\(\int \frac {x^2 \cosh (c+d x)}{(a+b x)^3} \, dx\) [34]

Optimal result

Integrand size = 17, antiderivative size = 241 \[ \int \frac {x^2 \cosh (c+d x)}{(a+b x)^3} \, dx=-\frac {a^2 \cosh (c+d x)}{2 b^3 (a+b x)^2}+\frac {2 a \cosh (c+d x)}{b^3 (a+b x)}+\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^3}+\frac {a^2 d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{2 b^5}-\frac {2 a d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{b^4}-\frac {a^2 d \sinh (c+d x)}{2 b^4 (a+b x)}-\frac {2 a d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^4}+\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^3}+\frac {a^2 d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{2 b^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6874, 3378, 3384, 3379, 3382} \[ \int \frac {x^2 \cosh (c+d x)}{(a+b x)^3} \, dx=\frac {a^2 d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{2 b^5}+\frac {a^2 d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{2 b^5}-\frac {a^2 d \sinh (c+d x)}{2 b^4 (a+b x)}-\frac {a^2 \cosh (c+d x)}{2 b^3 (a+b x)^2}-\frac {2 a d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^4}-\frac {2 a d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^4}+\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {2 a \cosh (c+d x)}{b^3 (a+b x)} \]

[In]

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

[Out]

-1/2*(a^2*Cosh[c + d*x])/(b^3*(a + b*x)^2) + (2*a*Cosh[c + d*x])/(b^3*(a + b*x)) + (Cosh[c - (a*d)/b]*CoshInte
gral[(a*d)/b + d*x])/b^3 + (a^2*d^2*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/(2*b^5) - (2*a*d*CoshIntegr
al[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/b^4 - (a^2*d*Sinh[c + d*x])/(2*b^4*(a + b*x)) - (2*a*d*Cosh[c - (a*d)/b]*
SinhIntegral[(a*d)/b + d*x])/b^4 + (Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^3 + (a^2*d^2*Sinh[c - (a*
d)/b]*SinhIntegral[(a*d)/b + d*x])/(2*b^5)

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

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.63 \[ \int \frac {x^2 \cosh (c+d x)}{(a+b x)^3} \, dx=\frac {\text {Chi}\left (d \left (\frac {a}{b}+x\right )\right ) \left (\left (2 b^2+a^2 d^2\right ) \cosh \left (c-\frac {a d}{b}\right )-4 a b d \sinh \left (c-\frac {a d}{b}\right )\right )-\frac {a b (-b (3 a+4 b x) \cosh (c+d x)+a d (a+b x) \sinh (c+d x))}{(a+b x)^2}+\left (-4 a b d \cosh \left (c-\frac {a d}{b}\right )+\left (2 b^2+a^2 d^2\right ) \sinh \left (c-\frac {a d}{b}\right )\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )}{2 b^5} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(916\) vs. \(2(240)=480\).

Time = 0.28 (sec) , antiderivative size = 917, normalized size of antiderivative = 3.80

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} d^{2} x^{2}+2 \,{\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} x +4 \,{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right ) a \,b^{3} d \,x^{2}+8 \,{\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 x -3 \,{\mathrm e}^{d x +c} a^{2} b^{2}-3 \,{\mathrm e}^{-d x -c} a^{2} b^{2}-{\mathrm e}^{-d x -c} a^{2} b^{2} d x +4 \,{\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 +4 \,{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right ) a \,b^{3} x -4 \,{\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 +4 \,{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right ) a \,b^{3} x +{\mathrm e}^{d x +c} a^{2} b^{2} d 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^{2}+2 \,{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right ) b^{4} x^{2}-{\mathrm e}^{-d x -c} a^{3} b d -4 \,{\mathrm e}^{-d x -c} a \,b^{3} x +2 \,{\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}+{\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^{2}+2 \,{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right ) b^{4} x^{2}+2 \,{\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}+{\mathrm e}^{d x +c} a^{3} b d -4 \,{\mathrm e}^{d x +c} a \,b^{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^{2} b^{2} d^{2} x^{2}+2 \,{\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} x -4 \,{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right ) a \,b^{3} d \,x^{2}-8 \,{\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 x}{4 b^{5} \left (b x +a \right )^{2}}\)	\(917\)

[In]

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

[Out]

-1/4*(exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*a^2*b^2*d^2*x^2+2*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*a^3*
b*d^2*x+4*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*a*b^3*d*x^2+8*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*a^2*
b^2*d*x-3*exp(d*x+c)*a^2*b^2-3*exp(-d*x-c)*a^2*b^2-exp(-d*x-c)*a^2*b^2*d*x+4*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-
b*c)/b)*a^3*b*d+4*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*a*b^3*x-4*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b
)*a^3*b*d+4*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a*b^3*x+exp(d*x+c)*a^2*b^2*d*x+exp((a*d-b*c)/b)*Ei(1,d*
x+c+(a*d-b*c)/b)*a^4*d^2+2*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*b^4*x^2-exp(-d*x-c)*a^3*b*d-4*exp(-d*x-c)*
a*b^3*x+2*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*a^2*b^2+exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a^4*d^2+
2*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*b^4*x^2+2*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a^2*b^2+exp(
d*x+c)*a^3*b*d-4*exp(d*x+c)*a*b^3*x+exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a^2*b^2*d^2*x^2+2*exp(-(a*d-b*c
)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a^3*b*d^2*x-4*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a*b^3*d*x^2-8*exp(-(a*d
-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a^2*b^2*d*x)/b^5/(b*x+a)^2

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.97 \[ \int \frac {x^2 \cosh (c+d x)}{(a+b x)^3} \, dx=\frac {2 \, {\left (4 \, a b^{3} x + 3 \, a^{2} b^{2}\right )} \cosh \left (d x + c\right ) + {\left ({\left (a^{4} d^{2} - 4 \, a^{3} b d + 2 \, a^{2} b^{2} + {\left (a^{2} b^{2} d^{2} - 4 \, a b^{3} d + 2 \, b^{4}\right )} x^{2} + 2 \, {\left (a^{3} b d^{2} - 4 \, a^{2} b^{2} d + 2 \, a b^{3}\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{4} d^{2} + 4 \, a^{3} b d + 2 \, a^{2} b^{2} + {\left (a^{2} b^{2} d^{2} + 4 \, a b^{3} d + 2 \, b^{4}\right )} x^{2} + 2 \, {\left (a^{3} b d^{2} + 4 \, a^{2} b^{2} d + 2 \, a b^{3}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) - 2 \, {\left (a^{2} b^{2} d x + a^{3} b d\right )} \sinh \left (d x + c\right ) - {\left ({\left (a^{4} d^{2} - 4 \, a^{3} b d + 2 \, a^{2} b^{2} + {\left (a^{2} b^{2} d^{2} - 4 \, a b^{3} d + 2 \, b^{4}\right )} x^{2} + 2 \, {\left (a^{3} b d^{2} - 4 \, a^{2} b^{2} d + 2 \, a b^{3}\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\left (a^{4} d^{2} + 4 \, a^{3} b d + 2 \, a^{2} b^{2} + {\left (a^{2} b^{2} d^{2} + 4 \, a b^{3} d + 2 \, b^{4}\right )} x^{2} + 2 \, {\left (a^{3} b d^{2} + 4 \, a^{2} b^{2} d + 2 \, a b^{3}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{4 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \]

[In]

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

[Out]

1/4*(2*(4*a*b^3*x + 3*a^2*b^2)*cosh(d*x + c) + ((a^4*d^2 - 4*a^3*b*d + 2*a^2*b^2 + (a^2*b^2*d^2 - 4*a*b^3*d +
2*b^4)*x^2 + 2*(a^3*b*d^2 - 4*a^2*b^2*d + 2*a*b^3)*x)*Ei((b*d*x + a*d)/b) + (a^4*d^2 + 4*a^3*b*d + 2*a^2*b^2 +
(a^2*b^2*d^2 + 4*a*b^3*d + 2*b^4)*x^2 + 2*(a^3*b*d^2 + 4*a^2*b^2*d + 2*a*b^3)*x)*Ei(-(b*d*x + a*d)/b))*cosh(-
(b*c - a*d)/b) - 2*(a^2*b^2*d*x + a^3*b*d)*sinh(d*x + c) - ((a^4*d^2 - 4*a^3*b*d + 2*a^2*b^2 + (a^2*b^2*d^2 -
4*a*b^3*d + 2*b^4)*x^2 + 2*(a^3*b*d^2 - 4*a^2*b^2*d + 2*a*b^3)*x)*Ei((b*d*x + a*d)/b) - (a^4*d^2 + 4*a^3*b*d +
2*a^2*b^2 + (a^2*b^2*d^2 + 4*a*b^3*d + 2*b^4)*x^2 + 2*(a^3*b*d^2 + 4*a^2*b^2*d + 2*a*b^3)*x)*Ei(-(b*d*x + a*d
)/b))*sinh(-(b*c - a*d)/b))/(b^7*x^2 + 2*a*b^6*x + a^2*b^5)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 741 vs. \(2 (240) = 480\).

Time = 0.32 (sec) , antiderivative size = 741, normalized size of antiderivative = 3.07 \[ \int \frac {x^2 \cosh (c+d x)}{(a+b x)^3} \, dx=\frac {a^{2} b^{2} d^{2} x^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + a^{2} b^{2} d^{2} x^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 2 \, a^{3} b d^{2} x {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} - 4 \, a b^{3} d x^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 2 \, a^{3} b d^{2} x {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 4 \, a b^{3} d x^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + a^{4} d^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} - 8 \, a^{2} b^{2} d x {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 2 \, b^{4} x^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + a^{4} d^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 8 \, a^{2} b^{2} d x {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 2 \, b^{4} x^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} - a^{2} b^{2} d x e^{\left (d x + c\right )} + a^{2} b^{2} d x e^{\left (-d x - c\right )} - 4 \, a^{3} b d {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 4 \, a b^{3} x {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 4 \, a^{3} b d {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 4 \, a b^{3} x {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} - a^{3} b d e^{\left (d x + c\right )} + 4 \, a b^{3} x e^{\left (d x + c\right )} + a^{3} b d e^{\left (-d x - c\right )} + 4 \, a b^{3} x e^{\left (-d x - c\right )} + 2 \, a^{2} b^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 2 \, a^{2} b^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 3 \, a^{2} b^{2} e^{\left (d x + c\right )} + 3 \, a^{2} b^{2} e^{\left (-d x - c\right )}}{4 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \]

[In]

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

[Out]

1/4*(a^2*b^2*d^2*x^2*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + a^2*b^2*d^2*x^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) +
2*a^3*b*d^2*x*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) - 4*a*b^3*d*x^2*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 2*a^3*b*d
^2*x*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + 4*a*b^3*d*x^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + a^4*d^2*Ei((b*d
*x + a*d)/b)*e^(c - a*d/b) - 8*a^2*b^2*d*x*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 2*b^4*x^2*Ei((b*d*x + a*d)/b)*e
^(c - a*d/b) + a^4*d^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + 8*a^2*b^2*d*x*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b)
+ 2*b^4*x^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) - a^2*b^2*d*x*e^(d*x + c) + a^2*b^2*d*x*e^(-d*x - c) - 4*a^3*
b*d*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 4*a*b^3*x*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 4*a^3*b*d*Ei(-(b*d*x + a
*d)/b)*e^(-c + a*d/b) + 4*a*b^3*x*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) - a^3*b*d*e^(d*x + c) + 4*a*b^3*x*e^(d*x
+ c) + a^3*b*d*e^(-d*x - c) + 4*a*b^3*x*e^(-d*x - c) + 2*a^2*b^2*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 2*a^2*b^
2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + 3*a^2*b^2*e^(d*x + c) + 3*a^2*b^2*e^(-d*x - c))/(b^7*x^2 + 2*a*b^6*x +
a^2*b^5)

Mupad [F(-1)]

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

[In]

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

[Out]

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