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

Optimal. Leaf size=28 \[ a \cosh (c) \text{Chi}(d x)+a \sinh (c) \text{Shi}(d x)+\frac{b \sinh (c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.148764, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6742, 2637, 3303, 3298, 3301} \[ a \cosh (c) \text{Chi}(d x)+a \sinh (c) \text{Shi}(d x)+\frac{b \sinh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*Cosh[c + d*x])/x,x]

[Out]

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

Rule 6742

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

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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

Mathematica [A]  time = 0.0279314, size = 39, normalized size = 1.39 \[ a \cosh (c) \text{Chi}(d x)+a \sinh (c) \text{Shi}(d x)+\frac{b \sinh (c) \cosh (d x)}{d}+\frac{b \cosh (c) \sinh (d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.038, size = 52, normalized size = 1.9 \begin{align*} -{\frac{a{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2}}-{\frac{b{{\rm e}^{-dx-c}}}{2\,d}}-{\frac{a{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2}}+{\frac{b{{\rm e}^{dx+c}}}{2\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.20109, size = 131, normalized size = 4.68 \begin{align*} -\frac{1}{2} \,{\left (b{\left (\frac{{\left (d x e^{c} - e^{c}\right )} e^{\left (d x\right )}}{d^{2}} + \frac{{\left (d x + 1\right )} e^{\left (-d x - c\right )}}{d^{2}}\right )} + \frac{2 \, a \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 )} a}{d}\right )} d +{\left (b x + a \log \left (x\right )\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,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.96747, size = 142, normalized size = 5.07 \begin{align*} \frac{{\left (a d{\rm Ei}\left (d x\right ) + a d{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \, b \sinh \left (d x + c\right ) +{\left (a d{\rm Ei}\left (d x\right ) - a d{\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 4.54689, size = 34, normalized size = 1.21 \begin{align*} a \sinh{\left (c \right )} \operatorname{Shi}{\left (d x \right )} + a \cosh{\left (c \right )} \operatorname{Chi}\left (d x\right ) + b \left (\begin{cases} \frac{\sinh{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \cosh{\left (c \right )} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*sinh(c)*Shi(d*x) + a*cosh(c)*Chi(d*x) + b*Piecewise((sinh(c + d*x)/d, Ne(d, 0)), (x*cosh(c), True))

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Giac [A]  time = 1.1822, size = 63, normalized size = 2.25 \begin{align*} \frac{a d{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a d{\rm Ei}\left (d x\right ) e^{c} + b e^{\left (d x + c\right )} - b e^{\left (-d x - c\right )}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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