\(\int \frac {(a+b x) \cosh (c+d x)}{x} \, dx\) [5]

Optimal result

Integrand size = 15, antiderivative size = 28 \[ \int \frac {(a+b x) \cosh (c+d x)}{x} \, dx=a \cosh (c) \text {Chi}(d x)+\frac {b \sinh (c+d x)}{d}+a \sinh (c) \text {Shi}(d x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6874, 2717, 3384, 3379, 3382} \[ \int \frac {(a+b x) \cosh (c+d x)}{x} \, dx=a \cosh (c) \text {Chi}(d x)+a \sinh (c) \text {Shi}(d x)+\frac {b \sinh (c+d x)}{d} \]

[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 2717

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

Rule 3379

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 3382

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 3384

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 6874

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

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.06 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {(a+b x) \cosh (c+d x)}{x} \, dx=a \cosh (c) \text {Chi}(d x)+\frac {b \cosh (d x) \sinh (c)}{d}+\frac {b \cosh (c) \sinh (d x)}{d}+a \sinh (c) \text {Shi}(d x) \]

[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]

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86

method	result	size
risch	\(-\frac {a \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right )}{2}-\frac {a \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right )}{2}-\frac {{\mathrm e}^{-d x -c} b}{2 d}+\frac {{\mathrm e}^{d x +c} b}{2 d}\)	\(52\)
meijerg	\(\frac {b \cosh \left (c \right ) \sinh \left (d x \right )}{d}-\frac {b \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (d x \right )}{\sqrt {\pi }}\right )}{d}+\frac {a \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {2 \gamma +2 \ln \left (x \right )+2 \ln \left (i d \right )}{\sqrt {\pi }}+\frac {2 \,\operatorname {Chi}\left (d x \right )-2 \ln \left (d x \right )-2 \gamma }{\sqrt {\pi }}\right )}{2}+a \,\operatorname {Shi}\left (d x \right ) \sinh \left (c \right )\)	\(92\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.93 \[ \int \frac {(a+b x) \cosh (c+d x)}{x} \, dx=\frac {{\left (a d {\rm Ei}\left (d x\right ) + a d {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \, b \sinh \left (d x + c\right ) + {\left (a d {\rm Ei}\left (d x\right ) - a d {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{2 \, d} \]

[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

Sympy [A] (verification not implemented)

Time = 1.48 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {(a+b x) \cosh (c+d x)}{x} \, dx=- a \left (- \sinh {\left (c \right )} \operatorname {Shi}{\left (d x \right )} - \cosh {\left (c \right )} \operatorname {Chi}\left (d x\right )\right ) - b \left (\begin {cases} - x \cosh {\left (c \right )} & \text {for}\: d = 0 \\- \frac {\sinh {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (28) = 56\).

Time = 0.27 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.46 \[ \int \frac {(a+b x) \cosh (c+d x)}{x} \, dx=-\frac {1}{2} \, {\left (b {\left (\frac {{\left (d x e^{c} - e^{c}\right )} e^{\left (d x\right )}}{d^{2}} + \frac {{\left (d x + 1\right )} e^{\left (-d x - c\right )}}{d^{2}}\right )} + \frac {2 \, a \cosh \left (d x + c\right ) \log \left (x\right )}{d} - \frac {{\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} a}{d}\right )} d + {\left (b x + a \log \left (x\right )\right )} \cosh \left (d x + c\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.49 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.68 \[ \int \frac {(a+b x) \cosh (c+d x)}{x} \, dx=\frac {a d {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a d {\rm Ei}\left (d x\right ) e^{c} + b e^{\left (d x + c\right )} - b e^{\left (-d x - c\right )}}{2 \, d} \]

[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

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x) \cosh (c+d x)}{x} \, dx=a\,\mathrm {coshint}\left (d\,x\right )\,\mathrm {cosh}\left (c\right )+a\,\mathrm {sinhint}\left (d\,x\right )\,\mathrm {sinh}\left (c\right )+\frac {b\,\mathrm {sinh}\left (c+d\,x\right )}{d} \]

[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