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*Chi(d*x)*cosh(c)+a*Shi(d*x)*sinh(c)+b*sinh(d*x+c)/d

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Rubi [A]  time = 0.15, antiderivative size = 28, normalized size of antiderivative = 1.00, 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 2637

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

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]

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 6742

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

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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 0.76, size = 54, normalized size = 1.93 \[ \frac {{\left (a d {\rm Ei}\left (d x\right ) + a d {\rm Ei}\left (-d x\right )\right )} \cosh \relax (c) + 2 \, b \sinh \left (d x + c\right ) + {\left (a d {\rm Ei}\left (d x\right ) - a d {\rm Ei}\left (-d x\right )\right )} \sinh \relax (c)}{2 \, d} \]

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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giac [A]  time = 0.14, size = 47, normalized size = 1.68 \[ \frac {a d {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a d {\rm Ei}\left (d x\right ) e^{c} + b e^{\left (d x + c\right )} - b e^{\left (-d x - c\right )}}{2 \, d} \]

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

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maple [A]  time = 0.09, size = 52, normalized size = 1.86 \[ -\frac {a \,{\mathrm e}^{-c} \Ei \left (1, d x \right )}{2}-\frac {b \,{\mathrm e}^{-d x -c}}{2 d}-\frac {a \,{\mathrm e}^{c} \Ei \left (1, -d x \right )}{2}+\frac {b \,{\mathrm e}^{d x +c}}{2 d} \]

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 = 0.53, size = 97, normalized size = 3.46 \[ -\frac {1}{2} \, {\left (b {\left (\frac {{\left (d x e^{c} - e^{c}\right )} e^{\left (d x\right )}}{d^{2}} + \frac {{\left (d x + 1\right )} e^{\left (-d x - c\right )}}{d^{2}}\right )} + \frac {2 \, a \cosh \left (d x + c\right ) \log \relax (x)}{d} - \frac {{\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} a}{d}\right )} d + {\left (b x + a \log \relax (x)\right )} \cosh \left (d x + c\right ) \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ a\,\mathrm {coshint}\left (d\,x\right )\,\mathrm {cosh}\relax (c)+a\,\mathrm {sinhint}\left (d\,x\right )\,\mathrm {sinh}\relax (c)+\frac {b\,\mathrm {sinh}\left (c+d\,x\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(c + d*x)*(a + b*x))/x,x)

[Out]

a*coshint(d*x)*cosh(c) + a*sinhint(d*x)*sinh(c) + (b*sinh(c + d*x))/d

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sympy [A]  time = 3.19, size = 34, normalized size = 1.21 \[ a \sinh {\relax (c )} \operatorname {Shi}{\left (d x \right )} + a \cosh {\relax (c )} \operatorname {Chi}\left (d x\right ) + b \left (\begin {cases} \frac {\sinh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \cosh {\relax (c )} & \text {otherwise} \end {cases}\right ) \]

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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