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

Optimal. Leaf size=113 \[ -\frac{b \cosh (c) \text{Chi}(d x)}{a^2}+\frac{b \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a^2}-\frac{b \sinh (c) \text{Shi}(d x)}{a^2}+\frac{b \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a^2}+\frac{d \sinh (c) \text{Chi}(d x)}{a}+\frac{d \cosh (c) \text{Shi}(d x)}{a}-\frac{\cosh (c+d x)}{a x} \]

[Out]

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

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Rubi [A]  time = 0.368679, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 12, 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} \[ -\frac{b \cosh (c) \text{Chi}(d x)}{a^2}+\frac{b \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a^2}-\frac{b \sinh (c) \text{Shi}(d x)}{a^2}+\frac{b \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a^2}+\frac{d \sinh (c) \text{Chi}(d x)}{a}+\frac{d \cosh (c) \text{Shi}(d x)}{a}-\frac{\cosh (c+d x)}{a x} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]/(x^2*(a + b*x)),x]

[Out]

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

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

Mathematica [A]  time = 0.380891, size = 101, normalized size = 0.89 \[ \frac{b x \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (d \left (\frac{a}{b}+x\right )\right )+\text{Chi}(d x) (a d x \sinh (c)-b x \cosh (c))+b x \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )+a d x \cosh (c) \text{Shi}(d x)-a \cosh (c+d x)-b x \sinh (c) \text{Shi}(d x)}{a^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(a*Cosh[c + d*x]) + b*x*Cosh[c - (a*d)/b]*CoshIntegral[d*(a/b + x)] + CoshIntegral[d*x]*(-(b*x*Cosh[c]) + a*
d*x*Sinh[c]) + a*d*x*Cosh[c]*SinhIntegral[d*x] - b*x*Sinh[c]*SinhIntegral[d*x] + b*x*Sinh[c - (a*d)/b]*SinhInt
egral[d*(a/b + x)])/(a^2*x)

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Maple [A]  time = 0.048, size = 172, normalized size = 1.5 \begin{align*} -{\frac{{{\rm e}^{-dx-c}}}{2\,ax}}+{\frac{d{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2\,a}}+{\frac{b{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2\,{a}^{2}}}-{\frac{b}{2\,{a}^{2}}{{\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 ) }{2\,{a}^{2}}}-{\frac{{{\rm e}^{dx+c}}}{2\,ax}}-{\frac{d{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2\,a}}-{\frac{b}{2\,{a}^{2}}{{\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),x)

[Out]

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

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Maxima [A]  time = 1.45567, size = 259, normalized size = 2.29 \begin{align*} -\frac{1}{2} \, d{\left (\frac{{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} -{\rm Ei}\left (d x\right ) e^{c}}{a} + \frac{b^{2}{\left (\frac{e^{\left (-c + \frac{a d}{b}\right )} E_{1}\left (\frac{{\left (b x + a\right )} d}{b}\right )}{b} + \frac{e^{\left (c - \frac{a d}{b}\right )} E_{1}\left (-\frac{{\left (b x + a\right )} d}{b}\right )}{b}\right )}}{a^{2} d} + \frac{2 \, b \cosh \left (d x + c\right ) \log \left (b x + a\right )}{a^{2} d} - \frac{2 \, b \cosh \left (d x + c\right ) \log \left (x\right )}{a^{2} d} + \frac{{\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} +{\rm Ei}\left (d x\right ) e^{c}\right )} b}{a^{2} d}\right )} +{\left (\frac{b \log \left (b x + a\right )}{a^{2}} - \frac{b \log \left (x\right )}{a^{2}} - \frac{1}{a x}\right )} \cosh \left (d x + c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*d*((Ei(-d*x)*e^(-c) - Ei(d*x)*e^c)/a + b^2*(e^(-c + a*d/b)*exp_integral_e(1, (b*x + a)*d/b)/b + e^(c - a*
d/b)*exp_integral_e(1, -(b*x + a)*d/b)/b)/(a^2*d) + 2*b*cosh(d*x + c)*log(b*x + a)/(a^2*d) - 2*b*cosh(d*x + c)
*log(x)/(a^2*d) + (Ei(-d*x)*e^(-c) + Ei(d*x)*e^c)*b/(a^2*d)) + (b*log(b*x + a)/a^2 - b*log(x)/a^2 - 1/(a*x))*c
osh(d*x + c)

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Fricas [A]  time = 2.04736, size = 400, normalized size = 3.54 \begin{align*} -\frac{2 \, a \cosh \left (d x + c\right ) -{\left ({\left (a d - b\right )} x{\rm Ei}\left (d x\right ) -{\left (a d + b\right )} x{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) -{\left (b x{\rm Ei}\left (\frac{b d x + a d}{b}\right ) + b x{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \cosh \left (-\frac{b c - a d}{b}\right ) -{\left ({\left (a d - b\right )} x{\rm Ei}\left (d x\right ) +{\left (a d + b\right )} x{\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right ) +{\left (b x{\rm Ei}\left (\frac{b d x + a d}{b}\right ) - b x{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \sinh \left (-\frac{b c - a d}{b}\right )}{2 \, a^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17191, size = 174, normalized size = 1.54 \begin{align*} -\frac{a d x{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a d x{\rm Ei}\left (d x\right ) e^{c} + b x{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - b x{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} + b x{\rm Ei}\left (d x\right ) e^{c} - b x{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} + a e^{\left (d x + c\right )} + a e^{\left (-d x - c\right )}}{2 \, a^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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