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

Optimal. Leaf size=190 \[ \frac{b^2 \cosh (c) \text{Chi}(d x)}{a^3}-\frac{b^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a^3}+\frac{b^2 \sinh (c) \text{Shi}(d x)}{a^3}-\frac{b^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a^3}-\frac{b d \sinh (c) \text{Chi}(d x)}{a^2}-\frac{b d \cosh (c) \text{Shi}(d x)}{a^2}+\frac{b \cosh (c+d x)}{a^2 x}+\frac{d^2 \cosh (c) \text{Chi}(d x)}{2 a}+\frac{d^2 \sinh (c) \text{Shi}(d x)}{2 a}-\frac{\cosh (c+d x)}{2 a x^2}-\frac{d \sinh (c+d x)}{2 a x} \]

[Out]

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

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Rubi [A]  time = 0.485121, antiderivative size = 190, 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, 3297, 3303, 3298, 3301} \[ \frac{b^2 \cosh (c) \text{Chi}(d x)}{a^3}-\frac{b^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a^3}+\frac{b^2 \sinh (c) \text{Shi}(d x)}{a^3}-\frac{b^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a^3}-\frac{b d \sinh (c) \text{Chi}(d x)}{a^2}-\frac{b d \cosh (c) \text{Shi}(d x)}{a^2}+\frac{b \cosh (c+d x)}{a^2 x}+\frac{d^2 \cosh (c) \text{Chi}(d x)}{2 a}+\frac{d^2 \sinh (c) \text{Shi}(d x)}{2 a}-\frac{\cosh (c+d x)}{2 a x^2}-\frac{d \sinh (c+d x)}{2 a x} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]/(x^3*(a + b*x)),x]

[Out]

-Cosh[c + d*x]/(2*a*x^2) + (b*Cosh[c + d*x])/(a^2*x) + (b^2*Cosh[c]*CoshIntegral[d*x])/a^3 + (d^2*Cosh[c]*Cosh
Integral[d*x])/(2*a) - (b^2*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/a^3 - (b*d*CoshIntegral[d*x]*Sinh[c
])/a^2 - (d*Sinh[c + d*x])/(2*a*x) - (b*d*Cosh[c]*SinhIntegral[d*x])/a^2 + (b^2*Sinh[c]*SinhIntegral[d*x])/a^3
+ (d^2*Sinh[c]*SinhIntegral[d*x])/(2*a) - (b^2*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^3 (a+b x)} \, dx &=\int \left (\frac{\cosh (c+d x)}{a x^3}-\frac{b \cosh (c+d x)}{a^2 x^2}+\frac{b^2 \cosh (c+d x)}{a^3 x}-\frac{b^3 \cosh (c+d x)}{a^3 (a+b x)}\right ) \, dx\\ &=\frac{\int \frac{\cosh (c+d x)}{x^3} \, dx}{a}-\frac{b \int \frac{\cosh (c+d x)}{x^2} \, dx}{a^2}+\frac{b^2 \int \frac{\cosh (c+d x)}{x} \, dx}{a^3}-\frac{b^3 \int \frac{\cosh (c+d x)}{a+b x} \, dx}{a^3}\\ &=-\frac{\cosh (c+d x)}{2 a x^2}+\frac{b \cosh (c+d x)}{a^2 x}+\frac{d \int \frac{\sinh (c+d x)}{x^2} \, dx}{2 a}-\frac{(b d) \int \frac{\sinh (c+d x)}{x} \, dx}{a^2}+\frac{\left (b^2 \cosh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx}{a^3}-\frac{\left (b^3 \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{\left (b^2 \sinh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx}{a^3}-\frac{\left (b^3 \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 x^2}+\frac{b \cosh (c+d x)}{a^2 x}+\frac{b^2 \cosh (c) \text{Chi}(d x)}{a^3}-\frac{b^2 \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 x}+\frac{b^2 \sinh (c) \text{Shi}(d x)}{a^3}-\frac{b^2 \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)}{x} \, dx}{2 a}-\frac{(b d \cosh (c)) \int \frac{\sinh (d x)}{x} \, dx}{a^2}-\frac{(b d \sinh (c)) \int \frac{\cosh (d x)}{x} \, dx}{a^2}\\ &=-\frac{\cosh (c+d x)}{2 a x^2}+\frac{b \cosh (c+d x)}{a^2 x}+\frac{b^2 \cosh (c) \text{Chi}(d x)}{a^3}-\frac{b^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{a^3}-\frac{b d \text{Chi}(d x) \sinh (c)}{a^2}-\frac{d \sinh (c+d x)}{2 a x}-\frac{b d \cosh (c) \text{Shi}(d x)}{a^2}+\frac{b^2 \sinh (c) \text{Shi}(d x)}{a^3}-\frac{b^2 \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 (c)\right ) \int \frac{\cosh (d x)}{x} \, dx}{2 a}+\frac{\left (d^2 \sinh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx}{2 a}\\ &=-\frac{\cosh (c+d x)}{2 a x^2}+\frac{b \cosh (c+d x)}{a^2 x}+\frac{b^2 \cosh (c) \text{Chi}(d x)}{a^3}+\frac{d^2 \cosh (c) \text{Chi}(d x)}{2 a}-\frac{b^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{a^3}-\frac{b d \text{Chi}(d x) \sinh (c)}{a^2}-\frac{d \sinh (c+d x)}{2 a x}-\frac{b d \cosh (c) \text{Shi}(d x)}{a^2}+\frac{b^2 \sinh (c) \text{Shi}(d x)}{a^3}+\frac{d^2 \sinh (c) \text{Shi}(d x)}{2 a}-\frac{b^2 \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 = 0.487479, size = 178, normalized size = 0.94 \[ \frac{x^2 \text{Chi}(d x) \left (\cosh (c) \left (a^2 d^2+2 b^2\right )-2 a b d \sinh (c)\right )+a^2 d^2 x^2 \sinh (c) \text{Shi}(d x)-a^2 d x \sinh (c+d x)+a^2 (-\cosh (c+d x))-2 b^2 x^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (d \left (\frac{a}{b}+x\right )\right )-2 b^2 x^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )-2 a b d x^2 \cosh (c) \text{Shi}(d x)+2 a b x \cosh (c+d x)+2 b^2 x^2 \sinh (c) \text{Shi}(d x)}{2 a^3 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.05, size = 281, normalized size = 1.5 \begin{align*}{\frac{d{{\rm e}^{-dx-c}}}{4\,ax}}+{\frac{{{\rm e}^{-dx-c}}b}{2\,{a}^{2}x}}-{\frac{{{\rm e}^{-dx-c}}}{4\,a{x}^{2}}}-{\frac{{d}^{2}{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{4\,a}}-{\frac{db{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2\,{a}^{2}}}-{\frac{{b}^{2}{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2\,{a}^{3}}}+{\frac{{b}^{2}}{2\,{a}^{3}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }-{\frac{{b}^{2}{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2\,{a}^{3}}}-{\frac{{{\rm e}^{dx+c}}}{4\,a{x}^{2}}}-{\frac{d{{\rm e}^{dx+c}}}{4\,ax}}-{\frac{{d}^{2}{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{4\,a}}+{\frac{b{{\rm e}^{dx+c}}}{2\,{a}^{2}x}}+{\frac{db{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2\,{a}^{2}}}+{\frac{{b}^{2}}{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^3/(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.62637, size = 327, normalized size = 1.72 \begin{align*} \frac{1}{4} \, d{\left (\frac{d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + d e^{c} \Gamma \left (-1, -d x\right )}{a} + \frac{2 \,{\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} -{\rm Ei}\left (d x\right ) e^{c}\right )} b}{a^{2}} + \frac{2 \, b^{3}{\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^{3} d} + \frac{4 \, b^{2} \cosh \left (d x + c\right ) \log \left (b x + a\right )}{a^{3} d} - \frac{4 \, b^{2} \cosh \left (d x + c\right ) \log \left (x\right )}{a^{3} d} + \frac{2 \,{\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} +{\rm Ei}\left (d x\right ) e^{c}\right )} b^{2}}{a^{3} d}\right )} - \frac{1}{2} \,{\left (\frac{2 \, b^{2} \log \left (b x + a\right )}{a^{3}} - \frac{2 \, b^{2} \log \left (x\right )}{a^{3}} - \frac{2 \, b x - a}{a^{2} x^{2}}\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^3/(b*x+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.08548, size = 591, normalized size = 3.11 \begin{align*} -\frac{2 \, a^{2} d x \sinh \left (d x + c\right ) - 2 \,{\left (2 \, a b x - a^{2}\right )} \cosh \left (d x + c\right ) -{\left ({\left (a^{2} d^{2} - 2 \, a b d + 2 \, b^{2}\right )} x^{2}{\rm Ei}\left (d x\right ) +{\left (a^{2} d^{2} + 2 \, a b d + 2 \, b^{2}\right )} x^{2}{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \,{\left (b^{2} x^{2}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) + b^{2} x^{2}{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \cosh \left (-\frac{b c - a d}{b}\right ) -{\left ({\left (a^{2} d^{2} - 2 \, a b d + 2 \, b^{2}\right )} x^{2}{\rm Ei}\left (d x\right ) -{\left (a^{2} d^{2} + 2 \, a b d + 2 \, b^{2}\right )} x^{2}{\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right ) - 2 \,{\left (b^{2} x^{2}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) - b^{2} x^{2}{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \sinh \left (-\frac{b c - a d}{b}\right )}{4 \, a^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Giac [A]  time = 1.20454, size = 335, normalized size = 1.76 \begin{align*} \frac{a^{2} d^{2} x^{2}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{2} x^{2}{\rm Ei}\left (d x\right ) e^{c} + 2 \, a b d x^{2}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 2 \, a b d x^{2}{\rm Ei}\left (d x\right ) e^{c} + 2 \, b^{2} x^{2}{\rm Ei}\left (-d x\right ) e^{\left (-c\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 \, b^{2} x^{2}{\rm Ei}\left (d x\right ) e^{c} - 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 e^{\left (d x + c\right )} + a^{2} d x e^{\left (-d x - c\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 )}}{4 \, a^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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