3.17 \(\int \frac{(a+b x)^2 \cosh (c+d x)}{x^5} \, dx\)

Optimal. Leaf size=248 \[ \frac{1}{24} a^2 d^4 \cosh (c) \text{Chi}(d x)+\frac{1}{24} a^2 d^4 \sinh (c) \text{Shi}(d x)-\frac{a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac{a^2 d^3 \sinh (c+d x)}{24 x}-\frac{a^2 d \sinh (c+d x)}{12 x^3}-\frac{a^2 \cosh (c+d x)}{4 x^4}+\frac{1}{3} a b d^3 \sinh (c) \text{Chi}(d x)+\frac{1}{3} a b d^3 \cosh (c) \text{Shi}(d x)-\frac{a b d^2 \cosh (c+d x)}{3 x}-\frac{a b d \sinh (c+d x)}{3 x^2}-\frac{2 a b \cosh (c+d x)}{3 x^3}+\frac{1}{2} b^2 d^2 \cosh (c) \text{Chi}(d x)+\frac{1}{2} b^2 d^2 \sinh (c) \text{Shi}(d x)-\frac{b^2 \cosh (c+d x)}{2 x^2}-\frac{b^2 d \sinh (c+d x)}{2 x} \]

[Out]

-(a^2*Cosh[c + d*x])/(4*x^4) - (2*a*b*Cosh[c + d*x])/(3*x^3) - (b^2*Cosh[c + d*x])/(2*x^2) - (a^2*d^2*Cosh[c +
d*x])/(24*x^2) - (a*b*d^2*Cosh[c + d*x])/(3*x) + (b^2*d^2*Cosh[c]*CoshIntegral[d*x])/2 + (a^2*d^4*Cosh[c]*Cos
hIntegral[d*x])/24 + (a*b*d^3*CoshIntegral[d*x]*Sinh[c])/3 - (a^2*d*Sinh[c + d*x])/(12*x^3) - (a*b*d*Sinh[c +
d*x])/(3*x^2) - (b^2*d*Sinh[c + d*x])/(2*x) - (a^2*d^3*Sinh[c + d*x])/(24*x) + (a*b*d^3*Cosh[c]*SinhIntegral[d
*x])/3 + (b^2*d^2*Sinh[c]*SinhIntegral[d*x])/2 + (a^2*d^4*Sinh[c]*SinhIntegral[d*x])/24

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Rubi [A]  time = 0.520699, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 20, 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} \[ \frac{1}{24} a^2 d^4 \cosh (c) \text{Chi}(d x)+\frac{1}{24} a^2 d^4 \sinh (c) \text{Shi}(d x)-\frac{a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac{a^2 d^3 \sinh (c+d x)}{24 x}-\frac{a^2 d \sinh (c+d x)}{12 x^3}-\frac{a^2 \cosh (c+d x)}{4 x^4}+\frac{1}{3} a b d^3 \sinh (c) \text{Chi}(d x)+\frac{1}{3} a b d^3 \cosh (c) \text{Shi}(d x)-\frac{a b d^2 \cosh (c+d x)}{3 x}-\frac{a b d \sinh (c+d x)}{3 x^2}-\frac{2 a b \cosh (c+d x)}{3 x^3}+\frac{1}{2} b^2 d^2 \cosh (c) \text{Chi}(d x)+\frac{1}{2} b^2 d^2 \sinh (c) \text{Shi}(d x)-\frac{b^2 \cosh (c+d x)}{2 x^2}-\frac{b^2 d \sinh (c+d x)}{2 x} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^2*Cosh[c + d*x])/x^5,x]

[Out]

-(a^2*Cosh[c + d*x])/(4*x^4) - (2*a*b*Cosh[c + d*x])/(3*x^3) - (b^2*Cosh[c + d*x])/(2*x^2) - (a^2*d^2*Cosh[c +
d*x])/(24*x^2) - (a*b*d^2*Cosh[c + d*x])/(3*x) + (b^2*d^2*Cosh[c]*CoshIntegral[d*x])/2 + (a^2*d^4*Cosh[c]*Cos
hIntegral[d*x])/24 + (a*b*d^3*CoshIntegral[d*x]*Sinh[c])/3 - (a^2*d*Sinh[c + d*x])/(12*x^3) - (a*b*d*Sinh[c +
d*x])/(3*x^2) - (b^2*d*Sinh[c + d*x])/(2*x) - (a^2*d^3*Sinh[c + d*x])/(24*x) + (a*b*d^3*Cosh[c]*SinhIntegral[d
*x])/3 + (b^2*d^2*Sinh[c]*SinhIntegral[d*x])/2 + (a^2*d^4*Sinh[c]*SinhIntegral[d*x])/24

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{(a+b x)^2 \cosh (c+d x)}{x^5} \, dx &=\int \left (\frac{a^2 \cosh (c+d x)}{x^5}+\frac{2 a b \cosh (c+d x)}{x^4}+\frac{b^2 \cosh (c+d x)}{x^3}\right ) \, dx\\ &=a^2 \int \frac{\cosh (c+d x)}{x^5} \, dx+(2 a b) \int \frac{\cosh (c+d x)}{x^4} \, dx+b^2 \int \frac{\cosh (c+d x)}{x^3} \, dx\\ &=-\frac{a^2 \cosh (c+d x)}{4 x^4}-\frac{2 a b \cosh (c+d x)}{3 x^3}-\frac{b^2 \cosh (c+d x)}{2 x^2}+\frac{1}{4} \left (a^2 d\right ) \int \frac{\sinh (c+d x)}{x^4} \, dx+\frac{1}{3} (2 a b d) \int \frac{\sinh (c+d x)}{x^3} \, dx+\frac{1}{2} \left (b^2 d\right ) \int \frac{\sinh (c+d x)}{x^2} \, dx\\ &=-\frac{a^2 \cosh (c+d x)}{4 x^4}-\frac{2 a b \cosh (c+d x)}{3 x^3}-\frac{b^2 \cosh (c+d x)}{2 x^2}-\frac{a^2 d \sinh (c+d x)}{12 x^3}-\frac{a b d \sinh (c+d x)}{3 x^2}-\frac{b^2 d \sinh (c+d x)}{2 x}+\frac{1}{12} \left (a^2 d^2\right ) \int \frac{\cosh (c+d x)}{x^3} \, dx+\frac{1}{3} \left (a b d^2\right ) \int \frac{\cosh (c+d x)}{x^2} \, dx+\frac{1}{2} \left (b^2 d^2\right ) \int \frac{\cosh (c+d x)}{x} \, dx\\ &=-\frac{a^2 \cosh (c+d x)}{4 x^4}-\frac{2 a b \cosh (c+d x)}{3 x^3}-\frac{b^2 \cosh (c+d x)}{2 x^2}-\frac{a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac{a b d^2 \cosh (c+d x)}{3 x}-\frac{a^2 d \sinh (c+d x)}{12 x^3}-\frac{a b d \sinh (c+d x)}{3 x^2}-\frac{b^2 d \sinh (c+d x)}{2 x}+\frac{1}{24} \left (a^2 d^3\right ) \int \frac{\sinh (c+d x)}{x^2} \, dx+\frac{1}{3} \left (a b d^3\right ) \int \frac{\sinh (c+d x)}{x} \, dx+\frac{1}{2} \left (b^2 d^2 \cosh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx+\frac{1}{2} \left (b^2 d^2 \sinh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx\\ &=-\frac{a^2 \cosh (c+d x)}{4 x^4}-\frac{2 a b \cosh (c+d x)}{3 x^3}-\frac{b^2 \cosh (c+d x)}{2 x^2}-\frac{a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac{a b d^2 \cosh (c+d x)}{3 x}+\frac{1}{2} b^2 d^2 \cosh (c) \text{Chi}(d x)-\frac{a^2 d \sinh (c+d x)}{12 x^3}-\frac{a b d \sinh (c+d x)}{3 x^2}-\frac{b^2 d \sinh (c+d x)}{2 x}-\frac{a^2 d^3 \sinh (c+d x)}{24 x}+\frac{1}{2} b^2 d^2 \sinh (c) \text{Shi}(d x)+\frac{1}{24} \left (a^2 d^4\right ) \int \frac{\cosh (c+d x)}{x} \, dx+\frac{1}{3} \left (a b d^3 \cosh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx+\frac{1}{3} \left (a b d^3 \sinh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx\\ &=-\frac{a^2 \cosh (c+d x)}{4 x^4}-\frac{2 a b \cosh (c+d x)}{3 x^3}-\frac{b^2 \cosh (c+d x)}{2 x^2}-\frac{a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac{a b d^2 \cosh (c+d x)}{3 x}+\frac{1}{2} b^2 d^2 \cosh (c) \text{Chi}(d x)+\frac{1}{3} a b d^3 \text{Chi}(d x) \sinh (c)-\frac{a^2 d \sinh (c+d x)}{12 x^3}-\frac{a b d \sinh (c+d x)}{3 x^2}-\frac{b^2 d \sinh (c+d x)}{2 x}-\frac{a^2 d^3 \sinh (c+d x)}{24 x}+\frac{1}{3} a b d^3 \cosh (c) \text{Shi}(d x)+\frac{1}{2} b^2 d^2 \sinh (c) \text{Shi}(d x)+\frac{1}{24} \left (a^2 d^4 \cosh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx+\frac{1}{24} \left (a^2 d^4 \sinh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx\\ &=-\frac{a^2 \cosh (c+d x)}{4 x^4}-\frac{2 a b \cosh (c+d x)}{3 x^3}-\frac{b^2 \cosh (c+d x)}{2 x^2}-\frac{a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac{a b d^2 \cosh (c+d x)}{3 x}+\frac{1}{2} b^2 d^2 \cosh (c) \text{Chi}(d x)+\frac{1}{24} a^2 d^4 \cosh (c) \text{Chi}(d x)+\frac{1}{3} a b d^3 \text{Chi}(d x) \sinh (c)-\frac{a^2 d \sinh (c+d x)}{12 x^3}-\frac{a b d \sinh (c+d x)}{3 x^2}-\frac{b^2 d \sinh (c+d x)}{2 x}-\frac{a^2 d^3 \sinh (c+d x)}{24 x}+\frac{1}{3} a b d^3 \cosh (c) \text{Shi}(d x)+\frac{1}{2} b^2 d^2 \sinh (c) \text{Shi}(d x)+\frac{1}{24} a^2 d^4 \sinh (c) \text{Shi}(d x)\\ \end{align*}

Mathematica [A]  time = 0.523918, size = 206, normalized size = 0.83 \[ -\frac{-d^2 x^4 \text{Chi}(d x) \left (\cosh (c) \left (a^2 d^2+12 b^2\right )+8 a b d \sinh (c)\right )-d^2 x^4 \text{Shi}(d x) \left (a^2 d^2 \sinh (c)+8 a b d \cosh (c)+12 b^2 \sinh (c)\right )+a^2 d^3 x^3 \sinh (c+d x)+a^2 d^2 x^2 \cosh (c+d x)+2 a^2 d x \sinh (c+d x)+6 a^2 \cosh (c+d x)+8 a b d^2 x^3 \cosh (c+d x)+8 a b d x^2 \sinh (c+d x)+16 a b x \cosh (c+d x)+12 b^2 d x^3 \sinh (c+d x)+12 b^2 x^2 \cosh (c+d x)}{24 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^2*Cosh[c + d*x])/x^5,x]

[Out]

-(6*a^2*Cosh[c + d*x] + 16*a*b*x*Cosh[c + d*x] + 12*b^2*x^2*Cosh[c + d*x] + a^2*d^2*x^2*Cosh[c + d*x] + 8*a*b*
d^2*x^3*Cosh[c + d*x] - d^2*x^4*CoshIntegral[d*x]*((12*b^2 + a^2*d^2)*Cosh[c] + 8*a*b*d*Sinh[c]) + 2*a^2*d*x*S
inh[c + d*x] + 8*a*b*d*x^2*Sinh[c + d*x] + 12*b^2*d*x^3*Sinh[c + d*x] + a^2*d^3*x^3*Sinh[c + d*x] - d^2*x^4*(8
*a*b*d*Cosh[c] + 12*b^2*Sinh[c] + a^2*d^2*Sinh[c])*SinhIntegral[d*x])/(24*x^4)

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Maple [A]  time = 0.072, size = 396, normalized size = 1.6 \begin{align*} -{\frac{{d}^{4}{a}^{2}{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{48}}+{\frac{d{a}^{2}{{\rm e}^{-dx-c}}}{24\,{x}^{3}}}-{\frac{{a}^{2}{{\rm e}^{-dx-c}}}{8\,{x}^{4}}}+{\frac{{a}^{2}{d}^{3}{{\rm e}^{-dx-c}}}{48\,x}}-{\frac{{a}^{2}{d}^{2}{{\rm e}^{-dx-c}}}{48\,{x}^{2}}}+{\frac{{d}^{3}ab{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{6}}-{\frac{ab{d}^{2}{{\rm e}^{-dx-c}}}{6\,x}}+{\frac{bda{{\rm e}^{-dx-c}}}{6\,{x}^{2}}}-{\frac{ab{{\rm e}^{-dx-c}}}{3\,{x}^{3}}}+{\frac{{b}^{2}d{{\rm e}^{-dx-c}}}{4\,x}}-{\frac{{b}^{2}{{\rm e}^{-dx-c}}}{4\,{x}^{2}}}-{\frac{{d}^{2}{b}^{2}{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{4}}-{\frac{{{\rm e}^{dx+c}}{b}^{2}}{4\,{x}^{2}}}-{\frac{{b}^{2}d{{\rm e}^{dx+c}}}{4\,x}}-{\frac{{d}^{2}{b}^{2}{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{4}}-{\frac{{d}^{4}{a}^{2}{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{48}}-{\frac{{{\rm e}^{dx+c}}{a}^{2}}{8\,{x}^{4}}}-{\frac{d{a}^{2}{{\rm e}^{dx+c}}}{24\,{x}^{3}}}-{\frac{{a}^{2}{d}^{2}{{\rm e}^{dx+c}}}{48\,{x}^{2}}}-{\frac{{a}^{2}{d}^{3}{{\rm e}^{dx+c}}}{48\,x}}-{\frac{ab{d}^{2}{{\rm e}^{dx+c}}}{6\,x}}-{\frac{{d}^{3}ab{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{6}}-{\frac{ab{{\rm e}^{dx+c}}}{3\,{x}^{3}}}-{\frac{bda{{\rm e}^{dx+c}}}{6\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2*cosh(d*x+c)/x^5,x)

[Out]

-1/48*d^4*a^2*exp(-c)*Ei(1,d*x)+1/24*d*a^2*exp(-d*x-c)/x^3-1/8*a^2*exp(-d*x-c)/x^4+1/48*d^3*a^2*exp(-d*x-c)/x-
1/48*d^2*a^2*exp(-d*x-c)/x^2+1/6*d^3*a*b*exp(-c)*Ei(1,d*x)-1/6*d^2*a*b*exp(-d*x-c)/x+1/6*d*a*b*exp(-d*x-c)/x^2
-1/3*a*b*exp(-d*x-c)/x^3+1/4*d*b^2*exp(-d*x-c)/x-1/4*b^2*exp(-d*x-c)/x^2-1/4*d^2*b^2*exp(-c)*Ei(1,d*x)-1/4*b^2
/x^2*exp(d*x+c)-1/4*d*b^2/x*exp(d*x+c)-1/4*d^2*b^2*exp(c)*Ei(1,-d*x)-1/48*d^4*a^2*exp(c)*Ei(1,-d*x)-1/8*a^2/x^
4*exp(d*x+c)-1/24*d*a^2/x^3*exp(d*x+c)-1/48*d^2*a^2/x^2*exp(d*x+c)-1/48*d^3*a^2/x*exp(d*x+c)-1/6*d^2*a*b/x*exp
(d*x+c)-1/6*d^3*a*b*exp(c)*Ei(1,-d*x)-1/3*a*b/x^3*exp(d*x+c)-1/6*d*a*b/x^2*exp(d*x+c)

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Maxima [A]  time = 1.39841, size = 173, normalized size = 0.7 \begin{align*} \frac{1}{24} \,{\left (3 \, a^{2} d^{3} e^{\left (-c\right )} \Gamma \left (-3, d x\right ) + 3 \, a^{2} d^{3} e^{c} \Gamma \left (-3, -d x\right ) + 8 \, a b d^{2} e^{\left (-c\right )} \Gamma \left (-2, d x\right ) - 8 \, a b d^{2} e^{c} \Gamma \left (-2, -d x\right ) + 6 \, b^{2} d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + 6 \, b^{2} d e^{c} \Gamma \left (-1, -d x\right )\right )} d - \frac{{\left (6 \, b^{2} x^{2} + 8 \, a b x + 3 \, a^{2}\right )} \cosh \left (d x + c\right )}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*cosh(d*x+c)/x^5,x, algorithm="maxima")

[Out]

1/24*(3*a^2*d^3*e^(-c)*gamma(-3, d*x) + 3*a^2*d^3*e^c*gamma(-3, -d*x) + 8*a*b*d^2*e^(-c)*gamma(-2, d*x) - 8*a*
b*d^2*e^c*gamma(-2, -d*x) + 6*b^2*d*e^(-c)*gamma(-1, d*x) + 6*b^2*d*e^c*gamma(-1, -d*x))*d - 1/12*(6*b^2*x^2 +
8*a*b*x + 3*a^2)*cosh(d*x + c)/x^4

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Fricas [A]  time = 1.97258, size = 510, normalized size = 2.06 \begin{align*} -\frac{2 \,{\left (8 \, a b d^{2} x^{3} + 16 \, a b x +{\left (a^{2} d^{2} + 12 \, b^{2}\right )} x^{2} + 6 \, a^{2}\right )} \cosh \left (d x + c\right ) -{\left ({\left (a^{2} d^{4} + 8 \, a b d^{3} + 12 \, b^{2} d^{2}\right )} x^{4}{\rm Ei}\left (d x\right ) +{\left (a^{2} d^{4} - 8 \, a b d^{3} + 12 \, b^{2} d^{2}\right )} x^{4}{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \,{\left (8 \, a b d x^{2} + 2 \, a^{2} d x +{\left (a^{2} d^{3} + 12 \, b^{2} d\right )} x^{3}\right )} \sinh \left (d x + c\right ) -{\left ({\left (a^{2} d^{4} + 8 \, a b d^{3} + 12 \, b^{2} d^{2}\right )} x^{4}{\rm Ei}\left (d x\right ) -{\left (a^{2} d^{4} - 8 \, a b d^{3} + 12 \, b^{2} d^{2}\right )} x^{4}{\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{48 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*cosh(d*x+c)/x^5,x, algorithm="fricas")

[Out]

-1/48*(2*(8*a*b*d^2*x^3 + 16*a*b*x + (a^2*d^2 + 12*b^2)*x^2 + 6*a^2)*cosh(d*x + c) - ((a^2*d^4 + 8*a*b*d^3 + 1
2*b^2*d^2)*x^4*Ei(d*x) + (a^2*d^4 - 8*a*b*d^3 + 12*b^2*d^2)*x^4*Ei(-d*x))*cosh(c) + 2*(8*a*b*d*x^2 + 2*a^2*d*x
+ (a^2*d^3 + 12*b^2*d)*x^3)*sinh(d*x + c) - ((a^2*d^4 + 8*a*b*d^3 + 12*b^2*d^2)*x^4*Ei(d*x) - (a^2*d^4 - 8*a*
b*d^3 + 12*b^2*d^2)*x^4*Ei(-d*x))*sinh(c))/x^4

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x\right )^{2} \cosh{\left (c + d x \right )}}{x^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2*cosh(d*x+c)/x**5,x)

[Out]

Integral((a + b*x)**2*cosh(c + d*x)/x**5, x)

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Giac [A]  time = 1.11518, size = 533, normalized size = 2.15 \begin{align*} \frac{a^{2} d^{4} x^{4}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{4} x^{4}{\rm Ei}\left (d x\right ) e^{c} - 8 \, a b d^{3} x^{4}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 8 \, a b d^{3} x^{4}{\rm Ei}\left (d x\right ) e^{c} + 12 \, b^{2} d^{2} x^{4}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 12 \, b^{2} d^{2} x^{4}{\rm Ei}\left (d x\right ) e^{c} - a^{2} d^{3} x^{3} e^{\left (d x + c\right )} + a^{2} d^{3} x^{3} e^{\left (-d x - c\right )} - 8 \, a b d^{2} x^{3} e^{\left (d x + c\right )} - 8 \, a b d^{2} x^{3} e^{\left (-d x - c\right )} - a^{2} d^{2} x^{2} e^{\left (d x + c\right )} - 12 \, b^{2} d x^{3} e^{\left (d x + c\right )} - a^{2} d^{2} x^{2} e^{\left (-d x - c\right )} + 12 \, b^{2} d x^{3} e^{\left (-d x - c\right )} - 8 \, a b d x^{2} e^{\left (d x + c\right )} + 8 \, a b d x^{2} e^{\left (-d x - c\right )} - 2 \, a^{2} d x e^{\left (d x + c\right )} - 12 \, b^{2} x^{2} e^{\left (d x + c\right )} + 2 \, a^{2} d x e^{\left (-d x - c\right )} - 12 \, b^{2} x^{2} e^{\left (-d x - c\right )} - 16 \, a b x e^{\left (d x + c\right )} - 16 \, a b x e^{\left (-d x - c\right )} - 6 \, a^{2} e^{\left (d x + c\right )} - 6 \, a^{2} e^{\left (-d x - c\right )}}{48 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*cosh(d*x+c)/x^5,x, algorithm="giac")

[Out]

1/48*(a^2*d^4*x^4*Ei(-d*x)*e^(-c) + a^2*d^4*x^4*Ei(d*x)*e^c - 8*a*b*d^3*x^4*Ei(-d*x)*e^(-c) + 8*a*b*d^3*x^4*Ei
(d*x)*e^c + 12*b^2*d^2*x^4*Ei(-d*x)*e^(-c) + 12*b^2*d^2*x^4*Ei(d*x)*e^c - a^2*d^3*x^3*e^(d*x + c) + a^2*d^3*x^
3*e^(-d*x - c) - 8*a*b*d^2*x^3*e^(d*x + c) - 8*a*b*d^2*x^3*e^(-d*x - c) - a^2*d^2*x^2*e^(d*x + c) - 12*b^2*d*x
^3*e^(d*x + c) - a^2*d^2*x^2*e^(-d*x - c) + 12*b^2*d*x^3*e^(-d*x - c) - 8*a*b*d*x^2*e^(d*x + c) + 8*a*b*d*x^2*
e^(-d*x - c) - 2*a^2*d*x*e^(d*x + c) - 12*b^2*x^2*e^(d*x + c) + 2*a^2*d*x*e^(-d*x - c) - 12*b^2*x^2*e^(-d*x -
c) - 16*a*b*x*e^(d*x + c) - 16*a*b*x*e^(-d*x - c) - 6*a^2*e^(d*x + c) - 6*a^2*e^(-d*x - c))/x^4