\(\int \frac {x^3 \cosh (c+d x)}{(a+b x)^2} \, dx\) [27]

Optimal result

Integrand size = 17, antiderivative size = 182 \[ \int \frac {x^3 \cosh (c+d x)}{(a+b x)^2} \, dx=-\frac {\cosh (c+d x)}{b^2 d^2}+\frac {a^3 \cosh (c+d x)}{b^4 (a+b x)}+\frac {3 a^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^4}-\frac {a^3 d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{b^5}-\frac {2 a \sinh (c+d x)}{b^3 d}+\frac {x \sinh (c+d x)}{b^2 d}-\frac {a^3 d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^5}+\frac {3 a^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^4} \]

[Out]

3*a^2*Chi(a*d/b+d*x)*cosh(-c+a*d/b)/b^4-cosh(d*x+c)/b^2/d^2+a^3*cosh(d*x+c)/b^4/(b*x+a)-a^3*d*cosh(-c+a*d/b)*S
hi(a*d/b+d*x)/b^5+a^3*d*Chi(a*d/b+d*x)*sinh(-c+a*d/b)/b^5-3*a^2*Shi(a*d/b+d*x)*sinh(-c+a*d/b)/b^4-2*a*sinh(d*x
+c)/b^3/d+x*sinh(d*x+c)/b^2/d

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {6874, 2717, 3377, 2718, 3378, 3384, 3379, 3382} \[ \int \frac {x^3 \cosh (c+d x)}{(a+b x)^2} \, dx=-\frac {a^3 d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^5}-\frac {a^3 d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^5}+\frac {a^3 \cosh (c+d x)}{b^4 (a+b x)}+\frac {3 a^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^4}+\frac {3 a^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^4}-\frac {2 a \sinh (c+d x)}{b^3 d}-\frac {\cosh (c+d x)}{b^2 d^2}+\frac {x \sinh (c+d x)}{b^2 d} \]

[In]

Int[(x^3*Cosh[c + d*x])/(a + b*x)^2,x]

[Out]

-(Cosh[c + d*x]/(b^2*d^2)) + (a^3*Cosh[c + d*x])/(b^4*(a + b*x)) + (3*a^2*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)
/b + d*x])/b^4 - (a^3*d*CoshIntegral[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/b^5 - (2*a*Sinh[c + d*x])/(b^3*d) + (x*
Sinh[c + d*x])/(b^2*d) - (a^3*d*Cosh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^5 + (3*a^2*Sinh[c - (a*d)/b]*
SinhIntegral[(a*d)/b + d*x])/b^4

Rule 2717

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

Rule 2718

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

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

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 {2 a \cosh (c+d x)}{b^3}+\frac {x \cosh (c+d x)}{b^2}-\frac {a^3 \cosh (c+d x)}{b^3 (a+b x)^2}+\frac {3 a^2 \cosh (c+d x)}{b^3 (a+b x)}\right ) \, dx \\ & = -\frac {(2 a) \int \cosh (c+d x) \, dx}{b^3}+\frac {\left (3 a^2\right ) \int \frac {\cosh (c+d x)}{a+b x} \, dx}{b^3}-\frac {a^3 \int \frac {\cosh (c+d x)}{(a+b x)^2} \, dx}{b^3}+\frac {\int x \cosh (c+d x) \, dx}{b^2} \\ & = \frac {a^3 \cosh (c+d x)}{b^4 (a+b x)}-\frac {2 a \sinh (c+d x)}{b^3 d}+\frac {x \sinh (c+d x)}{b^2 d}-\frac {\int \sinh (c+d x) \, dx}{b^2 d}-\frac {\left (a^3 d\right ) \int \frac {\sinh (c+d x)}{a+b x} \, dx}{b^4}+\frac {\left (3 a^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}{b^3}+\frac {\left (3 a^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}{b^3} \\ & = -\frac {\cosh (c+d x)}{b^2 d^2}+\frac {a^3 \cosh (c+d x)}{b^4 (a+b x)}+\frac {3 a^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^4}-\frac {2 a \sinh (c+d x)}{b^3 d}+\frac {x \sinh (c+d x)}{b^2 d}+\frac {3 a^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^4}-\frac {\left (a^3 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}{b^4}-\frac {\left (a^3 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}{b^4} \\ & = -\frac {\cosh (c+d x)}{b^2 d^2}+\frac {a^3 \cosh (c+d x)}{b^4 (a+b x)}+\frac {3 a^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^4}-\frac {a^3 d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{b^5}-\frac {2 a \sinh (c+d x)}{b^3 d}+\frac {x \sinh (c+d x)}{b^2 d}-\frac {a^3 d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^5}+\frac {3 a^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.86 \[ \int \frac {x^3 \cosh (c+d x)}{(a+b x)^2} \, dx=\frac {a^2 \text {Chi}\left (d \left (\frac {a}{b}+x\right )\right ) \left (3 b \cosh \left (c-\frac {a d}{b}\right )-a d \sinh \left (c-\frac {a d}{b}\right )\right )+\frac {b \left (\left (-a b^2+a^3 d^2-b^3 x\right ) \cosh (c+d x)+b d \left (-2 a^2-a b x+b^2 x^2\right ) \sinh (c+d x)\right )}{d^2 (a+b x)}-a^2 \left (a d \cosh \left (c-\frac {a d}{b}\right )-3 b \sinh \left (c-\frac {a d}{b}\right )\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )}{b^5} \]

[In]

Integrate[(x^3*Cosh[c + d*x])/(a + b*x)^2,x]

[Out]

(a^2*CoshIntegral[d*(a/b + x)]*(3*b*Cosh[c - (a*d)/b] - a*d*Sinh[c - (a*d)/b]) + (b*((-(a*b^2) + a^3*d^2 - b^3
*x)*Cosh[c + d*x] + b*d*(-2*a^2 - a*b*x + b^2*x^2)*Sinh[c + d*x]))/(d^2*(a + b*x)) - a^2*(a*d*Cosh[c - (a*d)/b
] - 3*b*Sinh[c - (a*d)/b])*SinhIntegral[d*(a/b + x)])/b^5

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(548\) vs. \(2(185)=370\).

Time = 0.20 (sec) , antiderivative size = 549, normalized size of antiderivative = 3.02

method	result	size
risch	\(\frac {{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right ) a^{3} b \,d^{3} x -{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right ) a^{3} b \,d^{3} x +{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right ) a^{4} d^{3}-3 \,{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right ) a^{2} b^{2} d^{2} x -{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right ) a^{4} d^{3}-3 \,{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right ) a^{2} b^{2} d^{2} x -3 \,{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right ) a^{3} b \,d^{2}-{\mathrm e}^{-d x -c} b^{4} d \,x^{2}-3 \,{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right ) a^{3} b \,d^{2}+{\mathrm e}^{d x +c} b^{4} d \,x^{2}+{\mathrm e}^{-d x -c} a^{3} b \,d^{2}+{\mathrm e}^{-d x -c} a \,b^{3} d x +{\mathrm e}^{d x +c} a^{3} b \,d^{2}-{\mathrm e}^{d x +c} a \,b^{3} d x +2 \,{\mathrm e}^{-d x -c} a^{2} b^{2} d -{\mathrm e}^{-d x -c} b^{4} x -2 \,{\mathrm e}^{d x +c} a^{2} b^{2} d -{\mathrm e}^{d x +c} b^{4} x -{\mathrm e}^{-d x -c} a \,b^{3}-{\mathrm e}^{d x +c} a \,b^{3}}{2 d^{2} b^{5} \left (b x +a \right )}\)	\(549\)

[In]

int(x^3*cosh(d*x+c)/(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/2/d^2*(exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a^3*b*d^3*x-exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*a^3*b
*d^3*x+exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a^4*d^3-3*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a^2*b^2
*d^2*x-exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*a^4*d^3-3*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*a^2*b^2*d^2
*x-3*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a^3*b*d^2-exp(-d*x-c)*b^4*d*x^2-3*exp((a*d-b*c)/b)*Ei(1,d*x+c+
(a*d-b*c)/b)*a^3*b*d^2+exp(d*x+c)*b^4*d*x^2+exp(-d*x-c)*a^3*b*d^2+exp(-d*x-c)*a*b^3*d*x+exp(d*x+c)*a^3*b*d^2-e
xp(d*x+c)*a*b^3*d*x+2*exp(-d*x-c)*a^2*b^2*d-exp(-d*x-c)*b^4*x-2*exp(d*x+c)*a^2*b^2*d-exp(d*x+c)*b^4*x-exp(-d*x
-c)*a*b^3-exp(d*x+c)*a*b^3)/b^5/(b*x+a)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.83 \[ \int \frac {x^3 \cosh (c+d x)}{(a+b x)^2} \, dx=\frac {2 \, {\left (a^{3} b d^{2} - b^{4} x - a b^{3}\right )} \cosh \left (d x + c\right ) - {\left ({\left (a^{4} d^{3} - 3 \, a^{3} b d^{2} + {\left (a^{3} b d^{3} - 3 \, a^{2} b^{2} d^{2}\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\left (a^{4} d^{3} + 3 \, a^{3} b d^{2} + {\left (a^{3} b d^{3} + 3 \, a^{2} b^{2} d^{2}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) + 2 \, {\left (b^{4} d x^{2} - a b^{3} d x - 2 \, a^{2} b^{2} d\right )} \sinh \left (d x + c\right ) + {\left ({\left (a^{4} d^{3} - 3 \, a^{3} b d^{2} + {\left (a^{3} b d^{3} - 3 \, a^{2} b^{2} d^{2}\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{4} d^{3} + 3 \, a^{3} b d^{2} + {\left (a^{3} b d^{3} + 3 \, a^{2} b^{2} d^{2}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{2 \, {\left (b^{6} d^{2} x + a b^{5} d^{2}\right )}} \]

[In]

integrate(x^3*cosh(d*x+c)/(b*x+a)^2,x, algorithm="fricas")

[Out]

1/2*(2*(a^3*b*d^2 - b^4*x - a*b^3)*cosh(d*x + c) - ((a^4*d^3 - 3*a^3*b*d^2 + (a^3*b*d^3 - 3*a^2*b^2*d^2)*x)*Ei
((b*d*x + a*d)/b) - (a^4*d^3 + 3*a^3*b*d^2 + (a^3*b*d^3 + 3*a^2*b^2*d^2)*x)*Ei(-(b*d*x + a*d)/b))*cosh(-(b*c -
a*d)/b) + 2*(b^4*d*x^2 - a*b^3*d*x - 2*a^2*b^2*d)*sinh(d*x + c) + ((a^4*d^3 - 3*a^3*b*d^2 + (a^3*b*d^3 - 3*a^
2*b^2*d^2)*x)*Ei((b*d*x + a*d)/b) + (a^4*d^3 + 3*a^3*b*d^2 + (a^3*b*d^3 + 3*a^2*b^2*d^2)*x)*Ei(-(b*d*x + a*d)/
b))*sinh(-(b*c - a*d)/b))/(b^6*d^2*x + a*b^5*d^2)

Sympy [F]

\[ \int \frac {x^3 \cosh (c+d x)}{(a+b x)^2} \, dx=\int \frac {x^{3} \cosh {\left (c + d x \right )}}{\left (a + b x\right )^{2}}\, dx \]

[In]

integrate(x**3*cosh(d*x+c)/(b*x+a)**2,x)

[Out]

Integral(x**3*cosh(c + d*x)/(a + b*x)**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.71 \[ \int \frac {x^3 \cosh (c+d x)}{(a+b x)^2} \, dx=-\frac {1}{4} \, {\left (2 \, a^{3} {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b^{5}} - \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b^{5}}\right )} + \frac {6 \, a^{2} {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b} + \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b}\right )}}{b^{3} d} - \frac {4 \, a {\left (\frac {{\left (d x e^{c} - e^{c}\right )} e^{\left (d x\right )}}{d^{2}} + \frac {{\left (d x + 1\right )} e^{\left (-d x - c\right )}}{d^{2}}\right )}}{b^{3}} + \frac {\frac {{\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} e^{\left (d x\right )}}{d^{3}} + \frac {{\left (d^{2} x^{2} + 2 \, d x + 2\right )} e^{\left (-d x - c\right )}}{d^{3}}}{b^{2}} + \frac {12 \, a^{2} \cosh \left (d x + c\right ) \log \left (b x + a\right )}{b^{4} d}\right )} d + \frac {1}{2} \, {\left (\frac {2 \, a^{3}}{b^{5} x + a b^{4}} + \frac {6 \, a^{2} \log \left (b x + a\right )}{b^{4}} + \frac {b x^{2} - 4 \, a x}{b^{3}}\right )} \cosh \left (d x + c\right ) \]

[In]

integrate(x^3*cosh(d*x+c)/(b*x+a)^2,x, algorithm="maxima")

[Out]

-1/4*(2*a^3*(e^(-c + a*d/b)*exp_integral_e(1, (b*x + a)*d/b)/b^5 - e^(c - a*d/b)*exp_integral_e(1, -(b*x + a)*
d/b)/b^5) + 6*a^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)/(b^3*d) - 4*a*((d*x*e^c - e^c)*e^(d*x)/d^2 + (d*x + 1)*e^(-d*x - c)/d^2)/b^3 + ((d^2*x^2*e^c - 2*d
*x*e^c + 2*e^c)*e^(d*x)/d^3 + (d^2*x^2 + 2*d*x + 2)*e^(-d*x - c)/d^3)/b^2 + 12*a^2*cosh(d*x + c)*log(b*x + a)/
(b^4*d))*d + 1/2*(2*a^3/(b^5*x + a*b^4) + 6*a^2*log(b*x + a)/b^4 + (b*x^2 - 4*a*x)/b^3)*cosh(d*x + c)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1991 vs. \(2 (185) = 370\).

Time = 0.33 (sec) , antiderivative size = 1991, normalized size of antiderivative = 10.94 \[ \int \frac {x^3 \cosh (c+d x)}{(a+b x)^2} \, dx=\text {Too large to display} \]

[In]

integrate(x^3*cosh(d*x+c)/(b*x+a)^2,x, algorithm="giac")

[Out]

-1/2*((b*x + a)*a^3*(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) - a^3*b*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) + a^4*d^4*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^3*(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) + a^3*b*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) - a^4*d^4*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) - 3*(b*x + a)*a^2*b*(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) + 3*a^2*b^2*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) - 3*a^3*b*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) - 3*(b*x + a)*a^2*b*(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) + 3*a^2*b^2*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) - 3*a^3*b*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) - a^3*b*d^3*e^((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b
) - a^3*b*d^3*e^(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) - (b*x + a)^2*b^2*(b*c/(b*x + a) - a*d/(b*x
+ a) + d)^2*e^((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + 2*(b*x + a)*b^3*(b*c/(b*x + a) - a*d/(b*x +
a) + d)*c*e^((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) - b^4*c^2*e^((b*x + a)*(b*c/(b*x + a) - a*d/(b*
x + a) + d)/b) + (b*x + a)*a*b^2*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d*e^((b*x + a)*(b*c/(b*x + a) - a*d/(b*x
+ a) + d)/b) - a*b^3*c*d*e^((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + 2*a^2*b^2*d^2*e^((b*x + a)*(b*c
/(b*x + a) - a*d/(b*x + a) + d)/b) + (b*x + a)^2*b^2*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*e^(-(b*x + a)*(b*c/
(b*x + a) - a*d/(b*x + a) + d)/b) - 2*(b*x + a)*b^3*(b*c/(b*x + a) - a*d/(b*x + a) + d)*c*e^(-(b*x + a)*(b*c/(
b*x + a) - a*d/(b*x + a) + d)/b) + b^4*c^2*e^(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) - (b*x + a)*a*
b^2*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d*e^(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + a*b^3*c*d*e^(
-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) - 2*a^2*b^2*d^2*e^(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a)
+ d)/b) + (b*x + a)*b^3*(b*c/(b*x + a) - a*d/(b*x + a) + d)*e^((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/
b) - b^4*c*e^((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + a*b^3*d*e^((b*x + a)*(b*c/(b*x + a) - a*d/(b*
x + a) + d)/b) + (b*x + a)*b^3*(b*c/(b*x + a) - a*d/(b*x + a) + d)*e^(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a
) + d)/b) - b^4*c*e^(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + a*b^3*d*e^(-(b*x + a)*(b*c/(b*x + a)
- a*d/(b*x + a) + d)/b))*b^2/(((b*x + a)*b^7*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d - b^8*c*d + a*b^7*d^2)*d)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \cosh (c+d x)}{(a+b x)^2} \, dx=\int \frac {x^3\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (a+b\,x\right )}^2} \,d x \]

[In]

int((x^3*cosh(c + d*x))/(a + b*x)^2,x)

[Out]

int((x^3*cosh(c + d*x))/(a + b*x)^2, x)