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

Optimal. Leaf size=298 \[ \frac{d^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{2 a^2 b}+\frac{2 d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a^3}-\frac{3 b \cosh (c) \text{Chi}(d x)}{a^4}+\frac{3 b \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a^4}+\frac{d^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{2 a^2 b}-\frac{3 b \sinh (c) \text{Shi}(d x)}{a^4}+\frac{3 b \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a^4}+\frac{2 d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a^3}-\frac{d \sinh (c+d x)}{2 a^2 (a+b x)}-\frac{2 b \cosh (c+d x)}{a^3 (a+b x)}-\frac{b \cosh (c+d x)}{2 a^2 (a+b x)^2}+\frac{d \sinh (c) \text{Chi}(d x)}{a^3}+\frac{d \cosh (c) \text{Shi}(d x)}{a^3}-\frac{\cosh (c+d x)}{a^3 x} \]

[Out]

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

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Rubi [A]  time = 0.689996, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 21, 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} \[ \frac{d^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{2 a^2 b}+\frac{2 d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a^3}-\frac{3 b \cosh (c) \text{Chi}(d x)}{a^4}+\frac{3 b \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a^4}+\frac{d^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{2 a^2 b}-\frac{3 b \sinh (c) \text{Shi}(d x)}{a^4}+\frac{3 b \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a^4}+\frac{2 d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a^3}-\frac{d \sinh (c+d x)}{2 a^2 (a+b x)}-\frac{2 b \cosh (c+d x)}{a^3 (a+b x)}-\frac{b \cosh (c+d x)}{2 a^2 (a+b x)^2}+\frac{d \sinh (c) \text{Chi}(d x)}{a^3}+\frac{d \cosh (c) \text{Shi}(d x)}{a^3}-\frac{\cosh (c+d x)}{a^3 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [B]  time = 1.43962, size = 710, normalized size = 2.38 \[ \frac{a^2 b^2 d^2 x^3 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{d (a+b x)}{b}\right )+8 a^2 b^2 d x^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{d (a+b x)}{b}\right )+a^2 b^2 d^2 x^3 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{d (a+b x)}{b}\right )+4 a^2 b^2 d x^2 \cosh (c) \text{Shi}(d x)+8 a^2 b^2 d x^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{d (a+b x)}{b}\right )-6 a^2 b^2 x \sinh (c) \text{Shi}(d x)+6 a^2 b^2 x \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )-a^2 b^2 d x^2 \sinh (c+d x)-9 a^2 b^2 x \cosh (c+d x)+2 a^3 b d^2 x^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{d (a+b x)}{b}\right )+a^4 d^2 x \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{d (a+b x)}{b}\right )+4 a^3 b d x \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{d (a+b x)}{b}\right )+2 a^3 b d^2 x^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{d (a+b x)}{b}\right )+a^4 d^2 x \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{d (a+b x)}{b}\right )+2 a^3 b d x \cosh (c) \text{Shi}(d x)+4 a^3 b d x \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{d (a+b x)}{b}\right )-a^3 b d x \sinh (c+d x)-2 a^3 b \cosh (c+d x)+4 a b^3 d x^3 \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{d (a+b x)}{b}\right )+6 b^2 x (a+b x)^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (d \left (\frac{a}{b}+x\right )\right )-12 a b^3 x^2 \sinh (c) \text{Shi}(d x)+12 a b^3 x^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )+6 b^4 x^3 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )+2 a b^3 d x^3 \cosh (c) \text{Shi}(d x)+4 a b^3 d x^3 \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{d (a+b x)}{b}\right )-6 a b^3 x^2 \cosh (c+d x)+2 b x (a+b x)^2 \text{Chi}(d x) (a d \sinh (c)-3 b \cosh (c))-6 b^4 x^3 \sinh (c) \text{Shi}(d x)}{2 a^4 b x (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^3*b*Cosh[c + d*x] - 9*a^2*b^2*x*Cosh[c + d*x] - 6*a*b^3*x^2*Cosh[c + d*x] + 6*b^2*x*(a + b*x)^2*Cosh[c -
(a*d)/b]*CoshIntegral[d*(a/b + x)] + a^4*d^2*x*Cosh[c - (a*d)/b]*CoshIntegral[(d*(a + b*x))/b] + 2*a^3*b*d^2*
x^2*Cosh[c - (a*d)/b]*CoshIntegral[(d*(a + b*x))/b] + a^2*b^2*d^2*x^3*Cosh[c - (a*d)/b]*CoshIntegral[(d*(a + b
*x))/b] + 2*b*x*(a + b*x)^2*CoshIntegral[d*x]*(-3*b*Cosh[c] + a*d*Sinh[c]) + 4*a^3*b*d*x*CoshIntegral[(d*(a +
b*x))/b]*Sinh[c - (a*d)/b] + 8*a^2*b^2*d*x^2*CoshIntegral[(d*(a + b*x))/b]*Sinh[c - (a*d)/b] + 4*a*b^3*d*x^3*C
oshIntegral[(d*(a + b*x))/b]*Sinh[c - (a*d)/b] - a^3*b*d*x*Sinh[c + d*x] - a^2*b^2*d*x^2*Sinh[c + d*x] + 2*a^3
*b*d*x*Cosh[c]*SinhIntegral[d*x] + 4*a^2*b^2*d*x^2*Cosh[c]*SinhIntegral[d*x] + 2*a*b^3*d*x^3*Cosh[c]*SinhInteg
ral[d*x] - 6*a^2*b^2*x*Sinh[c]*SinhIntegral[d*x] - 12*a*b^3*x^2*Sinh[c]*SinhIntegral[d*x] - 6*b^4*x^3*Sinh[c]*
SinhIntegral[d*x] + 6*a^2*b^2*x*Sinh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)] + 12*a*b^3*x^2*Sinh[c - (a*d)/b]*S
inhIntegral[d*(a/b + x)] + 6*b^4*x^3*Sinh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)] + 4*a^3*b*d*x*Cosh[c - (a*d)/
b]*SinhIntegral[(d*(a + b*x))/b] + 8*a^2*b^2*d*x^2*Cosh[c - (a*d)/b]*SinhIntegral[(d*(a + b*x))/b] + 4*a*b^3*d
*x^3*Cosh[c - (a*d)/b]*SinhIntegral[(d*(a + b*x))/b] + a^4*d^2*x*Sinh[c - (a*d)/b]*SinhIntegral[(d*(a + b*x))/
b] + 2*a^3*b*d^2*x^2*Sinh[c - (a*d)/b]*SinhIntegral[(d*(a + b*x))/b] + a^2*b^2*d^2*x^3*Sinh[c - (a*d)/b]*SinhI
ntegral[(d*(a + b*x))/b])/(2*a^4*b*x*(a + b*x)^2)

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Maple [B]  time = 0.073, size = 643, normalized size = 2.2 \begin{align*}{\frac{{{\rm e}^{-dx-c}}x{d}^{3}b}{4\,{a}^{2} \left ({b}^{2}{d}^{2}{x}^{2}+2\,ab{d}^{2}x+{a}^{2}{d}^{2} \right ) }}+{\frac{{{\rm e}^{-dx-c}}{d}^{3}}{4\,a \left ({b}^{2}{d}^{2}{x}^{2}+2\,ab{d}^{2}x+{a}^{2}{d}^{2} \right ) }}-{\frac{3\,{{\rm e}^{-dx-c}}x{d}^{2}{b}^{2}}{2\,{a}^{3} \left ({b}^{2}{d}^{2}{x}^{2}+2\,ab{d}^{2}x+{a}^{2}{d}^{2} \right ) }}-{\frac{9\,{{\rm e}^{-dx-c}}{d}^{2}b}{4\,{a}^{2} \left ({b}^{2}{d}^{2}{x}^{2}+2\,ab{d}^{2}x+{a}^{2}{d}^{2} \right ) }}-{\frac{{{\rm e}^{-dx-c}}{d}^{2}}{2\,ax \left ({b}^{2}{d}^{2}{x}^{2}+2\,ab{d}^{2}x+{a}^{2}{d}^{2} \right ) }}+{\frac{d{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2\,{a}^{3}}}+{\frac{3\,b{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2\,{a}^{4}}}-{\frac{{d}^{2}}{4\,{a}^{2}b}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }+{\frac{d}{{a}^{3}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }-{\frac{3\,b}{2\,{a}^{4}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }+{\frac{3\,b{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2\,{a}^{4}}}-{\frac{{{\rm e}^{dx+c}}}{2\,{a}^{3}x}}-{\frac{d{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2\,{a}^{3}}}-{\frac{{d}^{2}{{\rm e}^{dx+c}}}{4\,{a}^{2}b} \left ({\frac{da}{b}}+dx \right ) ^{-2}}-{\frac{{d}^{2}{{\rm e}^{dx+c}}}{4\,{a}^{2}b} \left ({\frac{da}{b}}+dx \right ) ^{-1}}-{\frac{{d}^{2}}{4\,{a}^{2}b}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) }-{\frac{d{{\rm e}^{dx+c}}}{{a}^{3}} \left ({\frac{da}{b}}+dx \right ) ^{-1}}-{\frac{d}{{a}^{3}}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) }-{\frac{3\,b}{2\,{a}^{4}}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*exp(-d*x-c)/a^2*x*d^3/(b^2*d^2*x^2+2*a*b*d^2*x+a^2*d^2)*b+1/4*exp(-d*x-c)/a*d^3/(b^2*d^2*x^2+2*a*b*d^2*x+a
^2*d^2)-3/2*exp(-d*x-c)/a^3*x*d^2/(b^2*d^2*x^2+2*a*b*d^2*x+a^2*d^2)*b^2-9/4*exp(-d*x-c)/a^2*d^2/(b^2*d^2*x^2+2
*a*b*d^2*x+a^2*d^2)*b-1/2*exp(-d*x-c)/a/x*d^2/(b^2*d^2*x^2+2*a*b*d^2*x+a^2*d^2)+1/2*d/a^3*exp(-c)*Ei(1,d*x)+3/
2/a^4*exp(-c)*Ei(1,d*x)*b-1/4/b/a^2*d^2*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)+d/a^3*exp((a*d-b*c)/b)*Ei(1,d
*x+c+(a*d-b*c)/b)-3/2*b/a^4*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)+3/2/a^4*b*exp(c)*Ei(1,-d*x)-1/2/a^3/x*exp
(d*x+c)-1/2*d/a^3*exp(c)*Ei(1,-d*x)-1/4*d^2/b/a^2*exp(d*x+c)/(1/b*d*a+d*x)^2-1/4*d^2/b/a^2*exp(d*x+c)/(1/b*d*a
+d*x)-1/4*d^2/b/a^2*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)-d/a^3*exp(d*x+c)/(1/b*d*a+d*x)-d/a^3*exp(-(a*d-
b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)-3/2*b/a^4*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^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.2, size = 1563, normalized size = 5.24 \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^2/(b*x+a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Giac [B]  time = 1.25263, size = 1358, normalized size = 4.56 \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^2/(b*x+a)^3,x, algorithm="giac")

[Out]

1/4*(a^2*b^2*d^2*x^3*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + a^2*b^2*d^2*x^3*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) -
2*a*b^3*d*x^3*Ei(-d*x)*e^(-c) + 2*a^3*b*d^2*x^2*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 4*a*b^3*d*x^3*Ei((b*d*x +
a*d)/b)*e^(c - a*d/b) + 2*a*b^3*d*x^3*Ei(d*x)*e^c + 2*a^3*b*d^2*x^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) - 4*a
*b^3*d*x^3*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) - 4*a^2*b^2*d*x^2*Ei(-d*x)*e^(-c) - 6*b^4*x^3*Ei(-d*x)*e^(-c) +
a^4*d^2*x*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 8*a^2*b^2*d*x^2*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 6*b^4*x^3*E
i((b*d*x + a*d)/b)*e^(c - a*d/b) + 4*a^2*b^2*d*x^2*Ei(d*x)*e^c - 6*b^4*x^3*Ei(d*x)*e^c + a^4*d^2*x*Ei(-(b*d*x
+ a*d)/b)*e^(-c + a*d/b) - 8*a^2*b^2*d*x^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + 6*b^4*x^3*Ei(-(b*d*x + a*d)/b
)*e^(-c + a*d/b) - a^2*b^2*d*x^2*e^(d*x + c) + a^2*b^2*d*x^2*e^(-d*x - c) - 2*a^3*b*d*x*Ei(-d*x)*e^(-c) - 12*a
*b^3*x^2*Ei(-d*x)*e^(-c) + 4*a^3*b*d*x*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 12*a*b^3*x^2*Ei((b*d*x + a*d)/b)*e^
(c - a*d/b) + 2*a^3*b*d*x*Ei(d*x)*e^c - 12*a*b^3*x^2*Ei(d*x)*e^c - 4*a^3*b*d*x*Ei(-(b*d*x + a*d)/b)*e^(-c + a*
d/b) + 12*a*b^3*x^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) - a^3*b*d*x*e^(d*x + c) - 6*a*b^3*x^2*e^(d*x + c) + a^
3*b*d*x*e^(-d*x - c) - 6*a*b^3*x^2*e^(-d*x - c) - 6*a^2*b^2*x*Ei(-d*x)*e^(-c) + 6*a^2*b^2*x*Ei((b*d*x + a*d)/b
)*e^(c - a*d/b) - 6*a^2*b^2*x*Ei(d*x)*e^c + 6*a^2*b^2*x*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) - 9*a^2*b^2*x*e^(d
*x + c) - 9*a^2*b^2*x*e^(-d*x - c) - 2*a^3*b*e^(d*x + c) - 2*a^3*b*e^(-d*x - c))/(a^4*b^3*x^3 + 2*a^5*b^2*x^2
+ a^6*b*x)