\(\int \frac {\cosh (c+d x)}{x^2 (a+b x)^3} \, dx\) [38]

Optimal result

Integrand size = 17, antiderivative size = 298 \[ \int \frac {\cosh (c+d x)}{x^2 (a+b x)^3} \, dx=-\frac {\cosh (c+d x)}{a^3 x}-\frac {b \cosh (c+d x)}{2 a^2 (a+b x)^2}-\frac {2 b \cosh (c+d x)}{a^3 (a+b x)}-\frac {3 b \cosh (c) \text {Chi}(d x)}{a^4}+\frac {3 b \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{a^4}+\frac {d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{2 a^2 b}+\frac {d \text {Chi}(d x) \sinh (c)}{a^3}+\frac {2 d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{a^3}-\frac {d \sinh (c+d x)}{2 a^2 (a+b x)}+\frac {d \cosh (c) \text {Shi}(d x)}{a^3}-\frac {3 b \sinh (c) \text {Shi}(d x)}{a^4}+\frac {2 d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^3}+\frac {3 b \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^4}+\frac {d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{2 a^2 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6874, 3378, 3384, 3379, 3382} \[ \int \frac {\cosh (c+d x)}{x^2 (a+b x)^3} \, dx=-\frac {3 b \cosh (c) \text {Chi}(d x)}{a^4}+\frac {3 b \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{a^4}-\frac {3 b \sinh (c) \text {Shi}(d x)}{a^4}+\frac {3 b \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{a^4}+\frac {2 d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{a^3}+\frac {2 d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{a^3}-\frac {2 b \cosh (c+d x)}{a^3 (a+b x)}+\frac {d \sinh (c) \text {Chi}(d x)}{a^3}+\frac {d \cosh (c) \text {Shi}(d x)}{a^3}-\frac {\cosh (c+d x)}{a^3 x}+\frac {d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{2 a^2 b}+\frac {d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{2 a^2 b}-\frac {d \sinh (c+d x)}{2 a^2 (a+b x)}-\frac {b \cosh (c+d x)}{2 a^2 (a+b x)^2} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.99 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.82 \[ \int \frac {\cosh (c+d x)}{x^2 (a+b x)^3} \, dx=\frac {-2 a^3 b \cosh (c+d x)-9 a^2 b^2 x \cosh (c+d x)-6 a b^3 x^2 \cosh (c+d x)+2 b x (a+b x)^2 \text {Chi}(d x) (-3 b \cosh (c)+a d \sinh (c))+x (a+b x)^2 \text {Chi}\left (d \left (\frac {a}{b}+x\right )\right ) \left (\left (6 b^2+a^2 d^2\right ) \cosh \left (c-\frac {a d}{b}\right )+4 a b d \sinh \left (c-\frac {a d}{b}\right )\right )-a^3 b d x \sinh (c+d x)-a^2 b^2 d x^2 \sinh (c+d x)+2 a^3 b d x \cosh (c) \text {Shi}(d x)+4 a^2 b^2 d x^2 \cosh (c) \text {Shi}(d x)+2 a b^3 d x^3 \cosh (c) \text {Shi}(d x)-6 a^2 b^2 x \sinh (c) \text {Shi}(d x)-12 a b^3 x^2 \sinh (c) \text {Shi}(d x)-6 b^4 x^3 \sinh (c) \text {Shi}(d x)+4 a^3 b d x \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+8 a^2 b^2 d x^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+4 a b^3 d x^3 \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+6 a^2 b^2 x \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+a^4 d^2 x \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+12 a b^3 x^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+2 a^3 b d^2 x^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+6 b^4 x^3 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+a^2 b^2 d^2 x^3 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )}{2 a^4 b x (a+b x)^2} \]

[In]

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

[Out]

(-2*a^3*b*Cosh[c + d*x] - 9*a^2*b^2*x*Cosh[c + d*x] - 6*a*b^3*x^2*Cosh[c + d*x] + 2*b*x*(a + b*x)^2*CoshIntegr
al[d*x]*(-3*b*Cosh[c] + a*d*Sinh[c]) + x*(a + b*x)^2*CoshIntegral[d*(a/b + x)]*((6*b^2 + a^2*d^2)*Cosh[c - (a*
d)/b] + 4*a*b*d*Sinh[c - (a*d)/b]) - a^3*b*d*x*Sinh[c + d*x] - a^2*b^2*d*x^2*Sinh[c + d*x] + 2*a^3*b*d*x*Cosh[
c]*SinhIntegral[d*x] + 4*a^2*b^2*d*x^2*Cosh[c]*SinhIntegral[d*x] + 2*a*b^3*d*x^3*Cosh[c]*SinhIntegral[d*x] - 6
*a^2*b^2*x*Sinh[c]*SinhIntegral[d*x] - 12*a*b^3*x^2*Sinh[c]*SinhIntegral[d*x] - 6*b^4*x^3*Sinh[c]*SinhIntegral
[d*x] + 4*a^3*b*d*x*Cosh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)] + 8*a^2*b^2*d*x^2*Cosh[c - (a*d)/b]*SinhIntegr
al[d*(a/b + x)] + 4*a*b^3*d*x^3*Cosh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)] + 6*a^2*b^2*x*Sinh[c - (a*d)/b]*Si
nhIntegral[d*(a/b + x)] + a^4*d^2*x*Sinh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)] + 12*a*b^3*x^2*Sinh[c - (a*d)/
b]*SinhIntegral[d*(a/b + x)] + 2*a^3*b*d^2*x^2*Sinh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)] + 6*b^4*x^3*Sinh[c
- (a*d)/b]*SinhIntegral[d*(a/b + x)] + a^2*b^2*d^2*x^3*Sinh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)])/(2*a^4*b*x
*(a + b*x)^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(642\) vs. \(2(296)=592\).

Time = 0.36 (sec) , antiderivative size = 643, normalized size of antiderivative = 2.16

method	result	size
risch	\(\frac {{\mathrm e}^{-d x -c} x \,d^{3} b}{4 a^{2} \left (x^{2} d^{2} b^{2}+2 a b \,d^{2} x +a^{2} d^{2}\right )}+\frac {{\mathrm e}^{-d x -c} d^{3}}{4 a \left (x^{2} d^{2} b^{2}+2 a b \,d^{2} x +a^{2} d^{2}\right )}-\frac {3 \,{\mathrm e}^{-d x -c} x \,d^{2} b^{2}}{2 a^{3} \left (x^{2} d^{2} b^{2}+2 a b \,d^{2} x +a^{2} d^{2}\right )}-\frac {9 \,{\mathrm e}^{-d x -c} d^{2} b}{4 a^{2} \left (x^{2} d^{2} b^{2}+2 a b \,d^{2} x +a^{2} d^{2}\right )}-\frac {{\mathrm e}^{-d x -c} d^{2}}{2 a x \left (x^{2} d^{2} b^{2}+2 a b \,d^{2} x +a^{2} d^{2}\right )}+\frac {d \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right )}{2 a^{3}}+\frac {3 \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) b}{2 a^{4}}-\frac {d^{2} {\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right )}{4 b \,a^{2}}+\frac {d \,{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right )}{a^{3}}-\frac {3 b \,{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right )}{2 a^{4}}-\frac {{\mathrm e}^{d x +c}}{2 a^{3} x}-\frac {d \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right )}{2 a^{3}}+\frac {3 b \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right )}{2 a^{4}}-\frac {d^{2} {\mathrm e}^{d x +c}}{4 a^{2} b \left (\frac {d a}{b}+d x \right )^{2}}-\frac {d^{2} {\mathrm e}^{d x +c}}{4 a^{2} b \left (\frac {d a}{b}+d x \right )}-\frac {d^{2} {\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right )}{4 a^{2} b}-\frac {d \,{\mathrm e}^{d x +c}}{a^{3} \left (\frac {d a}{b}+d x \right )}-\frac {d \,{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right )}{a^{3}}-\frac {3 b \,{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right )}{2 a^{4}}\)	\(643\)

[In]

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

[Out]

1/4*exp(-d*x-c)/a^2*x*d^3/(b^2*d^2*x^2+2*a*b*d^2*x+a^2*d^2)*b+1/4*exp(-d*x-c)/a*d^3/(b^2*d^2*x^2+2*a*b*d^2*x+a
^2*d^2)-3/2*exp(-d*x-c)/a^3*x*d^2/(b^2*d^2*x^2+2*a*b*d^2*x+a^2*d^2)*b^2-9/4*exp(-d*x-c)/a^2*d^2/(b^2*d^2*x^2+2
*a*b*d^2*x+a^2*d^2)*b-1/2*exp(-d*x-c)/a/x*d^2/(b^2*d^2*x^2+2*a*b*d^2*x+a^2*d^2)+1/2*d/a^3*exp(-c)*Ei(1,d*x)+3/
2/a^4*exp(-c)*Ei(1,d*x)*b-1/4/b/a^2*d^2*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)+d/a^3*exp((a*d-b*c)/b)*Ei(1,d
*x+c+(a*d-b*c)/b)-3/2*b/a^4*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)-1/2/a^3/x*exp(d*x+c)-1/2*d/a^3*exp(c)*Ei(
1,-d*x)+3/2/a^4*b*exp(c)*Ei(1,-d*x)-1/4/a^2*d^2/b*exp(d*x+c)/(d/b*a+d*x)^2-1/4/a^2*d^2/b*exp(d*x+c)/(d/b*a+d*x
)-1/4/a^2*d^2/b*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)-d/a^3*exp(d*x+c)/(d/b*a+d*x)-d/a^3*exp(-(a*d-b*c)/b
)*Ei(1,-d*x-c-(a*d-b*c)/b)-3/2*b/a^4*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 762 vs. \(2 (296) = 592\).

Time = 0.26 (sec) , antiderivative size = 762, normalized size of antiderivative = 2.56 \[ \int \frac {\cosh (c+d x)}{x^2 (a+b x)^3} \, dx=-\frac {2 \, {\left (6 \, a b^{3} x^{2} + 9 \, a^{2} b^{2} x + 2 \, a^{3} b\right )} \cosh \left (d x + c\right ) - 2 \, {\left ({\left ({\left (a b^{3} d - 3 \, b^{4}\right )} x^{3} + 2 \, {\left (a^{2} b^{2} d - 3 \, a b^{3}\right )} x^{2} + {\left (a^{3} b d - 3 \, a^{2} b^{2}\right )} x\right )} {\rm Ei}\left (d x\right ) - {\left ({\left (a b^{3} d + 3 \, b^{4}\right )} x^{3} + 2 \, {\left (a^{2} b^{2} d + 3 \, a b^{3}\right )} x^{2} + {\left (a^{3} b d + 3 \, a^{2} b^{2}\right )} x\right )} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) - {\left ({\left ({\left (a^{2} b^{2} d^{2} + 4 \, a b^{3} d + 6 \, b^{4}\right )} x^{3} + 2 \, {\left (a^{3} b d^{2} + 4 \, a^{2} b^{2} d + 6 \, a b^{3}\right )} x^{2} + {\left (a^{4} d^{2} + 4 \, a^{3} b d + 6 \, a^{2} b^{2}\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\left ({\left (a^{2} b^{2} d^{2} - 4 \, a b^{3} d + 6 \, b^{4}\right )} x^{3} + 2 \, {\left (a^{3} b d^{2} - 4 \, a^{2} b^{2} d + 6 \, a b^{3}\right )} x^{2} + {\left (a^{4} d^{2} - 4 \, a^{3} b d + 6 \, a^{2} b^{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 (a^{2} b^{2} d x^{2} + a^{3} b d x\right )} \sinh \left (d x + c\right ) - 2 \, {\left ({\left ({\left (a b^{3} d - 3 \, b^{4}\right )} x^{3} + 2 \, {\left (a^{2} b^{2} d - 3 \, a b^{3}\right )} x^{2} + {\left (a^{3} b d - 3 \, a^{2} b^{2}\right )} x\right )} {\rm Ei}\left (d x\right ) + {\left ({\left (a b^{3} d + 3 \, b^{4}\right )} x^{3} + 2 \, {\left (a^{2} b^{2} d + 3 \, a b^{3}\right )} x^{2} + {\left (a^{3} b d + 3 \, a^{2} b^{2}\right )} x\right )} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right ) + {\left ({\left ({\left (a^{2} b^{2} d^{2} + 4 \, a b^{3} d + 6 \, b^{4}\right )} x^{3} + 2 \, {\left (a^{3} b d^{2} + 4 \, a^{2} b^{2} d + 6 \, a b^{3}\right )} x^{2} + {\left (a^{4} d^{2} + 4 \, a^{3} b d + 6 \, a^{2} b^{2}\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\left ({\left (a^{2} b^{2} d^{2} - 4 \, a b^{3} d + 6 \, b^{4}\right )} x^{3} + 2 \, {\left (a^{3} b d^{2} - 4 \, a^{2} b^{2} d + 6 \, a b^{3}\right )} x^{2} + {\left (a^{4} d^{2} - 4 \, a^{3} b d + 6 \, a^{2} b^{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 )}{4 \, {\left (a^{4} b^{3} x^{3} + 2 \, a^{5} b^{2} x^{2} + a^{6} b x\right )}} \]

[In]

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

[Out]

-1/4*(2*(6*a*b^3*x^2 + 9*a^2*b^2*x + 2*a^3*b)*cosh(d*x + c) - 2*(((a*b^3*d - 3*b^4)*x^3 + 2*(a^2*b^2*d - 3*a*b
^3)*x^2 + (a^3*b*d - 3*a^2*b^2)*x)*Ei(d*x) - ((a*b^3*d + 3*b^4)*x^3 + 2*(a^2*b^2*d + 3*a*b^3)*x^2 + (a^3*b*d +
3*a^2*b^2)*x)*Ei(-d*x))*cosh(c) - (((a^2*b^2*d^2 + 4*a*b^3*d + 6*b^4)*x^3 + 2*(a^3*b*d^2 + 4*a^2*b^2*d + 6*a*
b^3)*x^2 + (a^4*d^2 + 4*a^3*b*d + 6*a^2*b^2)*x)*Ei((b*d*x + a*d)/b) + ((a^2*b^2*d^2 - 4*a*b^3*d + 6*b^4)*x^3 +
2*(a^3*b*d^2 - 4*a^2*b^2*d + 6*a*b^3)*x^2 + (a^4*d^2 - 4*a^3*b*d + 6*a^2*b^2)*x)*Ei(-(b*d*x + a*d)/b))*cosh(-
(b*c - a*d)/b) + 2*(a^2*b^2*d*x^2 + a^3*b*d*x)*sinh(d*x + c) - 2*(((a*b^3*d - 3*b^4)*x^3 + 2*(a^2*b^2*d - 3*a*
b^3)*x^2 + (a^3*b*d - 3*a^2*b^2)*x)*Ei(d*x) + ((a*b^3*d + 3*b^4)*x^3 + 2*(a^2*b^2*d + 3*a*b^3)*x^2 + (a^3*b*d
+ 3*a^2*b^2)*x)*Ei(-d*x))*sinh(c) + (((a^2*b^2*d^2 + 4*a*b^3*d + 6*b^4)*x^3 + 2*(a^3*b*d^2 + 4*a^2*b^2*d + 6*a
*b^3)*x^2 + (a^4*d^2 + 4*a^3*b*d + 6*a^2*b^2)*x)*Ei((b*d*x + a*d)/b) - ((a^2*b^2*d^2 - 4*a*b^3*d + 6*b^4)*x^3
+ 2*(a^3*b*d^2 - 4*a^2*b^2*d + 6*a*b^3)*x^2 + (a^4*d^2 - 4*a^3*b*d + 6*a^2*b^2)*x)*Ei(-(b*d*x + a*d)/b))*sinh(
-(b*c - a*d)/b))/(a^4*b^3*x^3 + 2*a^5*b^2*x^2 + a^6*b*x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1006 vs. \(2 (296) = 592\).

Time = 0.27 (sec) , antiderivative size = 1006, normalized size of antiderivative = 3.38 \[ \int \frac {\cosh (c+d x)}{x^2 (a+b x)^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/4*(a^2*b^2*d^2*x^3*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + a^2*b^2*d^2*x^3*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) -
2*a*b^3*d*x^3*Ei(-d*x)*e^(-c) + 2*a^3*b*d^2*x^2*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 4*a*b^3*d*x^3*Ei((b*d*x +
a*d)/b)*e^(c - a*d/b) + 2*a*b^3*d*x^3*Ei(d*x)*e^c + 2*a^3*b*d^2*x^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) - 4*a
*b^3*d*x^3*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) - 4*a^2*b^2*d*x^2*Ei(-d*x)*e^(-c) - 6*b^4*x^3*Ei(-d*x)*e^(-c) +
a^4*d^2*x*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 8*a^2*b^2*d*x^2*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 6*b^4*x^3*E
i((b*d*x + a*d)/b)*e^(c - a*d/b) + 4*a^2*b^2*d*x^2*Ei(d*x)*e^c - 6*b^4*x^3*Ei(d*x)*e^c + a^4*d^2*x*Ei(-(b*d*x
+ a*d)/b)*e^(-c + a*d/b) - 8*a^2*b^2*d*x^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + 6*b^4*x^3*Ei(-(b*d*x + a*d)/b
)*e^(-c + a*d/b) - a^2*b^2*d*x^2*e^(d*x + c) + a^2*b^2*d*x^2*e^(-d*x - c) - 2*a^3*b*d*x*Ei(-d*x)*e^(-c) - 12*a
*b^3*x^2*Ei(-d*x)*e^(-c) + 4*a^3*b*d*x*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 12*a*b^3*x^2*Ei((b*d*x + a*d)/b)*e^
(c - a*d/b) + 2*a^3*b*d*x*Ei(d*x)*e^c - 12*a*b^3*x^2*Ei(d*x)*e^c - 4*a^3*b*d*x*Ei(-(b*d*x + a*d)/b)*e^(-c + a*
d/b) + 12*a*b^3*x^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) - a^3*b*d*x*e^(d*x + c) - 6*a*b^3*x^2*e^(d*x + c) + a^
3*b*d*x*e^(-d*x - c) - 6*a*b^3*x^2*e^(-d*x - c) - 6*a^2*b^2*x*Ei(-d*x)*e^(-c) + 6*a^2*b^2*x*Ei((b*d*x + a*d)/b
)*e^(c - a*d/b) - 6*a^2*b^2*x*Ei(d*x)*e^c + 6*a^2*b^2*x*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) - 9*a^2*b^2*x*e^(d
*x + c) - 9*a^2*b^2*x*e^(-d*x - c) - 2*a^3*b*e^(d*x + c) - 2*a^3*b*e^(-d*x - c))/(a^4*b^3*x^3 + 2*a^5*b^2*x^2
+ a^6*b*x)

Mupad [F(-1)]

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

[In]

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

[Out]

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