\(\int \frac {\cosh (c+d x)}{x^2 (a+b x)^2} \, dx\) [32]

Optimal result

Integrand size = 17, antiderivative size = 186 \[ \int \frac {\cosh (c+d x)}{x^2 (a+b x)^2} \, dx=-\frac {\cosh (c+d x)}{a^2 x}-\frac {b \cosh (c+d x)}{a^2 (a+b x)}-\frac {2 b \cosh (c) \text {Chi}(d x)}{a^3}+\frac {2 b \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{a^3}+\frac {d \text {Chi}(d x) \sinh (c)}{a^2}+\frac {d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{a^2}+\frac {d \cosh (c) \text {Shi}(d x)}{a^2}-\frac {2 b \sinh (c) \text {Shi}(d x)}{a^3}+\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^2}+\frac {2 b \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^3} \]

[Out]

-2*b*Chi(d*x)*cosh(c)/a^3+2*b*Chi(a*d/b+d*x)*cosh(-c+a*d/b)/a^3-cosh(d*x+c)/a^2/x-b*cosh(d*x+c)/a^2/(b*x+a)+d*
cosh(c)*Shi(d*x)/a^2+d*cosh(-c+a*d/b)*Shi(a*d/b+d*x)/a^2+d*Chi(d*x)*sinh(c)/a^2-2*b*Shi(d*x)*sinh(c)/a^3-d*Chi
(a*d/b+d*x)*sinh(-c+a*d/b)/a^2-2*b*Shi(a*d/b+d*x)*sinh(-c+a*d/b)/a^3

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6874, 3378, 3384, 3379, 3382} \[ \int \frac {\cosh (c+d x)}{x^2 (a+b x)^2} \, dx=-\frac {2 b \cosh (c) \text {Chi}(d x)}{a^3}+\frac {2 b \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{a^3}-\frac {2 b \sinh (c) \text {Shi}(d x)}{a^3}+\frac {2 b \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{a^3}+\frac {d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{a^2}+\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{a^2}-\frac {b \cosh (c+d x)}{a^2 (a+b x)}+\frac {d \sinh (c) \text {Chi}(d x)}{a^2}+\frac {d \cosh (c) \text {Shi}(d x)}{a^2}-\frac {\cosh (c+d x)}{a^2 x} \]

[In]

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

[Out]

-(Cosh[c + d*x]/(a^2*x)) - (b*Cosh[c + d*x])/(a^2*(a + b*x)) - (2*b*Cosh[c]*CoshIntegral[d*x])/a^3 + (2*b*Cosh
[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/a^3 + (d*CoshIntegral[d*x]*Sinh[c])/a^2 + (d*CoshIntegral[(a*d)/b +
d*x]*Sinh[c - (a*d)/b])/a^2 + (d*Cosh[c]*SinhIntegral[d*x])/a^2 - (2*b*Sinh[c]*SinhIntegral[d*x])/a^3 + (d*Co
sh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/a^2 + (2*b*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/a^3

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^2 x^2}-\frac {2 b \cosh (c+d x)}{a^3 x}+\frac {b^2 \cosh (c+d x)}{a^2 (a+b x)^2}+\frac {2 b^2 \cosh (c+d x)}{a^3 (a+b x)}\right ) \, dx \\ & = \frac {\int \frac {\cosh (c+d x)}{x^2} \, dx}{a^2}-\frac {(2 b) \int \frac {\cosh (c+d x)}{x} \, dx}{a^3}+\frac {\left (2 b^2\right ) \int \frac {\cosh (c+d x)}{a+b x} \, dx}{a^3}+\frac {b^2 \int \frac {\cosh (c+d x)}{(a+b x)^2} \, dx}{a^2} \\ & = -\frac {\cosh (c+d x)}{a^2 x}-\frac {b \cosh (c+d x)}{a^2 (a+b x)}+\frac {d \int \frac {\sinh (c+d x)}{x} \, dx}{a^2}+\frac {(b d) \int \frac {\sinh (c+d x)}{a+b x} \, dx}{a^2}-\frac {(2 b \cosh (c)) \int \frac {\cosh (d x)}{x} \, dx}{a^3}+\frac {\left (2 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^3}-\frac {(2 b \sinh (c)) \int \frac {\sinh (d x)}{x} \, dx}{a^3}+\frac {\left (2 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^3} \\ & = -\frac {\cosh (c+d x)}{a^2 x}-\frac {b \cosh (c+d x)}{a^2 (a+b x)}-\frac {2 b \cosh (c) \text {Chi}(d x)}{a^3}+\frac {2 b \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{a^3}-\frac {2 b \sinh (c) \text {Shi}(d x)}{a^3}+\frac {2 b \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^3}+\frac {(d \cosh (c)) \int \frac {\sinh (d x)}{x} \, dx}{a^2}+\frac {\left (b d \cosh \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 {(d \sinh (c)) \int \frac {\cosh (d x)}{x} \, dx}{a^2}+\frac {\left (b d \sinh \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 {\cosh (c+d x)}{a^2 x}-\frac {b \cosh (c+d x)}{a^2 (a+b x)}-\frac {2 b \cosh (c) \text {Chi}(d x)}{a^3}+\frac {2 b \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{a^3}+\frac {d \text {Chi}(d x) \sinh (c)}{a^2}+\frac {d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{a^2}+\frac {d \cosh (c) \text {Shi}(d x)}{a^2}-\frac {2 b \sinh (c) \text {Shi}(d x)}{a^3}+\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^2}+\frac {2 b \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.92 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.98 \[ \int \frac {\cosh (c+d x)}{x^2 (a+b x)^2} \, dx=\frac {-\frac {a (a+2 b x) \cosh (c) \cosh (d x)}{x (a+b x)}-2 b \cosh (c) \text {Chi}(d x)+2 b \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (d \left (\frac {a}{b}+x\right )\right )+a d \text {Chi}(d x) \sinh (c)+a d \text {Chi}\left (d \left (\frac {a}{b}+x\right )\right ) \sinh \left (c-\frac {a d}{b}\right )-\frac {a (a+2 b x) \sinh (c) \sinh (d x)}{x (a+b x)}+a d \cosh (c) \text {Shi}(d x)-2 b \sinh (c) \text {Shi}(d x)+a d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+2 b \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )}{a^3} \]

[In]

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

[Out]

(-((a*(a + 2*b*x)*Cosh[c]*Cosh[d*x])/(x*(a + b*x))) - 2*b*Cosh[c]*CoshIntegral[d*x] + 2*b*Cosh[c - (a*d)/b]*Co
shIntegral[d*(a/b + x)] + a*d*CoshIntegral[d*x]*Sinh[c] + a*d*CoshIntegral[d*(a/b + x)]*Sinh[c - (a*d)/b] - (a
*(a + 2*b*x)*Sinh[c]*Sinh[d*x])/(x*(a + b*x)) + a*d*Cosh[c]*SinhIntegral[d*x] - 2*b*Sinh[c]*SinhIntegral[d*x]
+ a*d*Cosh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)] + 2*b*Sinh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)])/a^3

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.68

method	result	size
risch	\(-\frac {d \,{\mathrm e}^{-d x -c} b}{a^{2} \left (d x b +d a \right )}-\frac {d \,{\mathrm e}^{-d x -c}}{2 a x \left (d x b +d a \right )}+\frac {d \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right )}{2 a^{2}}+\frac {{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) b}{a^{3}}+\frac {d \,{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right )}{2 a^{2}}-\frac {{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right ) b}{a^{3}}-\frac {{\mathrm e}^{d x +c}}{2 a^{2} x}-\frac {d \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right )}{2 a^{2}}+\frac {b \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right )}{a^{3}}-\frac {d \,{\mathrm e}^{d x +c}}{2 a^{2} \left (\frac {d a}{b}+d x \right )}-\frac {d \,{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right )}{2 a^{2}}-\frac {b \,{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right )}{a^{3}}\)	\(312\)

[In]

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

[Out]

-d*exp(-d*x-c)/a^2/(b*d*x+a*d)*b-1/2*d*exp(-d*x-c)/a/x/(b*d*x+a*d)+1/2*d/a^2*exp(-c)*Ei(1,d*x)+1/a^3*exp(-c)*E
i(1,d*x)*b+1/2*d/a^2*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)-1/a^3*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*b
-1/2/a^2/x*exp(d*x+c)-1/2*d/a^2*exp(c)*Ei(1,-d*x)+1/a^3*b*exp(c)*Ei(1,-d*x)-1/2*d/a^2*exp(d*x+c)/(d/b*a+d*x)-1
/2*d/a^2*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)-b/a^3*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 = 377, normalized size of antiderivative = 2.03 \[ \int \frac {\cosh (c+d x)}{x^2 (a+b x)^2} \, dx=-\frac {2 \, {\left (2 \, a b x + a^{2}\right )} \cosh \left (d x + c\right ) - {\left ({\left ({\left (a b d - 2 \, b^{2}\right )} x^{2} + {\left (a^{2} d - 2 \, a b\right )} x\right )} {\rm Ei}\left (d x\right ) - {\left ({\left (a b d + 2 \, b^{2}\right )} x^{2} + {\left (a^{2} d + 2 \, a b\right )} x\right )} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) - {\left ({\left ({\left (a b d + 2 \, b^{2}\right )} x^{2} + {\left (a^{2} d + 2 \, a b\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\left ({\left (a b d - 2 \, b^{2}\right )} x^{2} + {\left (a^{2} d - 2 \, a b\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) - {\left ({\left ({\left (a b d - 2 \, b^{2}\right )} x^{2} + {\left (a^{2} d - 2 \, a b\right )} x\right )} {\rm Ei}\left (d x\right ) + {\left ({\left (a b d + 2 \, b^{2}\right )} x^{2} + {\left (a^{2} d + 2 \, a b\right )} x\right )} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right ) + {\left ({\left ({\left (a b d + 2 \, b^{2}\right )} x^{2} + {\left (a^{2} d + 2 \, a b\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\left ({\left (a b d - 2 \, b^{2}\right )} x^{2} + {\left (a^{2} d - 2 \, a b\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{2 \, {\left (a^{3} b x^{2} + a^{4} x\right )}} \]

[In]

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

[Out]

-1/2*(2*(2*a*b*x + a^2)*cosh(d*x + c) - (((a*b*d - 2*b^2)*x^2 + (a^2*d - 2*a*b)*x)*Ei(d*x) - ((a*b*d + 2*b^2)*
x^2 + (a^2*d + 2*a*b)*x)*Ei(-d*x))*cosh(c) - (((a*b*d + 2*b^2)*x^2 + (a^2*d + 2*a*b)*x)*Ei((b*d*x + a*d)/b) -
((a*b*d - 2*b^2)*x^2 + (a^2*d - 2*a*b)*x)*Ei(-(b*d*x + a*d)/b))*cosh(-(b*c - a*d)/b) - (((a*b*d - 2*b^2)*x^2 +
(a^2*d - 2*a*b)*x)*Ei(d*x) + ((a*b*d + 2*b^2)*x^2 + (a^2*d + 2*a*b)*x)*Ei(-d*x))*sinh(c) + (((a*b*d + 2*b^2)*
x^2 + (a^2*d + 2*a*b)*x)*Ei((b*d*x + a*d)/b) + ((a*b*d - 2*b^2)*x^2 + (a^2*d - 2*a*b)*x)*Ei(-(b*d*x + a*d)/b))
*sinh(-(b*c - a*d)/b))/(a^3*b*x^2 + a^4*x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3353 vs. \(2 (191) = 382\).

Time = 0.33 (sec) , antiderivative size = 3353, normalized size of antiderivative = 18.03 \[ \int \frac {\cosh (c+d x)}{x^2 (a+b x)^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/2*((b*x + a)^2*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*d^2*Ei(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d
)/b + c)*e^(-c)/b - 2*(b*x + a)*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)*c*d^2*Ei(-(b*x + a)*(b*c/(b*x + a) - a*d
/(b*x + a) + d)/b + c)*e^(-c) + a*b*c^2*d^2*Ei(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b + c)*e^(-c) +
(b*x + a)*a^2*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^3*Ei(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b + c)
*e^(-c)/b - a^2*c*d^3*Ei(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b + c)*e^(-c) - (b*x + a)^2*a*(b*c/(b*
x + a) - a*d/(b*x + a) + d)^2*d^2*Ei((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b - c)*e^c/b + 2*(b*x + a)*
a*(b*c/(b*x + a) - a*d/(b*x + a) + d)*c*d^2*Ei((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b - c)*e^c - a*b*
c^2*d^2*Ei((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b - c)*e^c - (b*x + a)*a^2*(b*c/(b*x + a) - a*d/(b*x
+ a) + d)*d^3*Ei((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b - c)*e^c/b + a^2*c*d^3*Ei((b*x + a)*(b*c/(b*x
+ a) - a*d/(b*x + a) + d)/b - c)*e^c - (b*x + a)^2*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*d^2*Ei(((b*x + a)*
(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*e^((b*c - a*d)/b)/b + 2*(b*x + a)*a*(b*c/(b*x + a) - a*d/(
b*x + a) + d)*c*d^2*Ei(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*e^((b*c - a*d)/b) - a*b*
c^2*d^2*Ei(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*e^((b*c - a*d)/b) - (b*x + a)*a^2*(b
*c/(b*x + a) - a*d/(b*x + a) + d)*d^3*Ei(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*e^((b*
c - a*d)/b)/b + a^2*c*d^3*Ei(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*e^((b*c - a*d)/b)
+ (b*x + a)^2*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*d^2*Ei(-((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) -
b*c + a*d)/b)*e^(-(b*c - a*d)/b)/b - 2*(b*x + a)*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)*c*d^2*Ei(-((b*x + a)*(
b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*e^(-(b*c - a*d)/b) + a*b*c^2*d^2*Ei(-((b*x + a)*(b*c/(b*x +
a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*e^(-(b*c - a*d)/b) + (b*x + a)*a^2*(b*c/(b*x + a) - a*d/(b*x + a) + d
)*d^3*Ei(-((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*e^(-(b*c - a*d)/b)/b - a^2*c*d^3*Ei(-
((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*e^(-(b*c - a*d)/b) + 2*(b*x + a)^2*(b*c/(b*x +
a) - a*d/(b*x + a) + d)^2*d*Ei(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b + c)*e^(-c) - 4*(b*x + a)*b*(b
*c/(b*x + a) - a*d/(b*x + a) + d)*c*d*Ei(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b + c)*e^(-c) + 2*b^2*
c^2*d*Ei(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b + c)*e^(-c) + 2*(b*x + a)*a*(b*c/(b*x + a) - a*d/(b*
x + a) + d)*d^2*Ei(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b + c)*e^(-c) - 2*a*b*c*d^2*Ei(-(b*x + a)*(b
*c/(b*x + a) - a*d/(b*x + a) + d)/b + c)*e^(-c) + 2*(b*x + a)^2*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*d*Ei((b*
x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b - c)*e^c - 4*(b*x + a)*b*(b*c/(b*x + a) - a*d/(b*x + a) + d)*c*d*
Ei((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b - c)*e^c + 2*b^2*c^2*d*Ei((b*x + a)*(b*c/(b*x + a) - a*d/(b
*x + a) + d)/b - c)*e^c + 2*(b*x + a)*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^2*Ei((b*x + a)*(b*c/(b*x + a) -
a*d/(b*x + a) + d)/b - c)*e^c - 2*a*b*c*d^2*Ei((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b - c)*e^c - 2*(b
*x + a)^2*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*d*Ei(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*
d)/b)*e^((b*c - a*d)/b) + 4*(b*x + a)*b*(b*c/(b*x + a) - a*d/(b*x + a) + d)*c*d*Ei(((b*x + a)*(b*c/(b*x + a) -
a*d/(b*x + a) + d) - b*c + a*d)/b)*e^((b*c - a*d)/b) - 2*b^2*c^2*d*Ei(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x +
a) + d) - b*c + a*d)/b)*e^((b*c - a*d)/b) - 2*(b*x + a)*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^2*Ei(((b*x + a
)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*e^((b*c - a*d)/b) + 2*a*b*c*d^2*Ei(((b*x + a)*(b*c/(b*x
+ a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*e^((b*c - a*d)/b) - 2*(b*x + a)^2*(b*c/(b*x + a) - a*d/(b*x + a) + d
)^2*d*Ei(-((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*e^(-(b*c - a*d)/b) + 4*(b*x + a)*b*(b
*c/(b*x + a) - a*d/(b*x + a) + d)*c*d*Ei(-((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*e^(-(
b*c - a*d)/b) - 2*b^2*c^2*d*Ei(-((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*e^(-(b*c - a*d)
/b) - 2*(b*x + a)*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^2*Ei(-((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)
- b*c + a*d)/b)*e^(-(b*c - a*d)/b) + 2*a*b*c*d^2*Ei(-((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a
*d)/b)*e^(-(b*c - a*d)/b) + 2*(b*x + a)*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^2*e^((b*x + a)*(b*c/(b*x + a)
- a*d/(b*x + a) + d)/b) - 2*a*b*c*d^2*e^((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + a^2*d^3*e^((b*x +
a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + 2*(b*x + a)*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^2*e^(-(b*x + a
)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) - 2*a*b*c*d^2*e^(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) +
a^2*d^3*e^(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b))*b^2/(((b*x + a)^2*a^3*b*(b*c/(b*x + a) - a*d/(b*
x + a) + d)^2 - 2*(b*x + a)*a^3*b^2*(b*c/(b*x + a) - a*d/(b*x + a) + d)*c + a^3*b^3*c^2 + (b*x + a)*a^4*b*(b*c
/(b*x + a) - a*d/(b*x + a) + d)*d - a^4*b^2*c*d)*d)

Mupad [F(-1)]

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

[In]

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

[Out]

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