\(\int \frac {\cosh (c+d x)}{a+b x} \, dx\) [22]

Optimal result

Integrand size = 14, antiderivative size = 51 \[ \int \frac {\cosh (c+d x)}{a+b x} \, dx=\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b}+\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3384, 3379, 3382} \[ \int \frac {\cosh (c+d x)}{a+b x} \, dx=\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b}+\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b} \]

[In]

Int[Cosh[c + d*x]/(a + b*x),x]

[Out]

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

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]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.96 \[ \int \frac {\cosh (c+d x)}{a+b x} \, dx=\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )+\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b} \]

[In]

Integrate[Cosh[c + d*x]/(a + b*x),x]

[Out]

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

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.59

method	result	size
risch	\(-\frac {{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right )}{2 b}-\frac {{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right )}{2 b}\)	\(81\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.86 \[ \int \frac {\cosh (c+d x)}{a+b x} \, dx=\frac {{\left ({\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) - {\left ({\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{2 \, b} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(cosh(c + d*x)/(a + b*x), x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.12 \[ \int \frac {\cosh (c+d x)}{a+b x} \, dx=-\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{2 \, b} - \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{2 \, b} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.10 \[ \int \frac {\cosh (c+d x)}{a+b x} \, dx=\frac {{\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )}}{2 \, b} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(cosh(c + d*x)/(a + b*x),x)

[Out]

int(cosh(c + d*x)/(a + b*x), x)