\(\int \frac {x^4 \cosh (c+d x)}{(a+b x)^2} \, dx\) [26]

Optimal result

Integrand size = 17, antiderivative size = 231 \[ \int \frac {x^4 \cosh (c+d x)}{(a+b x)^2} \, dx=\frac {2 a \cosh (c+d x)}{b^3 d^2}-\frac {2 x \cosh (c+d x)}{b^2 d^2}-\frac {a^4 \cosh (c+d x)}{b^5 (a+b x)}-\frac {4 a^3 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^5}+\frac {a^4 d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{b^6}+\frac {2 \sinh (c+d x)}{b^2 d^3}+\frac {3 a^2 \sinh (c+d x)}{b^4 d}-\frac {2 a x \sinh (c+d x)}{b^3 d}+\frac {x^2 \sinh (c+d x)}{b^2 d}+\frac {a^4 d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^6}-\frac {4 a^3 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^5} \]

[Out]

-4*a^3*Chi(a*d/b+d*x)*cosh(-c+a*d/b)/b^5+2*a*cosh(d*x+c)/b^3/d^2-2*x*cosh(d*x+c)/b^2/d^2-a^4*cosh(d*x+c)/b^5/(
b*x+a)+a^4*d*cosh(-c+a*d/b)*Shi(a*d/b+d*x)/b^6-a^4*d*Chi(a*d/b+d*x)*sinh(-c+a*d/b)/b^6+4*a^3*Shi(a*d/b+d*x)*si
nh(-c+a*d/b)/b^5+2*sinh(d*x+c)/b^2/d^3+3*a^2*sinh(d*x+c)/b^4/d-2*a*x*sinh(d*x+c)/b^3/d+x^2*sinh(d*x+c)/b^2/d

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 15, 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} \[ \int \frac {x^4 \cosh (c+d x)}{(a+b x)^2} \, dx=\frac {a^4 d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^6}+\frac {a^4 d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^6}-\frac {a^4 \cosh (c+d x)}{b^5 (a+b x)}-\frac {4 a^3 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^5}-\frac {4 a^3 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^5}+\frac {3 a^2 \sinh (c+d x)}{b^4 d}+\frac {2 a \cosh (c+d x)}{b^3 d^2}-\frac {2 a x \sinh (c+d x)}{b^3 d}+\frac {2 \sinh (c+d x)}{b^2 d^3}-\frac {2 x \cosh (c+d x)}{b^2 d^2}+\frac {x^2 \sinh (c+d x)}{b^2 d} \]

[In]

Int[(x^4*Cosh[c + d*x])/(a + b*x)^2,x]

[Out]

(2*a*Cosh[c + d*x])/(b^3*d^2) - (2*x*Cosh[c + d*x])/(b^2*d^2) - (a^4*Cosh[c + d*x])/(b^5*(a + b*x)) - (4*a^3*C
osh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/b^5 + (a^4*d*CoshIntegral[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/b^6
+ (2*Sinh[c + d*x])/(b^2*d^3) + (3*a^2*Sinh[c + d*x])/(b^4*d) - (2*a*x*Sinh[c + d*x])/(b^3*d) + (x^2*Sinh[c +
d*x])/(b^2*d) + (a^4*d*Cosh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^6 - (4*a^3*Sinh[c - (a*d)/b]*SinhInteg
ral[(a*d)/b + d*x])/b^5

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 {3 a^2 \cosh (c+d x)}{b^4}-\frac {2 a x \cosh (c+d x)}{b^3}+\frac {x^2 \cosh (c+d x)}{b^2}+\frac {a^4 \cosh (c+d x)}{b^4 (a+b x)^2}-\frac {4 a^3 \cosh (c+d x)}{b^4 (a+b x)}\right ) \, dx \\ & = \frac {\left (3 a^2\right ) \int \cosh (c+d x) \, dx}{b^4}-\frac {\left (4 a^3\right ) \int \frac {\cosh (c+d x)}{a+b x} \, dx}{b^4}+\frac {a^4 \int \frac {\cosh (c+d x)}{(a+b x)^2} \, dx}{b^4}-\frac {(2 a) \int x \cosh (c+d x) \, dx}{b^3}+\frac {\int x^2 \cosh (c+d x) \, dx}{b^2} \\ & = -\frac {a^4 \cosh (c+d x)}{b^5 (a+b x)}+\frac {3 a^2 \sinh (c+d x)}{b^4 d}-\frac {2 a x \sinh (c+d x)}{b^3 d}+\frac {x^2 \sinh (c+d x)}{b^2 d}+\frac {(2 a) \int \sinh (c+d x) \, dx}{b^3 d}-\frac {2 \int x \sinh (c+d x) \, dx}{b^2 d}+\frac {\left (a^4 d\right ) \int \frac {\sinh (c+d x)}{a+b x} \, dx}{b^5}-\frac {\left (4 a^3 \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^4}-\frac {\left (4 a^3 \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^4} \\ & = \frac {2 a \cosh (c+d x)}{b^3 d^2}-\frac {2 x \cosh (c+d x)}{b^2 d^2}-\frac {a^4 \cosh (c+d x)}{b^5 (a+b x)}-\frac {4 a^3 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^5}+\frac {3 a^2 \sinh (c+d x)}{b^4 d}-\frac {2 a x \sinh (c+d x)}{b^3 d}+\frac {x^2 \sinh (c+d x)}{b^2 d}-\frac {4 a^3 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^5}+\frac {2 \int \cosh (c+d x) \, dx}{b^2 d^2}+\frac {\left (a^4 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^5}+\frac {\left (a^4 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^5} \\ & = \frac {2 a \cosh (c+d x)}{b^3 d^2}-\frac {2 x \cosh (c+d x)}{b^2 d^2}-\frac {a^4 \cosh (c+d x)}{b^5 (a+b x)}-\frac {4 a^3 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^5}+\frac {a^4 d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{b^6}+\frac {2 \sinh (c+d x)}{b^2 d^3}+\frac {3 a^2 \sinh (c+d x)}{b^4 d}-\frac {2 a x \sinh (c+d x)}{b^3 d}+\frac {x^2 \sinh (c+d x)}{b^2 d}+\frac {a^4 d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^6}-\frac {4 a^3 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.77 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.75 \[ \int \frac {x^4 \cosh (c+d x)}{(a+b x)^2} \, dx=\frac {-\frac {b \left (-2 a^2 b^2+a^4 d^2+2 b^4 x^2\right ) \cosh (c+d x)}{d^2 (a+b x)}+a^3 \text {Chi}\left (d \left (\frac {a}{b}+x\right )\right ) \left (-4 b \cosh \left (c-\frac {a d}{b}\right )+a d \sinh \left (c-\frac {a d}{b}\right )\right )+\frac {b^2 \left (3 a^2 d^2-2 a b d^2 x+b^2 \left (2+d^2 x^2\right )\right ) \sinh (c+d x)}{d^3}+a^3 \left (a d \cosh \left (c-\frac {a d}{b}\right )-4 b \sinh \left (c-\frac {a d}{b}\right )\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )}{b^6} \]

[In]

Integrate[(x^4*Cosh[c + d*x])/(a + b*x)^2,x]

[Out]

(-((b*(-2*a^2*b^2 + a^4*d^2 + 2*b^4*x^2)*Cosh[c + d*x])/(d^2*(a + b*x))) + a^3*CoshIntegral[d*(a/b + x)]*(-4*b
*Cosh[c - (a*d)/b] + a*d*Sinh[c - (a*d)/b]) + (b^2*(3*a^2*d^2 - 2*a*b*d^2*x + b^2*(2 + d^2*x^2))*Sinh[c + d*x]
)/d^3 + a^3*(a*d*Cosh[c - (a*d)/b] - 4*b*Sinh[c - (a*d)/b])*SinhIntegral[d*(a/b + x)])/b^6

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(668\) vs. \(2(236)=472\).

Time = 0.29 (sec) , antiderivative size = 669, normalized size of antiderivative = 2.90

method	result	size
risch	\(-\frac {2 \,{\mathrm e}^{-d x -c} b^{5} x +2 \,{\mathrm e}^{-d x -c} a \,b^{4}-4 \,{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right ) a^{3} b^{2} d^{3} x -{\mathrm e}^{-d x -c} a \,b^{4} d^{2} x^{2}-2 \,{\mathrm e}^{d x +c} b^{5} x -2 \,{\mathrm e}^{d x +c} a \,b^{4}+{\mathrm e}^{d x +c} a \,b^{4} d^{2} x^{2}-4 \,{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right ) a^{4} b \,d^{3}-{\mathrm e}^{d x +c} a^{2} b^{3} d^{2} x +{\mathrm e}^{-d x -c} b^{5} d^{2} x^{3}-{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right ) a^{5} d^{4}+{\mathrm e}^{-d x -c} a^{4} b \,d^{3}+2 \,{\mathrm e}^{-d x -c} b^{5} d \,x^{2}+3 \,{\mathrm e}^{-d x -c} a^{3} b^{2} d^{2}-2 \,{\mathrm e}^{-d x -c} a^{2} b^{3} d -4 \,{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right ) a^{4} b \,d^{3}+{\mathrm e}^{-d x -c} a^{2} b^{3} d^{2} x +{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right ) a^{4} b \,d^{4} x -{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right ) a^{4} b \,d^{4} x -4 \,{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right ) a^{3} b^{2} d^{3} x -{\mathrm e}^{d x +c} b^{5} d^{2} x^{3}+{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right ) a^{5} d^{4}+{\mathrm e}^{d x +c} a^{4} b \,d^{3}+2 \,{\mathrm e}^{d x +c} b^{5} d \,x^{2}-3 \,{\mathrm e}^{d x +c} a^{3} b^{2} d^{2}-2 \,{\mathrm e}^{d x +c} a^{2} b^{3} d}{2 d^{3} b^{6} \left (b x +a \right )}\)	\(669\)

[In]

int(x^4*cosh(d*x+c)/(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.61 \[ \int \frac {x^4 \cosh (c+d x)}{(a+b x)^2} \, dx=-\frac {2 \, {\left (a^{4} b d^{3} + 2 \, b^{5} d x^{2} - 2 \, a^{2} b^{3} d\right )} \cosh \left (d x + c\right ) - {\left ({\left (a^{5} d^{4} - 4 \, a^{4} b d^{3} + {\left (a^{4} b d^{4} - 4 \, a^{3} b^{2} d^{3}\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\left (a^{5} d^{4} + 4 \, a^{4} b d^{3} + {\left (a^{4} b d^{4} + 4 \, a^{3} b^{2} d^{3}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) - 2 \, {\left (b^{5} d^{2} x^{3} - a b^{4} d^{2} x^{2} + 3 \, a^{3} b^{2} d^{2} + 2 \, a b^{4} + {\left (a^{2} b^{3} d^{2} + 2 \, b^{5}\right )} x\right )} \sinh \left (d x + c\right ) + {\left ({\left (a^{5} d^{4} - 4 \, a^{4} b d^{3} + {\left (a^{4} b d^{4} - 4 \, a^{3} b^{2} d^{3}\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{5} d^{4} + 4 \, a^{4} b d^{3} + {\left (a^{4} b d^{4} + 4 \, a^{3} b^{2} d^{3}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{2 \, {\left (b^{7} d^{3} x + a b^{6} d^{3}\right )}} \]

[In]

integrate(x^4*cosh(d*x+c)/(b*x+a)^2,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {x^4 \cosh (c+d x)}{(a+b x)^2} \, dx=\int \frac {x^{4} \cosh {\left (c + d x \right )}}{\left (a + b x\right )^{2}}\, dx \]

[In]

integrate(x**4*cosh(d*x+c)/(b*x+a)**2,x)

[Out]

Integral(x**4*cosh(c + d*x)/(a + b*x)**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.76 \[ \int \frac {x^4 \cosh (c+d x)}{(a+b x)^2} \, dx=\frac {1}{6} \, {\left (3 \, a^{4} {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b^{6}} - \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b^{6}}\right )} + \frac {12 \, a^{3} {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b} + \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b}\right )}}{b^{4} d} - \frac {9 \, a^{2} {\left (\frac {{\left (d x e^{c} - e^{c}\right )} e^{\left (d x\right )}}{d^{2}} + \frac {{\left (d x + 1\right )} e^{\left (-d x - c\right )}}{d^{2}}\right )}}{b^{4}} + \frac {3 \, a {\left (\frac {{\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} e^{\left (d x\right )}}{d^{3}} + \frac {{\left (d^{2} x^{2} + 2 \, d x + 2\right )} e^{\left (-d x - c\right )}}{d^{3}}\right )}}{b^{3}} - \frac {\frac {{\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} e^{\left (d x\right )}}{d^{4}} + \frac {{\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} e^{\left (-d x - c\right )}}{d^{4}}}{b^{2}} + \frac {24 \, a^{3} \cosh \left (d x + c\right ) \log \left (b x + a\right )}{b^{5} d}\right )} d - \frac {1}{3} \, {\left (\frac {3 \, a^{4}}{b^{6} x + a b^{5}} + \frac {12 \, a^{3} \log \left (b x + a\right )}{b^{5}} - \frac {b^{2} x^{3} - 3 \, a b x^{2} + 9 \, a^{2} x}{b^{4}}\right )} \cosh \left (d x + c\right ) \]

[In]

integrate(x^4*cosh(d*x+c)/(b*x+a)^2,x, algorithm="maxima")

[Out]

1/6*(3*a^4*(e^(-c + a*d/b)*exp_integral_e(1, (b*x + a)*d/b)/b^6 - e^(c - a*d/b)*exp_integral_e(1, -(b*x + a)*d
/b)/b^6) + 12*a^3*(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^4*d) - 9*a^2*((d*x*e^c - e^c)*e^(d*x)/d^2 + (d*x + 1)*e^(-d*x - c)/d^2)/b^4 + 3*a*((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^3 - ((d^3*x^3*e^c - 3*d^2*x^2*e^
c + 6*d*x*e^c - 6*e^c)*e^(d*x)/d^4 + (d^3*x^3 + 3*d^2*x^2 + 6*d*x + 6)*e^(-d*x - c)/d^4)/b^2 + 24*a^3*cosh(d*x
+ c)*log(b*x + a)/(b^5*d))*d - 1/3*(3*a^4/(b^6*x + a*b^5) + 12*a^3*log(b*x + a)/b^5 - (b^2*x^3 - 3*a*b*x^2 +
9*a^2*x)/b^4)*cosh(d*x + c)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2979 vs. \(2 (236) = 472\).

Time = 0.33 (sec) , antiderivative size = 2979, normalized size of antiderivative = 12.90 \[ \int \frac {x^4 \cosh (c+d x)}{(a+b x)^2} \, dx=\text {Too large to display} \]

[In]

integrate(x^4*cosh(d*x+c)/(b*x+a)^2,x, algorithm="giac")

[Out]

1/2*((b*x + a)*a^4*(b*c/(b*x + a) - a*d/(b*x + a) + d)*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) - a^4*b*c*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) + a^5*d^5*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^4*(b*c/(b*x + a) - a*d/(b*x + a) + d)*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) + a^4*b*c*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) - a^5*d^5*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)*a^3*b*(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) + 4*a^3*b^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) - 4*a^4*b*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) - 4*(b*x + a)*a^3*b*(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) + 4*a^3*b^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) - 4*a^4*b*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) - a^4*b*d^4*e^((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b)
- a^4*b*d^4*e^(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + (b*x + a)^3*b^2*(b*c/(b*x + a) - a*d/(b*x
+ a) + d)^3*e^((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) - 3*(b*x + a)^2*b^3*(b*c/(b*x + a) - a*d/(b*x
+ a) + d)^2*c*e^((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + 3*(b*x + a)*b^4*(b*c/(b*x + a) - a*d/(b*x
+ a) + d)*c^2*e^((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) - b^5*c^3*e^((b*x + a)*(b*c/(b*x + a) - a*d/
(b*x + a) + d)/b) - (b*x + a)^2*a*b^2*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*d*e^((b*x + a)*(b*c/(b*x + a) - a*
d/(b*x + a) + d)/b) + 2*(b*x + a)*a*b^3*(b*c/(b*x + a) - a*d/(b*x + a) + d)*c*d*e^((b*x + a)*(b*c/(b*x + a) -
a*d/(b*x + a) + d)/b) - a*b^4*c^2*d*e^((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + (b*x + a)*a^2*b^2*(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) - a^2*b^3*c*d^2*e^((
b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + 3*a^3*b^2*d^3*e^((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) +
d)/b) - (b*x + a)^3*b^2*(b*c/(b*x + a) - a*d/(b*x + a) + d)^3*e^(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d
)/b) + 3*(b*x + a)^2*b^3*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*c*e^(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a)
+ d)/b) - 3*(b*x + a)*b^4*(b*c/(b*x + a) - a*d/(b*x + a) + d)*c^2*e^(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a)
+ d)/b) + b^5*c^3*e^(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + (b*x + a)^2*a*b^2*(b*c/(b*x + a) - a
*d/(b*x + a) + d)^2*d*e^(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) - 2*(b*x + a)*a*b^3*(b*c/(b*x + a)
- a*d/(b*x + a) + d)*c*d*e^(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + a*b^4*c^2*d*e^(-(b*x + a)*(b*c
/(b*x + a) - a*d/(b*x + a) + d)/b) - (b*x + a)*a^2*b^2*(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) + a^2*b^3*c*d^2*e^(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) - 3
*a^3*b^2*d^3*e^(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) - 2*(b*x + a)^2*b^3*(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) + 4*(b*x + a)*b^4*(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) - 2*b^5*c^2*e^((b*x + a)*(b*c/(b*x + a) - a*d/
(b*x + a) + d)/b) + 2*a^2*b^3*d^2*e^((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) - 2*(b*x + a)^2*b^3*(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) + 4*(b*x + a)*b^4*(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) - 2*b^5*c^2*e^(-(b*x + a
)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + 2*a^2*b^3*d^2*e^(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b)
+ 2*(b*x + a)*b^4*(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) - 2*
b^5*c*e^((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + 2*a*b^4*d*e^((b*x + a)*(b*c/(b*x + a) - a*d/(b*x +
a) + d)/b) - 2*(b*x + a)*b^4*(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) + 2*b^5*c*e^(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) - 2*a*b^4*d*e^(-(b*x + a)*(b*c/(b*x +
a) - a*d/(b*x + a) + d)/b))*b^2/(((b*x + a)*b^8*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^2 - b^9*c*d^2 + a*b^8*d^
3)*d)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \cosh (c+d x)}{(a+b x)^2} \, dx=\int \frac {x^4\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (a+b\,x\right )}^2} \,d x \]

[In]

int((x^4*cosh(c + d*x))/(a + b*x)^2,x)

[Out]

int((x^4*cosh(c + d*x))/(a + b*x)^2, x)