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

Optimal. Leaf size=186 \[ \frac{d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a^2}-\frac{2 b \cosh (c) \text{Chi}(d x)}{a^3}+\frac{2 b \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a^3}-\frac{2 b \sinh (c) \text{Shi}(d x)}{a^3}+\frac{2 b \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a^3}+\frac{d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a^2}-\frac{b \cosh (c+d x)}{a^2 (a+b x)}+\frac{d \sinh (c) \text{Chi}(d x)}{a^2}+\frac{d \cosh (c) \text{Shi}(d x)}{a^2}-\frac{\cosh (c+d x)}{a^2 x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.501074, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 16, 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} \[ \frac{d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a^2}-\frac{2 b \cosh (c) \text{Chi}(d x)}{a^3}+\frac{2 b \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a^3}-\frac{2 b \sinh (c) \text{Shi}(d x)}{a^3}+\frac{2 b \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a^3}+\frac{d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a^2}-\frac{b \cosh (c+d x)}{a^2 (a+b x)}+\frac{d \sinh (c) \text{Chi}(d x)}{a^2}+\frac{d \cosh (c) \text{Shi}(d x)}{a^2}-\frac{\cosh (c+d x)}{a^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Cosh[c + d*x]/(a^2*x)) - (b*Cosh[c + d*x])/(a^2*(a + b*x)) - (2*b*Cosh[c]*CoshIntegral[d*x])/a^3 + (2*b*Cosh
[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/a^3 + (d*CoshIntegral[d*x]*Sinh[c])/a^2 + (d*CoshIntegral[(a*d)/b +
d*x]*Sinh[c - (a*d)/b])/a^2 + (d*Cosh[c]*SinhIntegral[d*x])/a^2 - (2*b*Sinh[c]*SinhIntegral[d*x])/a^3 + (d*Co
sh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/a^2 + (2*b*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/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^2 (a+b x)^2} \, dx &=\int \left (\frac{\cosh (c+d x)}{a^2 x^2}-\frac{2 b \cosh (c+d x)}{a^3 x}+\frac{b^2 \cosh (c+d x)}{a^2 (a+b x)^2}+\frac{2 b^2 \cosh (c+d x)}{a^3 (a+b x)}\right ) \, dx\\ &=\frac{\int \frac{\cosh (c+d x)}{x^2} \, dx}{a^2}-\frac{(2 b) \int \frac{\cosh (c+d x)}{x} \, dx}{a^3}+\frac{\left (2 b^2\right ) \int \frac{\cosh (c+d x)}{a+b x} \, dx}{a^3}+\frac{b^2 \int \frac{\cosh (c+d x)}{(a+b x)^2} \, dx}{a^2}\\ &=-\frac{\cosh (c+d x)}{a^2 x}-\frac{b \cosh (c+d x)}{a^2 (a+b x)}+\frac{d \int \frac{\sinh (c+d x)}{x} \, dx}{a^2}+\frac{(b d) \int \frac{\sinh (c+d x)}{a+b x} \, dx}{a^2}-\frac{(2 b \cosh (c)) \int \frac{\cosh (d x)}{x} \, dx}{a^3}+\frac{\left (2 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^3}-\frac{(2 b \sinh (c)) \int \frac{\sinh (d x)}{x} \, dx}{a^3}+\frac{\left (2 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^3}\\ &=-\frac{\cosh (c+d x)}{a^2 x}-\frac{b \cosh (c+d x)}{a^2 (a+b x)}-\frac{2 b \cosh (c) \text{Chi}(d x)}{a^3}+\frac{2 b \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{a^3}-\frac{2 b \sinh (c) \text{Shi}(d x)}{a^3}+\frac{2 b \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{a^3}+\frac{(d \cosh (c)) \int \frac{\sinh (d x)}{x} \, dx}{a^2}+\frac{\left (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^2}+\frac{(d \sinh (c)) \int \frac{\cosh (d x)}{x} \, dx}{a^2}+\frac{\left (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^2}\\ &=-\frac{\cosh (c+d x)}{a^2 x}-\frac{b \cosh (c+d x)}{a^2 (a+b x)}-\frac{2 b \cosh (c) \text{Chi}(d x)}{a^3}+\frac{2 b \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{a^3}+\frac{d \text{Chi}(d x) \sinh (c)}{a^2}+\frac{d \text{Chi}\left (\frac{a d}{b}+d x\right ) \sinh \left (c-\frac{a d}{b}\right )}{a^2}+\frac{d \cosh (c) \text{Shi}(d x)}{a^2}-\frac{2 b \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}+\frac{2 b \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{a^3}\\ \end{align*}

Mathematica [A]  time = 1.41446, size = 183, normalized size = 0.98 \[ \frac{a d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (d \left (\frac{a}{b}+x\right )\right )+2 b \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (d \left (\frac{a}{b}+x\right )\right )+2 b \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )+a d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )-\frac{a \sinh (c) (a+2 b x) \sinh (d x)}{x (a+b x)}-\frac{a \cosh (c) (a+2 b x) \cosh (d x)}{x (a+b x)}+a d \sinh (c) \text{Chi}(d x)+a d \cosh (c) \text{Shi}(d x)-2 b \cosh (c) \text{Chi}(d x)-2 b \sinh (c) \text{Shi}(d x)}{a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.058, size = 312, normalized size = 1.7 \begin{align*} -{\frac{d{{\rm e}^{-dx-c}}b}{{a}^{2} \left ( bdx+da \right ) }}-{\frac{d{{\rm e}^{-dx-c}}}{2\,ax \left ( bdx+da \right ) }}+{\frac{d{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2\,{a}^{2}}}+{\frac{b{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{{a}^{3}}}+{\frac{d}{2\,{a}^{2}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }-{\frac{b}{{a}^{3}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }+{\frac{b{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{{a}^{3}}}-{\frac{{{\rm e}^{dx+c}}}{2\,{a}^{2}x}}-{\frac{d{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2\,{a}^{2}}}-{\frac{d{{\rm e}^{dx+c}}}{2\,{a}^{2}} \left ({\frac{da}{b}}+dx \right ) ^{-1}}-{\frac{d}{2\,{a}^{2}}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) }-{\frac{b}{{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^2/(b*x+a)^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (d x + c\right )}{{\left (b x + a\right )}^{2} 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)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.11014, size = 802, normalized size = 4.31 \begin{align*} -\frac{2 \,{\left (2 \, a b x + a^{2}\right )} \cosh \left (d x + c\right ) -{\left ({\left ({\left (a b d - 2 \, b^{2}\right )} x^{2} +{\left (a^{2} d - 2 \, a b\right )} x\right )}{\rm Ei}\left (d x\right ) -{\left ({\left (a b d + 2 \, b^{2}\right )} x^{2} +{\left (a^{2} d + 2 \, a b\right )} x\right )}{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) -{\left ({\left ({\left (a b d + 2 \, b^{2}\right )} x^{2} +{\left (a^{2} d + 2 \, a b\right )} x\right )}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) -{\left ({\left (a b d - 2 \, b^{2}\right )} x^{2} +{\left (a^{2} d - 2 \, a b\right )} x\right )}{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \cosh \left (-\frac{b c - a d}{b}\right ) -{\left ({\left ({\left (a b d - 2 \, b^{2}\right )} x^{2} +{\left (a^{2} d - 2 \, a b\right )} x\right )}{\rm Ei}\left (d x\right ) +{\left ({\left (a b d + 2 \, b^{2}\right )} x^{2} +{\left (a^{2} d + 2 \, a b\right )} x\right )}{\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right ) +{\left ({\left ({\left (a b d + 2 \, b^{2}\right )} x^{2} +{\left (a^{2} d + 2 \, a b\right )} x\right )}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) +{\left ({\left (a b d - 2 \, b^{2}\right )} x^{2} +{\left (a^{2} d - 2 \, a b\right )} x\right )}{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \sinh \left (-\frac{b c - a d}{b}\right )}{2 \,{\left (a^{3} b x^{2} + a^{4} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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)**2,x)

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Giac [B]  time = 1.28598, size = 576, normalized size = 3.1 \begin{align*} -\frac{a b d x^{2}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a b d x^{2}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - a b d x^{2}{\rm Ei}\left (d x\right ) e^{c} + a b d x^{2}{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} + a^{2} d x{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 2 \, b^{2} x^{2}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a^{2} d x{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - 2 \, b^{2} x^{2}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - a^{2} d x{\rm Ei}\left (d x\right ) e^{c} + 2 \, b^{2} x^{2}{\rm Ei}\left (d x\right ) e^{c} + a^{2} d x{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} - 2 \, b^{2} x^{2}{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} + 2 \, a b x{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 2 \, a b x{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} + 2 \, a b x{\rm Ei}\left (d x\right ) e^{c} - 2 \, a b x{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} + 2 \, a b x e^{\left (d x + c\right )} + 2 \, a b x e^{\left (-d x - c\right )} + a^{2} e^{\left (d x + c\right )} + a^{2} e^{\left (-d x - c\right )}}{2 \,{\left (a^{3} b x^{2} + a^{4} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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