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

Optimal. Leaf size=262 \[ -\frac{d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a^2 b}-\frac{\cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a^3}-\frac{\sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a^3}-\frac{d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a^2 b}+\frac{\cosh (c+d x)}{a^2 (a+b x)}+\frac{\cosh (c) \text{Chi}(d x)}{a^3}+\frac{\sinh (c) \text{Shi}(d x)}{a^3}-\frac{d^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{2 a b^2}-\frac{d^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{2 a b^2}+\frac{d \sinh (c+d x)}{2 a b (a+b x)}+\frac{\cosh (c+d x)}{2 a (a+b x)^2} \]

[Out]

Cosh[c + d*x]/(2*a*(a + b*x)^2) + Cosh[c + d*x]/(a^2*(a + b*x)) + (Cosh[c]*CoshIntegral[d*x])/a^3 - (Cosh[c -
(a*d)/b]*CoshIntegral[(a*d)/b + d*x])/a^3 - (d^2*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/(2*a*b^2) - (d
*CoshIntegral[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/(a^2*b) + (d*Sinh[c + d*x])/(2*a*b*(a + b*x)) + (Sinh[c]*SinhI
ntegral[d*x])/a^3 - (d*Cosh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/(a^2*b) - (Sinh[c - (a*d)/b]*SinhIntegra
l[(a*d)/b + d*x])/a^3 - (d^2*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/(2*a*b^2)

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Rubi [A]  time = 0.561227, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {6742, 3303, 3298, 3301, 3297} \[ -\frac{d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a^2 b}-\frac{\cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a^3}-\frac{\sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a^3}-\frac{d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a^2 b}+\frac{\cosh (c+d x)}{a^2 (a+b x)}+\frac{\cosh (c) \text{Chi}(d x)}{a^3}+\frac{\sinh (c) \text{Shi}(d x)}{a^3}-\frac{d^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{2 a b^2}-\frac{d^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{2 a b^2}+\frac{d \sinh (c+d x)}{2 a b (a+b x)}+\frac{\cosh (c+d x)}{2 a (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]/(x*(a + b*x)^3),x]

[Out]

Cosh[c + d*x]/(2*a*(a + b*x)^2) + Cosh[c + d*x]/(a^2*(a + b*x)) + (Cosh[c]*CoshIntegral[d*x])/a^3 - (Cosh[c -
(a*d)/b]*CoshIntegral[(a*d)/b + d*x])/a^3 - (d^2*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/(2*a*b^2) - (d
*CoshIntegral[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/(a^2*b) + (d*Sinh[c + d*x])/(2*a*b*(a + b*x)) + (Sinh[c]*SinhI
ntegral[d*x])/a^3 - (d*Cosh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/(a^2*b) - (Sinh[c - (a*d)/b]*SinhIntegra
l[(a*d)/b + d*x])/a^3 - (d^2*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/(2*a*b^2)

Rule 6742

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

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]

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]

Rubi steps

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

Mathematica [B]  time = 4.59218, size = 614, normalized size = 2.34 \[ -\frac{a^2 b^2 d^2 x^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{d (a+b x)}{b}\right )+4 a^2 b^2 d x \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{d (a+b x)}{b}\right )+a^2 b^2 d^2 x^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{d (a+b x)}{b}\right )-2 a^2 b^2 \sinh (c) \text{Shi}(d x)+2 a^2 b^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )+4 a^2 b^2 d x \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{d (a+b x)}{b}\right )-a^2 b^2 d x \sinh (c+d x)-3 a^2 b^2 \cosh (c+d x)+a^4 d^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{d (a+b x)}{b}\right )+2 a^3 b d^2 x \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{d (a+b x)}{b}\right )+2 a^3 b d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{d (a+b x)}{b}\right )+a^4 d^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{d (a+b x)}{b}\right )+2 a^3 b d^2 x \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{d (a+b x)}{b}\right )+2 a^3 b d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{d (a+b x)}{b}\right )-a^3 b d \sinh (c+d x)+2 a b^3 d x^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{d (a+b x)}{b}\right )-2 b^2 \cosh (c) (a+b x)^2 \text{Chi}(d x)+2 b^2 (a+b x)^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (d \left (\frac{a}{b}+x\right )\right )+2 b^4 x^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )+2 a b^3 d x^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{d (a+b x)}{b}\right )-4 a b^3 x \sinh (c) \text{Shi}(d x)+4 a b^3 x \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )-2 a b^3 x \cosh (c+d x)-2 b^4 x^2 \sinh (c) \text{Shi}(d x)}{2 a^3 b^2 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-3*a^2*b^2*Cosh[c + d*x] - 2*a*b^3*x*Cosh[c + d*x] - 2*b^2*(a + b*x)^2*Cosh[c]*CoshIntegral[d*x] + 2*b^2*(a
+ b*x)^2*Cosh[c - (a*d)/b]*CoshIntegral[d*(a/b + x)] + a^4*d^2*Cosh[c - (a*d)/b]*CoshIntegral[(d*(a + b*x))/b]
+ 2*a^3*b*d^2*x*Cosh[c - (a*d)/b]*CoshIntegral[(d*(a + b*x))/b] + a^2*b^2*d^2*x^2*Cosh[c - (a*d)/b]*CoshInteg
ral[(d*(a + b*x))/b] + 2*a^3*b*d*CoshIntegral[(d*(a + b*x))/b]*Sinh[c - (a*d)/b] + 4*a^2*b^2*d*x*CoshIntegral[
(d*(a + b*x))/b]*Sinh[c - (a*d)/b] + 2*a*b^3*d*x^2*CoshIntegral[(d*(a + b*x))/b]*Sinh[c - (a*d)/b] - a^3*b*d*S
inh[c + d*x] - a^2*b^2*d*x*Sinh[c + d*x] - 2*a^2*b^2*Sinh[c]*SinhIntegral[d*x] - 4*a*b^3*x*Sinh[c]*SinhIntegra
l[d*x] - 2*b^4*x^2*Sinh[c]*SinhIntegral[d*x] + 2*a^2*b^2*Sinh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)] + 4*a*b^3
*x*Sinh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)] + 2*b^4*x^2*Sinh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)] + 2*a^3
*b*d*Cosh[c - (a*d)/b]*SinhIntegral[(d*(a + b*x))/b] + 4*a^2*b^2*d*x*Cosh[c - (a*d)/b]*SinhIntegral[(d*(a + b*
x))/b] + 2*a*b^3*d*x^2*Cosh[c - (a*d)/b]*SinhIntegral[(d*(a + b*x))/b] + a^4*d^2*Sinh[c - (a*d)/b]*SinhIntegra
l[(d*(a + b*x))/b] + 2*a^3*b*d^2*x*Sinh[c - (a*d)/b]*SinhIntegral[(d*(a + b*x))/b] + a^2*b^2*d^2*x^2*Sinh[c -
(a*d)/b]*SinhIntegral[(d*(a + b*x))/b])/(2*a^3*b^2*(a + b*x)^2)

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Maple [A]  time = 0.055, size = 488, normalized size = 1.9 \begin{align*} -{\frac{{{\rm e}^{-dx-c}}d \left ( \left ( dx+c \right ) abd+{a}^{2}{d}^{2}-cdba-2\,{b}^{2} \left ( dx+c \right ) -3\,bda+2\,c{b}^{2} \right ) }{4\,{a}^{2}b \left ({b}^{2} \left ( dx+c \right ) ^{2}+2\, \left ( dx+c \right ) abd-2\, \left ( dx+c \right ){b}^{2}c+{a}^{2}{d}^{2}-2\,cdba+{c}^{2}{b}^{2} \right ) }}-{\frac{{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2\,{a}^{3}}}+{\frac{{d}^{2}}{4\,a{b}^{2}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }-{\frac{d}{2\,{a}^{2}b}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }+{\frac{1}{2\,{a}^{3}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }-{\frac{{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2\,{a}^{3}}}+{\frac{{d}^{2}{{\rm e}^{dx+c}}}{4\,a{b}^{2}} \left ({\frac{da}{b}}+dx \right ) ^{-2}}+{\frac{{d}^{2}{{\rm e}^{dx+c}}}{4\,a{b}^{2}} \left ({\frac{da}{b}}+dx \right ) ^{-1}}+{\frac{{d}^{2}}{4\,a{b}^{2}}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) }+{\frac{d{{\rm e}^{dx+c}}}{2\,{a}^{2}b} \left ({\frac{da}{b}}+dx \right ) ^{-1}}+{\frac{d}{2\,{a}^{2}b}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) }+{\frac{1}{2\,{a}^{3}}{{\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/(b*x+a)^3,x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.17532, size = 1243, normalized size = 4.74 \begin{align*} \frac{2 \,{\left (2 \, a b^{3} x + 3 \, a^{2} b^{2}\right )} \cosh \left (d x + c\right ) + 2 \,{\left ({\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}{\rm Ei}\left (d x\right ) +{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) -{\left ({\left (a^{4} d^{2} + 2 \, a^{3} b d + 2 \, a^{2} b^{2} +{\left (a^{2} b^{2} d^{2} + 2 \, a b^{3} d + 2 \, b^{4}\right )} x^{2} + 2 \,{\left (a^{3} b d^{2} + 2 \, a^{2} b^{2} d + 2 \, a b^{3}\right )} x\right )}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) +{\left (a^{4} d^{2} - 2 \, a^{3} b d + 2 \, a^{2} b^{2} +{\left (a^{2} b^{2} d^{2} - 2 \, a b^{3} d + 2 \, b^{4}\right )} x^{2} + 2 \,{\left (a^{3} b d^{2} - 2 \, a^{2} b^{2} d + 2 \, a b^{3}\right )} x\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^{2} b^{2} d x + a^{3} b d\right )} \sinh \left (d x + c\right ) + 2 \,{\left ({\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}{\rm Ei}\left (d x\right ) -{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}{\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right ) +{\left ({\left (a^{4} d^{2} + 2 \, a^{3} b d + 2 \, a^{2} b^{2} +{\left (a^{2} b^{2} d^{2} + 2 \, a b^{3} d + 2 \, b^{4}\right )} x^{2} + 2 \,{\left (a^{3} b d^{2} + 2 \, a^{2} b^{2} d + 2 \, a b^{3}\right )} x\right )}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) -{\left (a^{4} d^{2} - 2 \, a^{3} b d + 2 \, a^{2} b^{2} +{\left (a^{2} b^{2} d^{2} - 2 \, a b^{3} d + 2 \, b^{4}\right )} x^{2} + 2 \,{\left (a^{3} b d^{2} - 2 \, a^{2} b^{2} d + 2 \, a b^{3}\right )} x\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^{3} b^{4} x^{2} + 2 \, a^{4} b^{3} x + a^{5} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.2106, size = 1130, normalized size = 4.31 \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/(b*x+a)^3,x, algorithm="giac")

[Out]

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