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

Optimal. Leaf size=377 \[ \frac{6 b^2 \cosh (c) \text{Chi}(d x)}{a^5}-\frac{6 b^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a^5}+\frac{6 b^2 \sinh (c) \text{Shi}(d x)}{a^5}-\frac{6 b^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a^5}+\frac{3 b^2 \cosh (c+d x)}{a^4 (a+b x)}+\frac{b^2 \cosh (c+d x)}{2 a^3 (a+b x)^2}-\frac{d^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{2 a^3}-\frac{3 b d \sinh (c) \text{Chi}(d x)}{a^4}-\frac{3 b d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a^4}-\frac{d^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{2 a^3}-\frac{3 b d \cosh (c) \text{Shi}(d x)}{a^4}-\frac{3 b d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a^4}+\frac{b d \sinh (c+d x)}{2 a^3 (a+b x)}+\frac{3 b \cosh (c+d x)}{a^4 x}+\frac{d^2 \cosh (c) \text{Chi}(d x)}{2 a^3}+\frac{d^2 \sinh (c) \text{Shi}(d x)}{2 a^3}-\frac{\cosh (c+d x)}{2 a^3 x^2}-\frac{d \sinh (c+d x)}{2 a^3 x} \]

[Out]

-Cosh[c + d*x]/(2*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*Co
sh[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)

________________________________________________________________________________________

Rubi [A]  time = 0.821274, antiderivative size = 377, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {6742, 3297, 3303, 3298, 3301} \[ \frac{6 b^2 \cosh (c) \text{Chi}(d x)}{a^5}-\frac{6 b^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a^5}+\frac{6 b^2 \sinh (c) \text{Shi}(d x)}{a^5}-\frac{6 b^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a^5}+\frac{3 b^2 \cosh (c+d x)}{a^4 (a+b x)}+\frac{b^2 \cosh (c+d x)}{2 a^3 (a+b x)^2}-\frac{d^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{2 a^3}-\frac{3 b d \sinh (c) \text{Chi}(d x)}{a^4}-\frac{3 b d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a^4}-\frac{d^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{2 a^3}-\frac{3 b d \cosh (c) \text{Shi}(d x)}{a^4}-\frac{3 b d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a^4}+\frac{b d \sinh (c+d x)}{2 a^3 (a+b x)}+\frac{3 b \cosh (c+d x)}{a^4 x}+\frac{d^2 \cosh (c) \text{Chi}(d x)}{2 a^3}+\frac{d^2 \sinh (c) \text{Shi}(d x)}{2 a^3}-\frac{\cosh (c+d x)}{2 a^3 x^2}-\frac{d \sinh (c+d x)}{2 a^3 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cosh[c + d*x]/(2*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*Co
sh[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 6742

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

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 (c+d x)}{x^3 (a+b x)^3} \, dx &=\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]  time = 1.80699, size = 627, normalized size = 1.66 \[ -\frac{-x^2 (a+b x)^2 \text{Chi}(d x) \left (\cosh (c) \left (a^2 d^2+12 b^2\right )-6 a b d \sinh (c)\right )+x^2 (a+b x)^2 \text{Chi}\left (d \left (\frac{a}{b}+x\right )\right ) \left (\left (a^2 d^2+12 b^2\right ) \cosh \left (c-\frac{a d}{b}\right )+6 a b d \sinh \left (c-\frac{a d}{b}\right )\right )-a^2 b^2 d^2 x^4 \sinh (c) \text{Shi}(d x)+a^2 b^2 d^2 x^4 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )-12 a^2 b^2 x^2 \sinh (c) \text{Shi}(d x)+12 a^2 b^2 x^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )+12 a^2 b^2 d x^3 \cosh (c) \text{Shi}(d x)+12 a^2 b^2 d x^3 \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )-18 a^2 b^2 x^2 \cosh (c+d x)+a^4 d^2 x^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )-2 a^3 b d^2 x^3 \sinh (c) \text{Shi}(d x)+2 a^3 b d^2 x^3 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )+6 a^3 b d x^2 \cosh (c) \text{Shi}(d x)+6 a^3 b d x^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )+a^3 b d x^2 \sinh (c+d x)-4 a^3 b x \cosh (c+d x)-a^4 d^2 x^2 \sinh (c) \text{Shi}(d x)+a^4 d x \sinh (c+d x)+a^4 \cosh (c+d x)-24 a b^3 x^3 \sinh (c) \text{Shi}(d x)+24 a b^3 x^3 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )+12 b^4 x^4 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )+6 a b^3 d x^4 \cosh (c) \text{Shi}(d x)+6 a b^3 d x^4 \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )-12 a b^3 x^3 \cosh (c+d x)-12 b^4 x^4 \sinh (c) \text{Shi}(d x)}{2 a^5 x^2 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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*CoshIntegral[
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]*Sinh
Integral[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]*Sinh
Integral[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)])/(2*a^5*x^2*(a + b*x)^2)

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Maple [B]  time = 0.082, size = 760, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 2.18189, size = 1845, normalized size = 4.89 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.31946, size = 1578, normalized size = 4.19 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)