\(\int \frac {x \cosh (c+d x)}{a+b x} \, dx\) [21]

Optimal result

Integrand size = 15, antiderivative size = 68 \[ \int \frac {x \cosh (c+d x)}{a+b x} \, dx=-\frac {a \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^2}+\frac {\sinh (c+d x)}{b d}-\frac {a \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^2} \]

[Out]

-a*Chi(a*d/b+d*x)*cosh(-c+a*d/b)/b^2+a*Shi(a*d/b+d*x)*sinh(-c+a*d/b)/b^2+sinh(d*x+c)/b/d

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 68, 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} \[ \int \frac {x \cosh (c+d x)}{a+b x} \, dx=-\frac {a \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^2}-\frac {a \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^2}+\frac {\sinh (c+d x)}{b d} \]

[In]

Int[(x*Cosh[c + d*x])/(a + b*x),x]

[Out]

-((a*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/b^2) + Sinh[c + d*x]/(b*d) - (a*Sinh[c - (a*d)/b]*SinhInte
gral[(a*d)/b + d*x])/b^2

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.94 \[ \int \frac {x \cosh (c+d x)}{a+b x} \, dx=\frac {-a d \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (d \left (\frac {a}{b}+x\right )\right )+b \sinh (c+d x)-a d \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )}{b^2 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.68

method	result	size
risch	\(\frac {{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right ) a}{2 b^{2}}+\frac {{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right ) a}{2 b^{2}}-\frac {{\mathrm e}^{-d x -c}}{2 d b}+\frac {{\mathrm e}^{d x +c}}{2 d b}\)	\(114\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.74 \[ \int \frac {x \cosh (c+d x)}{a+b x} \, dx=-\frac {{\left (a d {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + a d {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) - 2 \, b \sinh \left (d x + c\right ) - {\left (a d {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - a d {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{2 \, b^{2} d} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (69) = 138\).

Time = 0.23 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.29 \[ \int \frac {x \cosh (c+d x)}{a+b x} \, dx=\frac {1}{2} \, d {\left (\frac {a {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b} + \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b}\right )}}{b d} - \frac {\frac {{\left (d x e^{c} - e^{c}\right )} e^{\left (d x\right )}}{d^{2}} + \frac {{\left (d x + 1\right )} e^{\left (-d x - c\right )}}{d^{2}}}{b} + \frac {2 \, a \cosh \left (d x + c\right ) \log \left (b x + a\right )}{b^{2} d}\right )} + {\left (\frac {x}{b} - \frac {a \log \left (b x + a\right )}{b^{2}}\right )} \cosh \left (d x + c\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.22 \[ \int \frac {x \cosh (c+d x)}{a+b x} \, dx=-\frac {a d {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + a d {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} - b e^{\left (d x + c\right )} + b e^{\left (-d x - c\right )}}{2 \, b^{2} d} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((x*cosh(c + d*x))/(a + b*x),x)

[Out]

int((x*cosh(c + d*x))/(a + b*x), x)