh[c + d*x])/(2*a^3*(a + b*x)) - (3*b*d*Cosh[c]*SinhIntegral[d*x])
/a^4 + (6*b^2*Sinh[c]*SinhIntegral[d*x])/a^5 + (d^2*Sinh[c]*SinhIntegral[d*x])/(2*a^3) - (3*b*d*Cosh[c - (a*d)
/b]*SinhIntegral[(a*d)/b + d*x])/a^4 - (6*b^2*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/a^5 - (d^2*Sinh[c
- (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/(2*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 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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^2}+\frac {6 b^2 \cosh (c+d x)}{a^5 x}-\frac {b^3 \cosh (c+d x)}{a^3 (a+b x)^3}-\frac {3 b^3 \cosh (c+d x)}{a^4 (a+b x)^2}-\frac {6 b^3 \cosh (c+d x)}{a^5 (a+b x)}\right ) \, dx \\ & = \frac {\int \frac {\cosh (c+d x)}{x^3} \, dx}{a^3}-\frac {(3 b) \int \frac {\cosh (c+d x)}{x^2} \, dx}{a^4}+\frac {\left (6 b^2\right ) \int \frac {\cosh (c+d x)}{x} \, dx}{a^5}-\frac {\left (6 b^3\right ) \int \frac {\cosh (c+d x)}{a+b x} \, dx}{a^5}-\frac {\left (3 b^3\right ) \int \frac {\cosh (c+d x)}{(a+b x)^2} \, dx}{a^4}-\frac {b^3 \int \frac {\cosh (c+d x)}{(a+b x)^3} \, dx}{a^3} \\ & = -\frac {\cosh (c+d x)}{2 a^3 x^2}+\frac {3 b \cosh (c+d x)}{a^4 x}+\frac {b^2 \cosh (c+d x)}{2 a^3 (a+b x)^2}+\frac {3 b^2 \cosh (c+d x)}{a^4 (a+b x)}+\frac {d \int \frac {\sinh (c+d x)}{x^2} \, dx}{2 a^3}-\frac {(3 b d) \int \frac {\sinh (c+d x)}{x} \, dx}{a^4}-\frac {\left (3 b^2 d\right ) \int \frac {\sinh (c+d x)}{a+b x} \, dx}{a^4}-\frac {\left (b^2 d\right ) \int \frac {\sinh (c+d x)}{(a+b x)^2} \, dx}{2 a^3}+\frac {\left (6 b^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx}{a^5}-\frac {\left (6 b^3 \cosh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cosh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{a^5}+\frac {\left (6 b^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx}{a^5}-\frac {\left (6 b^3 \sinh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sinh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{a^5} \\ & = -\frac {\cosh (c+d x)}{2 a^3 x^2}+\frac {3 b \cosh (c+d x)}{a^4 x}+\frac {b^2 \cosh (c+d x)}{2 a^3 (a+b x)^2}+\frac {3 b^2 \cosh (c+d x)}{a^4 (a+b x)}+\frac {6 b^2 \cosh (c) \text {Chi}(d x)}{a^5}-\frac {6 b^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{a^5}-\frac {d \sinh (c+d x)}{2 a^3 x}+\frac {b d \sinh (c+d x)}{2 a^3 (a+b x)}+\frac {6 b^2 \sinh (c) \text {Shi}(d x)}{a^5}-\frac {6 b^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^5}+\frac {d^2 \int \frac {\cosh (c+d x)}{x} \, dx}{2 a^3}-\frac {\left (b d^2\right ) \int \frac {\cosh (c+d x)}{a+b x} \, dx}{2 a^3}-\frac {(3 b d \cosh (c)) \int \frac {\sinh (d x)}{x} \, dx}{a^4}-\frac {\left (3 b^2 d \cosh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sinh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{a^4}-\frac {(3 b d \sinh (c)) \int \frac {\cosh (d x)}{x} \, dx}{a^4}-\frac {\left (3 b^2 d \sinh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cosh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{a^4} \\ & = -\frac {\cosh (c+d x)}{2 a^3 x^2}+\frac {3 b \cosh (c+d x)}{a^4 x}+\frac {b^2 \cosh (c+d x)}{2 a^3 (a+b x)^2}+\frac {3 b^2 \cosh (c+d x)}{a^4 (a+b x)}+\frac {6 b^2 \cosh (c) \text {Chi}(d x)}{a^5}-\frac {6 b^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{a^5}-\frac {3 b d \text {Chi}(d x) \sinh (c)}{a^4}-\frac {3 b d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{a^4}-\frac {d \sinh (c+d x)}{2 a^3 x}+\frac {b d \sinh (c+d x)}{2 a^3 (a+b x)}-\frac {3 b d \cosh (c) \text {Shi}(d x)}{a^4}+\frac {6 b^2 \sinh (c) \text {Shi}(d x)}{a^5}-\frac {3 b d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^4}-\frac {6 b^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^5}+\frac {\left (d^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx}{2 a^3}-\frac {\left (b d^2 \cosh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cosh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{2 a^3}+\frac {\left (d^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx}{2 a^3}-\frac {\left (b d^2 \sinh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sinh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{2 a^3} \\ & = -\frac {\cosh (c+d x)}{2 a^3 x^2}+\frac {3 b \cosh (c+d x)}{a^4 x}+\frac {b^2 \cosh (c+d x)}{2 a^3 (a+b x)^2}+\frac {3 b^2 \cosh (c+d x)}{a^4 (a+b x)}+\frac {6 b^2 \cosh (c) \text {Chi}(d x)}{a^5}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a^3}-\frac {6 b^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{a^5}-\frac {d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{2 a^3}-\frac {3 b d \text {Chi}(d x) \sinh (c)}{a^4}-\frac {3 b d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{a^4}-\frac {d \sinh (c+d x)}{2 a^3 x}+\frac {b d \sinh (c+d x)}{2 a^3 (a+b x)}-\frac {3 b d \cosh (c) \text {Shi}(d x)}{a^4}+\frac {6 b^2 \sinh (c) \text {Shi}(d x)}{a^5}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a^3}-\frac {3 b d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^4}-\frac {6 b^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^5}-\frac {d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{2 a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.15 (sec) , antiderivative size = 627, normalized size of antiderivative = 1.66 \[ \int \frac {\cosh (c+d x)}{x^3 (a+b x)^3} \, dx=-\frac {a^4 \cosh (c+d x)-4 a^3 b x \cosh (c+d x)-18 a^2 b^2 x^2 \cosh (c+d x)-12 a b^3 x^3 \cosh (c+d x)-x^2 (a+b x)^2 \text {Chi}(d x) \left (\left (12 b^2+a^2 d^2\right ) \cosh (c)-6 a b d \sinh (c)\right )+x^2 (a+b x)^2 \text {Chi}\left (d \left (\frac {a}{b}+x\right )\right ) \left (\left (12 b^2+a^2 d^2\right ) \cosh \left (c-\frac {a d}{b}\right )+6 a b d \sinh \left (c-\frac {a d}{b}\right )\right )+a^4 d x \sinh (c+d x)+a^3 b d x^2 \sinh (c+d x)+6 a^3 b d x^2 \cosh (c) \text {Shi}(d x)+12 a^2 b^2 d x^3 \cosh (c) \text {Shi}(d x)+6 a b^3 d x^4 \cosh (c) \text {Shi}(d x)-12 a^2 b^2 x^2 \sinh (c) \text {Shi}(d x)-a^4 d^2 x^2 \sinh (c) \text {Shi}(d x)-24 a b^3 x^3 \sinh (c) \text {Shi}(d x)-2 a^3 b d^2 x^3 \sinh (c) \text {Shi}(d x)-12 b^4 x^4 \sinh (c) \text {Shi}(d x)-a^2 b^2 d^2 x^4 \sinh (c) \text {Shi}(d x)+6 a^3 b d x^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+12 a^2 b^2 d x^3 \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+6 a b^3 d x^4 \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+12 a^2 b^2 x^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+a^4 d^2 x^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+24 a b^3 x^3 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+2 a^3 b d^2 x^3 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+12 b^4 x^4 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+a^2 b^2 d^2 x^4 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )}{2 a^5 x^2 (a+b x)^2} \]

[In]

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

[Out]

-1/2*(a^4*Cosh[c + d*x] - 4*a^3*b*x*Cosh[c + d*x] - 18*a^2*b^2*x^2*Cosh[c + d*x] - 12*a*b^3*x^3*Cosh[c + d*x]
- x^2*(a + b*x)^2*CoshIntegral[d*x]*((12*b^2 + a^2*d^2)*Cosh[c] - 6*a*b*d*Sinh[c]) + x^2*(a + b*x)^2*CoshInteg
ral[d*(a/b + x)]*((12*b^2 + a^2*d^2)*Cosh[c - (a*d)/b] + 6*a*b*d*Sinh[c - (a*d)/b]) + a^4*d*x*Sinh[c + d*x] +
a^3*b*d*x^2*Sinh[c + d*x] + 6*a^3*b*d*x^2*Cosh[c]*SinhIntegral[d*x] + 12*a^2*b^2*d*x^3*Cosh[c]*SinhIntegral[d*
x] + 6*a*b^3*d*x^4*Cosh[c]*SinhIntegral[d*x] - 12*a^2*b^2*x^2*Sinh[c]*SinhIntegral[d*x] - a^4*d^2*x^2*Sinh[c]*
SinhIntegral[d*x] - 24*a*b^3*x^3*Sinh[c]*SinhIntegral[d*x] - 2*a^3*b*d^2*x^3*Sinh[c]*SinhIntegral[d*x] - 12*b^
4*x^4*Sinh[c]*SinhIntegral[d*x] - a^2*b^2*d^2*x^4*Sinh[c]*SinhIntegral[d*x] + 6*a^3*b*d*x^2*Cosh[c - (a*d)/b]*
SinhIntegral[d*(a/b + x)] + 12*a^2*b^2*d*x^3*Cosh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)] + 6*a*b^3*d*x^4*Cosh[
c - (a*d)/b]*SinhIntegral[d*(a/b + x)] + 12*a^2*b^2*x^2*Sinh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)] + a^4*d^2*
x^2*Sinh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)] + 24*a*b^3*x^3*Sinh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)] + 2
*a^3*b*d^2*x^3*Sinh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)] + 12*b^4*x^4*Sinh[c - (a*d)/b]*SinhIntegral[d*(a/b
+ x)] + a^2*b^2*d^2*x^4*Sinh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)])/(a^5*x^2*(a + b*x)^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(759\) vs. \(2(367)=734\).

Time = 0.35 (sec) , antiderivative size = 760, normalized size of antiderivative = 2.02

method	result	size
risch	\(\frac {{\mathrm e}^{-d x -c} d^{3} b}{4 a^{2} \left (x^{2} d^{2} b^{2}+2 a b \,d^{2} x +a^{2} d^{2}\right )}+\frac {3 d^{2} {\mathrm e}^{-d x -c} x \,b^{3}}{a^{4} \left (x^{2} d^{2} b^{2}+2 a b \,d^{2} x +a^{2} d^{2}\right )}+\frac {d^{3} {\mathrm e}^{-d x -c}}{4 a x \left (x^{2} d^{2} b^{2}+2 a b \,d^{2} x +a^{2} d^{2}\right )}+\frac {9 \,{\mathrm e}^{-d x -c} d^{2} b^{2}}{2 a^{3} \left (x^{2} d^{2} b^{2}+2 a b \,d^{2} x +a^{2} d^{2}\right )}+\frac {d^{2} {\mathrm e}^{-d x -c} b}{a^{2} x \left (x^{2} d^{2} b^{2}+2 a b \,d^{2} x +a^{2} d^{2}\right )}-\frac {{\mathrm e}^{-d x -c} d^{2}}{4 a \,x^{2} \left (x^{2} d^{2} b^{2}+2 a b \,d^{2} x +a^{2} d^{2}\right )}-\frac {d^{2} {\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right )}{4 a^{3}}-\frac {3 d \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) b}{2 a^{4}}-\frac {3 \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) b^{2}}{a^{5}}+\frac {d^{2} {\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right )}{4 a^{3}}-\frac {3 d \,{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right ) b}{2 a^{4}}+\frac {3 \,{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right ) b^{2}}{a^{5}}+\frac {3 b \,{\mathrm e}^{d x +c}}{2 a^{4} x}+\frac {3 d b \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right )}{2 a^{4}}-\frac {3 b^{2} {\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right )}{a^{5}}+\frac {d^{2} {\mathrm e}^{d x +c}}{4 a^{3} \left (\frac {d a}{b}+d x \right )^{2}}+\frac {d^{2} {\mathrm e}^{d x +c}}{4 a^{3} \left (\frac {d a}{b}+d x \right )}+\frac {d^{2} {\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right )}{4 a^{3}}+\frac {3 d b \,{\mathrm e}^{d x +c}}{2 a^{4} \left (\frac {d a}{b}+d x \right )}+\frac {3 d b \,{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right )}{2 a^{4}}+\frac {3 b^{2} {\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right )}{a^{5}}-\frac {{\mathrm e}^{d x +c}}{4 a^{3} x^{2}}-\frac {d \,{\mathrm e}^{d x +c}}{4 a^{3} x}-\frac {d^{2} {\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right )}{4 a^{3}}\)	\(760\)

[In]

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

[Out]

1/4*exp(-d*x-c)/a^2*d^3/(b^2*d^2*x^2+2*a*b*d^2*x+a^2*d^2)*b+3*d^2*exp(-d*x-c)/a^4*x/(b^2*d^2*x^2+2*a*b*d^2*x+a
^2*d^2)*b^3+1/4*d^3*exp(-d*x-c)/a/x/(b^2*d^2*x^2+2*a*b*d^2*x+a^2*d^2)+9/2*exp(-d*x-c)/a^3*d^2/(b^2*d^2*x^2+2*a
*b*d^2*x+a^2*d^2)*b^2+d^2*exp(-d*x-c)/a^2/x/(b^2*d^2*x^2+2*a*b*d^2*x+a^2*d^2)*b-1/4*exp(-d*x-c)/a/x^2*d^2/(b^2
*d^2*x^2+2*a*b*d^2*x+a^2*d^2)-1/4*d^2/a^3*exp(-c)*Ei(1,d*x)-3/2*d/a^4*exp(-c)*Ei(1,d*x)*b-3/a^5*exp(-c)*Ei(1,d
*x)*b^2+1/4*d^2/a^3*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)-3/2*d/a^4*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b
)*b+3/a^5*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*b^2+3/2/a^4*b/x*exp(d*x+c)+3/2*d/a^4*b*exp(c)*Ei(1,-d*x)-3/
a^5*b^2*exp(c)*Ei(1,-d*x)+1/4*d^2/a^3*exp(d*x+c)/(d/b*a+d*x)^2+1/4*d^2/a^3*exp(d*x+c)/(d/b*a+d*x)+1/4*d^2/a^3*
exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)+3/2*d/a^4*b*exp(d*x+c)/(d/b*a+d*x)+3/2*d/a^4*b*exp(-(a*d-b*c)/b)*Ei
(1,-d*x-c-(a*d-b*c)/b)+3*b^2/a^5*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)-1/4/a^3/x^2*exp(d*x+c)-1/4/a^3/x*d
*exp(d*x+c)-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. 892 vs. \(2 (367) = 734\).

Time = 0.27 (sec) , antiderivative size = 892, normalized size of antiderivative = 2.37 \[ \int \frac {\cosh (c+d x)}{x^3 (a+b x)^3} \, dx=\frac {2 \, {\left (12 \, a b^{3} x^{3} + 18 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x - a^{4}\right )} \cosh \left (d x + c\right ) + {\left ({\left ({\left (a^{2} b^{2} d^{2} - 6 \, a b^{3} d + 12 \, b^{4}\right )} x^{4} + 2 \, {\left (a^{3} b d^{2} - 6 \, a^{2} b^{2} d + 12 \, a b^{3}\right )} x^{3} + {\left (a^{4} d^{2} - 6 \, a^{3} b d + 12 \, a^{2} b^{2}\right )} x^{2}\right )} {\rm Ei}\left (d x\right ) + {\left ({\left (a^{2} b^{2} d^{2} + 6 \, a b^{3} d + 12 \, b^{4}\right )} x^{4} + 2 \, {\left (a^{3} b d^{2} + 6 \, a^{2} b^{2} d + 12 \, a b^{3}\right )} x^{3} + {\left (a^{4} d^{2} + 6 \, a^{3} b d + 12 \, a^{2} b^{2}\right )} x^{2}\right )} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) - {\left ({\left ({\left (a^{2} b^{2} d^{2} + 6 \, a b^{3} d + 12 \, b^{4}\right )} x^{4} + 2 \, {\left (a^{3} b d^{2} + 6 \, a^{2} b^{2} d + 12 \, a b^{3}\right )} x^{3} + {\left (a^{4} d^{2} + 6 \, a^{3} b d + 12 \, a^{2} b^{2}\right )} x^{2}\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\left ({\left (a^{2} b^{2} d^{2} - 6 \, a b^{3} d + 12 \, b^{4}\right )} x^{4} + 2 \, {\left (a^{3} b d^{2} - 6 \, a^{2} b^{2} d + 12 \, a b^{3}\right )} x^{3} + {\left (a^{4} d^{2} - 6 \, a^{3} b d + 12 \, a^{2} b^{2}\right )} x^{2}\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) - 2 \, {\left (a^{3} b d x^{2} + a^{4} d x\right )} \sinh \left (d x + c\right ) + {\left ({\left ({\left (a^{2} b^{2} d^{2} - 6 \, a b^{3} d + 12 \, b^{4}\right )} x^{4} + 2 \, {\left (a^{3} b d^{2} - 6 \, a^{2} b^{2} d + 12 \, a b^{3}\right )} x^{3} + {\left (a^{4} d^{2} - 6 \, a^{3} b d + 12 \, a^{2} b^{2}\right )} x^{2}\right )} {\rm Ei}\left (d x\right ) - {\left ({\left (a^{2} b^{2} d^{2} + 6 \, a b^{3} d + 12 \, b^{4}\right )} x^{4} + 2 \, {\left (a^{3} b d^{2} + 6 \, a^{2} b^{2} d + 12 \, a b^{3}\right )} x^{3} + {\left (a^{4} d^{2} + 6 \, a^{3} b d + 12 \, a^{2} b^{2}\right )} x^{2}\right )} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right ) + {\left ({\left ({\left (a^{2} b^{2} d^{2} + 6 \, a b^{3} d + 12 \, b^{4}\right )} x^{4} + 2 \, {\left (a^{3} b d^{2} + 6 \, a^{2} b^{2} d + 12 \, a b^{3}\right )} x^{3} + {\left (a^{4} d^{2} + 6 \, a^{3} b d + 12 \, a^{2} b^{2}\right )} x^{2}\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\left ({\left (a^{2} b^{2} d^{2} - 6 \, a b^{3} d + 12 \, b^{4}\right )} x^{4} + 2 \, {\left (a^{3} b d^{2} - 6 \, a^{2} b^{2} d + 12 \, a b^{3}\right )} x^{3} + {\left (a^{4} d^{2} - 6 \, a^{3} b d + 12 \, a^{2} b^{2}\right )} x^{2}\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{4 \, {\left (a^{5} b^{2} x^{4} + 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}} \]

[In]

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

[Out]

1/4*(2*(12*a*b^3*x^3 + 18*a^2*b^2*x^2 + 4*a^3*b*x - a^4)*cosh(d*x + c) + (((a^2*b^2*d^2 - 6*a*b^3*d + 12*b^4)*
x^4 + 2*(a^3*b*d^2 - 6*a^2*b^2*d + 12*a*b^3)*x^3 + (a^4*d^2 - 6*a^3*b*d + 12*a^2*b^2)*x^2)*Ei(d*x) + ((a^2*b^2
*d^2 + 6*a*b^3*d + 12*b^4)*x^4 + 2*(a^3*b*d^2 + 6*a^2*b^2*d + 12*a*b^3)*x^3 + (a^4*d^2 + 6*a^3*b*d + 12*a^2*b^
2)*x^2)*Ei(-d*x))*cosh(c) - (((a^2*b^2*d^2 + 6*a*b^3*d + 12*b^4)*x^4 + 2*(a^3*b*d^2 + 6*a^2*b^2*d + 12*a*b^3)*
x^3 + (a^4*d^2 + 6*a^3*b*d + 12*a^2*b^2)*x^2)*Ei((b*d*x + a*d)/b) + ((a^2*b^2*d^2 - 6*a*b^3*d + 12*b^4)*x^4 +
2*(a^3*b*d^2 - 6*a^2*b^2*d + 12*a*b^3)*x^3 + (a^4*d^2 - 6*a^3*b*d + 12*a^2*b^2)*x^2)*Ei(-(b*d*x + a*d)/b))*cos
h(-(b*c - a*d)/b) - 2*(a^3*b*d*x^2 + a^4*d*x)*sinh(d*x + c) + (((a^2*b^2*d^2 - 6*a*b^3*d + 12*b^4)*x^4 + 2*(a^
3*b*d^2 - 6*a^2*b^2*d + 12*a*b^3)*x^3 + (a^4*d^2 - 6*a^3*b*d + 12*a^2*b^2)*x^2)*Ei(d*x) - ((a^2*b^2*d^2 + 6*a*
b^3*d + 12*b^4)*x^4 + 2*(a^3*b*d^2 + 6*a^2*b^2*d + 12*a*b^3)*x^3 + (a^4*d^2 + 6*a^3*b*d + 12*a^2*b^2)*x^2)*Ei(
-d*x))*sinh(c) + (((a^2*b^2*d^2 + 6*a*b^3*d + 12*b^4)*x^4 + 2*(a^3*b*d^2 + 6*a^2*b^2*d + 12*a*b^3)*x^3 + (a^4*
d^2 + 6*a^3*b*d + 12*a^2*b^2)*x^2)*Ei((b*d*x + a*d)/b) - ((a^2*b^2*d^2 - 6*a*b^3*d + 12*b^4)*x^4 + 2*(a^3*b*d^
2 - 6*a^2*b^2*d + 12*a*b^3)*x^3 + (a^4*d^2 - 6*a^3*b*d + 12*a^2*b^2)*x^2)*Ei(-(b*d*x + a*d)/b))*sinh(-(b*c - a
*d)/b))/(a^5*b^2*x^4 + 2*a^6*b*x^3 + a^7*x^2)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1169 vs. \(2 (367) = 734\).

Time = 0.28 (sec) , antiderivative size = 1169, normalized size of antiderivative = 3.10 \[ \int \frac {\cosh (c+d x)}{x^3 (a+b x)^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/4*(a^2*b^2*d^2*x^4*Ei(-d*x)*e^(-c) - a^2*b^2*d^2*x^4*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + a^2*b^2*d^2*x^4*Ei(
d*x)*e^c - a^2*b^2*d^2*x^4*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + 2*a^3*b*d^2*x^3*Ei(-d*x)*e^(-c) + 6*a*b^3*d*x
^4*Ei(-d*x)*e^(-c) - 2*a^3*b*d^2*x^3*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) - 6*a*b^3*d*x^4*Ei((b*d*x + a*d)/b)*e^(
c - a*d/b) + 2*a^3*b*d^2*x^3*Ei(d*x)*e^c - 6*a*b^3*d*x^4*Ei(d*x)*e^c - 2*a^3*b*d^2*x^3*Ei(-(b*d*x + a*d)/b)*e^
(-c + a*d/b) + 6*a*b^3*d*x^4*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + a^4*d^2*x^2*Ei(-d*x)*e^(-c) + 12*a^2*b^2*d*
x^3*Ei(-d*x)*e^(-c) + 12*b^4*x^4*Ei(-d*x)*e^(-c) - a^4*d^2*x^2*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) - 12*a^2*b^2*
d*x^3*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) - 12*b^4*x^4*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + a^4*d^2*x^2*Ei(d*x)*e
^c - 12*a^2*b^2*d*x^3*Ei(d*x)*e^c + 12*b^4*x^4*Ei(d*x)*e^c - a^4*d^2*x^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) +
12*a^2*b^2*d*x^3*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) - 12*b^4*x^4*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + 6*a^3
*b*d*x^2*Ei(-d*x)*e^(-c) + 24*a*b^3*x^3*Ei(-d*x)*e^(-c) - 6*a^3*b*d*x^2*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) - 24
*a*b^3*x^3*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) - 6*a^3*b*d*x^2*Ei(d*x)*e^c + 24*a*b^3*x^3*Ei(d*x)*e^c + 6*a^3*b*
d*x^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) - 24*a*b^3*x^3*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) - a^3*b*d*x^2*e^(
d*x + c) + 12*a*b^3*x^3*e^(d*x + c) + a^3*b*d*x^2*e^(-d*x - c) + 12*a*b^3*x^3*e^(-d*x - c) + 12*a^2*b^2*x^2*Ei
(-d*x)*e^(-c) - 12*a^2*b^2*x^2*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 12*a^2*b^2*x^2*Ei(d*x)*e^c - 12*a^2*b^2*x^2
*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) - a^4*d*x*e^(d*x + c) + 18*a^2*b^2*x^2*e^(d*x + c) + a^4*d*x*e^(-d*x - c)
+ 18*a^2*b^2*x^2*e^(-d*x - c) + 4*a^3*b*x*e^(d*x + c) + 4*a^3*b*x*e^(-d*x - c) - a^4*e^(d*x + c) - a^4*e^(-d*
x - c))/(a^5*b^2*x^4 + 2*a^6*b*x^3 + a^7*x^2)

Mupad [F(-1)]

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

[In]

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

[Out]

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