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

Optimal. Leaf size=47 \[ a d \sinh (c) \text{Chi}(d x)+a d \cosh (c) \text{Shi}(d x)-\frac{a \cosh (c+d x)}{x}+b \cosh (c) \text{Chi}(d x)+b \sinh (c) \text{Shi}(d x) \]

[Out]

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

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Rubi [A]  time = 0.227998, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6742, 3297, 3303, 3298, 3301} \[ a d \sinh (c) \text{Chi}(d x)+a d \cosh (c) \text{Shi}(d x)-\frac{a \cosh (c+d x)}{x}+b \cosh (c) \text{Chi}(d x)+b \sinh (c) \text{Shi}(d x) \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*Cosh[c + d*x])/x^2,x]

[Out]

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

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

Mathematica [A]  time = 0.127054, size = 59, normalized size = 1.26 \[ a d (\sinh (c) \text{Chi}(d x)+\cosh (c) \text{Shi}(d x))-\frac{a \sinh (c) \sinh (d x)}{x}-\frac{a \cosh (c) \cosh (d x)}{x}+b \cosh (c) \text{Chi}(d x)+b \sinh (c) \text{Shi}(d x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.039, size = 77, normalized size = 1.6 \begin{align*} -{\frac{a{{\rm e}^{-dx-c}}}{2\,x}}+{\frac{da{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2}}-{\frac{b{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2}}-{\frac{a{{\rm e}^{dx+c}}}{2\,x}}-{\frac{da{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2}}-{\frac{b{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.39651, size = 111, normalized size = 2.36 \begin{align*} -\frac{1}{2} \,{\left ({\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} -{\rm Ei}\left (d x\right ) e^{c}\right )} a + \frac{2 \, b \cosh \left (d x + c\right ) \log \left (x\right )}{d} - \frac{{\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} +{\rm Ei}\left (d x\right ) e^{c}\right )} b}{d}\right )} d +{\left (b \log \left (x\right ) - \frac{a}{x}\right )} \cosh \left (d x + c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.96812, size = 186, normalized size = 3.96 \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 ({\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 )}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15134, size = 97, normalized size = 2.06 \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 (d x\right ) e^{c} + a e^{\left (d x + c\right )} + a e^{\left (-d x - c\right )}}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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