\(\int \frac {\cosh (c+d x)}{x^2 (a+b x)} \, dx\) [24]

Optimal result

Integrand size = 17, antiderivative size = 113 \[ \int \frac {\cosh (c+d x)}{x^2 (a+b x)} \, dx=-\frac {\cosh (c+d x)}{a x}-\frac {b \cosh (c) \text {Chi}(d x)}{a^2}+\frac {b \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{a^2}+\frac {d \text {Chi}(d x) \sinh (c)}{a}+\frac {d \cosh (c) \text {Shi}(d x)}{a}-\frac {b \sinh (c) \text {Shi}(d x)}{a^2}+\frac {b \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6874, 3378, 3384, 3379, 3382} \[ \int \frac {\cosh (c+d x)}{x^2 (a+b x)} \, dx=-\frac {b \cosh (c) \text {Chi}(d x)}{a^2}+\frac {b \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{a^2}-\frac {b \sinh (c) \text {Shi}(d x)}{a^2}+\frac {b \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{a^2}+\frac {d \sinh (c) \text {Chi}(d x)}{a}+\frac {d \cosh (c) \text {Shi}(d x)}{a}-\frac {\cosh (c+d x)}{a x} \]

[In]

Int[Cosh[c + d*x]/(x^2*(a + b*x)),x]

[Out]

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

Rule 3378

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m
+ 1))), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

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

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.89 \[ \int \frac {\cosh (c+d x)}{x^2 (a+b x)} \, dx=\frac {-a \cosh (c+d x)+b x \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (d \left (\frac {a}{b}+x\right )\right )+\text {Chi}(d x) (-b x \cosh (c)+a d x \sinh (c))+a d x \cosh (c) \text {Shi}(d x)-b x \sinh (c) \text {Shi}(d x)+b x \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )}{a^2 x} \]

[In]

Integrate[Cosh[c + d*x]/(x^2*(a + b*x)),x]

[Out]

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

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.37

method	result	size
risch	\(-\frac {{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a d x -{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a d x -{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) b x +b \,{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right ) x -{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) b x +b \,{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right ) x +{\mathrm e}^{-d x -c} a +a \,{\mathrm e}^{d x +c}}{2 a^{2} x}\)	\(155\)

[In]

int(cosh(d*x+c)/x^2/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.58 \[ \int \frac {\cosh (c+d x)}{x^2 (a+b x)} \, dx=-\frac {2 \, a \cosh \left (d x + c\right ) - {\left ({\left (a d - b\right )} x {\rm Ei}\left (d x\right ) - {\left (a d + b\right )} x {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) - {\left (b x {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + b x {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) - {\left ({\left (a d - b\right )} x {\rm Ei}\left (d x\right ) + {\left (a d + b\right )} x {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right ) + {\left (b x {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - b x {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{2 \, a^{2} x} \]

[In]

integrate(cosh(d*x+c)/x^2/(b*x+a),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\cosh (c+d x)}{x^2 (a+b x)} \, dx=\int \frac {\cosh {\left (c + d x \right )}}{x^{2} \left (a + b x\right )}\, dx \]

[In]

integrate(cosh(d*x+c)/x**2/(b*x+a),x)

[Out]

Integral(cosh(c + d*x)/(x**2*(a + b*x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.70 \[ \int \frac {\cosh (c+d x)}{x^2 (a+b x)} \, dx=-\frac {1}{2} \, d {\left (\frac {{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - {\rm Ei}\left (d x\right ) e^{c}}{a} + \frac {b^{2} {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b} + \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b}\right )}}{a^{2} d} + \frac {2 \, b \cosh \left (d x + c\right ) \log \left (b x + a\right )}{a^{2} d} - \frac {2 \, b \cosh \left (d x + c\right ) \log \left (x\right )}{a^{2} d} + \frac {{\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} b}{a^{2} d}\right )} + {\left (\frac {b \log \left (b x + a\right )}{a^{2}} - \frac {b \log \left (x\right )}{a^{2}} - \frac {1}{a x}\right )} \cosh \left (d x + c\right ) \]

[In]

integrate(cosh(d*x+c)/x^2/(b*x+a),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.14 \[ \int \frac {\cosh (c+d x)}{x^2 (a+b x)} \, dx=-\frac {a d x {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a d x {\rm Ei}\left (d x\right ) e^{c} + b x {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - b x {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + b x {\rm Ei}\left (d x\right ) e^{c} - b x {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + a e^{\left (d x + c\right )} + a e^{\left (-d x - c\right )}}{2 \, a^{2} x} \]

[In]

integrate(cosh(d*x+c)/x^2/(b*x+a),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\cosh (c+d x)}{x^2 (a+b x)} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )}{x^2\,\left (a+b\,x\right )} \,d x \]

[In]

int(cosh(c + d*x)/(x^2*(a + b*x)),x)

[Out]

int(cosh(c + d*x)/(x^2*(a + b*x)), x)