\(\int \frac {\cosh (c+d x)}{x (a+b x)^3} \, dx\) [37]

Optimal result

Integrand size = 17, antiderivative size = 262 \[ \int \frac {\cosh (c+d x)}{x (a+b x)^3} \, dx=\frac {\cosh (c+d x)}{2 a (a+b x)^2}+\frac {\cosh (c+d x)}{a^2 (a+b x)}+\frac {\cosh (c) \text {Chi}(d x)}{a^3}-\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{a^3}-\frac {d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{2 a b^2}-\frac {d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{a^2 b}+\frac {d \sinh (c+d x)}{2 a b (a+b x)}+\frac {\sinh (c) \text {Shi}(d x)}{a^3}-\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^2 b}-\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^3}-\frac {d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{2 a b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6874, 3384, 3379, 3382, 3378} \[ \int \frac {\cosh (c+d x)}{x (a+b x)^3} \, dx=-\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{a^3}-\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{a^3}+\frac {\cosh (c) \text {Chi}(d x)}{a^3}+\frac {\sinh (c) \text {Shi}(d x)}{a^3}-\frac {d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{a^2 b}-\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{a^2 b}+\frac {\cosh (c+d x)}{a^2 (a+b x)}-\frac {d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{2 a b^2}-\frac {d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{2 a b^2}+\frac {d \sinh (c+d x)}{2 a b (a+b x)}+\frac {\cosh (c+d x)}{2 a (a+b x)^2} \]

[In]

Int[Cosh[c + d*x]/(x*(a + b*x)^3),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.87 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.72 \[ \int \frac {\cosh (c+d x)}{x (a+b x)^3} \, dx=-\frac {-3 a^2 b^2 \cosh (c+d x)-2 a b^3 x \cosh (c+d x)-2 b^2 (a+b x)^2 \cosh (c) \text {Chi}(d x)+(a+b x)^2 \text {Chi}\left (d \left (\frac {a}{b}+x\right )\right ) \left (\left (2 b^2+a^2 d^2\right ) \cosh \left (c-\frac {a d}{b}\right )+2 a b d \sinh \left (c-\frac {a d}{b}\right )\right )-a^3 b d \sinh (c+d x)-a^2 b^2 d x \sinh (c+d x)-2 a^2 b^2 \sinh (c) \text {Shi}(d x)-4 a b^3 x \sinh (c) \text {Shi}(d x)-2 b^4 x^2 \sinh (c) \text {Shi}(d x)+2 a^3 b d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+4 a^2 b^2 d x \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+2 a b^3 d x^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+2 a^2 b^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+a^4 d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+4 a b^3 x \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+2 a^3 b d^2 x \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+2 b^4 x^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+a^2 b^2 d^2 x^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )}{2 a^3 b^2 (a+b x)^2} \]

[In]

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

[Out]

-1/2*(-3*a^2*b^2*Cosh[c + d*x] - 2*a*b^3*x*Cosh[c + d*x] - 2*b^2*(a + b*x)^2*Cosh[c]*CoshIntegral[d*x] + (a +
b*x)^2*CoshIntegral[d*(a/b + x)]*((2*b^2 + a^2*d^2)*Cosh[c - (a*d)/b] + 2*a*b*d*Sinh[c - (a*d)/b]) - a^3*b*d*S
inh[c + d*x] - a^2*b^2*d*x*Sinh[c + d*x] - 2*a^2*b^2*Sinh[c]*SinhIntegral[d*x] - 4*a*b^3*x*Sinh[c]*SinhIntegra
l[d*x] - 2*b^4*x^2*Sinh[c]*SinhIntegral[d*x] + 2*a^3*b*d*Cosh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)] + 4*a^2*b
^2*d*x*Cosh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)] + 2*a*b^3*d*x^2*Cosh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)]
+ 2*a^2*b^2*Sinh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)] + a^4*d^2*Sinh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)]
+ 4*a*b^3*x*Sinh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)] + 2*a^3*b*d^2*x*Sinh[c - (a*d)/b]*SinhIntegral[d*(a/b
+ x)] + 2*b^4*x^2*Sinh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)] + a^2*b^2*d^2*x^2*Sinh[c - (a*d)/b]*SinhIntegra
l[d*(a/b + x)])/(a^3*b^2*(a + b*x)^2)

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.86

method	result	size
risch	\(-\frac {{\mathrm e}^{-d x -c} d \left (\left (d x +c \right ) a b d +a^{2} d^{2}-b c d a -2 \left (d x +c \right ) b^{2}-3 d a b +2 c \,b^{2}\right )}{4 a^{2} b \left (\left (d x +c \right )^{2} b^{2}+2 \left (d x +c \right ) a b d -2 \left (d x +c \right ) b^{2} c +a^{2} d^{2}-2 b c d a +b^{2} c^{2}\right )}-\frac {{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right )}{2 a^{3}}+\frac {{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right ) d^{2}}{4 b^{2} a}-\frac {{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right ) d}{2 b \,a^{2}}+\frac {{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right )}{2 a^{3}}-\frac {{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right )}{2 a^{3}}+\frac {d^{2} {\mathrm e}^{d x +c}}{4 a \,b^{2} \left (\frac {d a}{b}+d x \right )^{2}}+\frac {d^{2} {\mathrm e}^{d x +c}}{4 a \,b^{2} \left (\frac {d a}{b}+d x \right )}+\frac {d^{2} {\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right )}{4 a \,b^{2}}+\frac {d \,{\mathrm e}^{d x +c}}{2 a^{2} b \left (\frac {d a}{b}+d x \right )}+\frac {d \,{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right )}{2 a^{2} b}+\frac {{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right )}{2 a^{3}}\)	\(488\)

[In]

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

[Out]

-1/4*exp(-d*x-c)*d*((d*x+c)*a*b*d+a^2*d^2-b*c*d*a-2*(d*x+c)*b^2-3*d*a*b+2*c*b^2)/a^2/b/((d*x+c)^2*b^2+2*(d*x+c
)*a*b*d-2*(d*x+c)*b^2*c+a^2*d^2-2*b*c*d*a+b^2*c^2)-1/2/a^3*exp(-c)*Ei(1,d*x)+1/4/b^2/a*exp((a*d-b*c)/b)*Ei(1,d
*x+c+(a*d-b*c)/b)*d^2-1/2/b/a^2*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*d+1/2/a^3*exp((a*d-b*c)/b)*Ei(1,d*x+c
+(a*d-b*c)/b)-1/2/a^3*exp(c)*Ei(1,-d*x)+1/4/a/b^2*d^2*exp(d*x+c)/(d/b*a+d*x)^2+1/4/a/b^2*d^2*exp(d*x+c)/(d/b*a
+d*x)+1/4/a/b^2*d^2*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)+1/2/a^2*d/b*exp(d*x+c)/(d/b*a+d*x)+1/2/a^2*d/b*
exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)+1/2/a^3*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. 601 vs. \(2 (258) = 516\).

Time = 0.29 (sec) , antiderivative size = 601, normalized size of antiderivative = 2.29 \[ \int \frac {\cosh (c+d x)}{x (a+b x)^3} \, dx=\frac {2 \, {\left (2 \, a b^{3} x + 3 \, a^{2} b^{2}\right )} \cosh \left (d x + c\right ) + 2 \, {\left ({\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} {\rm Ei}\left (d x\right ) + {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) - {\left ({\left (a^{4} d^{2} + 2 \, a^{3} b d + 2 \, a^{2} b^{2} + {\left (a^{2} b^{2} d^{2} + 2 \, a b^{3} d + 2 \, b^{4}\right )} x^{2} + 2 \, {\left (a^{3} b d^{2} + 2 \, a^{2} b^{2} d + 2 \, a b^{3}\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{4} d^{2} - 2 \, a^{3} b d + 2 \, a^{2} b^{2} + {\left (a^{2} b^{2} d^{2} - 2 \, a b^{3} d + 2 \, b^{4}\right )} x^{2} + 2 \, {\left (a^{3} b d^{2} - 2 \, a^{2} b^{2} d + 2 \, a b^{3}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) + 2 \, {\left (a^{2} b^{2} d x + a^{3} b d\right )} \sinh \left (d x + c\right ) + 2 \, {\left ({\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} {\rm Ei}\left (d x\right ) - {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right ) + {\left ({\left (a^{4} d^{2} + 2 \, a^{3} b d + 2 \, a^{2} b^{2} + {\left (a^{2} b^{2} d^{2} + 2 \, a b^{3} d + 2 \, b^{4}\right )} x^{2} + 2 \, {\left (a^{3} b d^{2} + 2 \, a^{2} b^{2} d + 2 \, a b^{3}\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\left (a^{4} d^{2} - 2 \, a^{3} b d + 2 \, a^{2} b^{2} + {\left (a^{2} b^{2} d^{2} - 2 \, a b^{3} d + 2 \, b^{4}\right )} x^{2} + 2 \, {\left (a^{3} b d^{2} - 2 \, a^{2} b^{2} d + 2 \, a b^{3}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{4 \, {\left (a^{3} b^{4} x^{2} + 2 \, a^{4} b^{3} x + a^{5} b^{2}\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 837 vs. \(2 (258) = 516\).

Time = 0.28 (sec) , antiderivative size = 837, normalized size of antiderivative = 3.19 \[ \int \frac {\cosh (c+d x)}{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 )} + 2 \, 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 )} - 2 \, 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 \, b^{4} x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{4} d^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 4 \, 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 )} - 2 \, b^{4} x^{2} {\rm Ei}\left (d x\right ) e^{c} + a^{4} d^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} - 4 \, 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 b^{3} x {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 2 \, 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 b^{3} x {\rm Ei}\left (d x\right ) e^{c} - 2 \, 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 )} - 2 \, a b^{3} x e^{\left (d x + c\right )} + a^{3} b d e^{\left (-d x - c\right )} - 2 \, a b^{3} x e^{\left (-d x - c\right )} - 2 \, a^{2} b^{2} {\rm Ei}\left (-d x\right ) e^{\left (-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 (d x\right ) e^{c} + 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 (a^{3} b^{4} x^{2} + 2 \, a^{4} b^{3} x + a^{5} b^{2}\right )}} \]

[In]

integrate(cosh(d*x+c)/x/(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) + 2*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) - 2*a*b^3*d*x^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) - 2*b^4*x^2*Ei(-
d*x)*e^(-c) + a^4*d^2*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 4*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) - 2*b^4*x^2*Ei(d*x)*e^c + a^4*d^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b
) - 4*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*b^3*x*Ei(-d*x)*e^(-c) + 2*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*b^3*x*Ei(d*x)*e^c - 2*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) - 2*a*b^3*x*e^(d*x + c)
+ a^3*b*d*e^(-d*x - c) - 2*a*b^3*x*e^(-d*x - c) - 2*a^2*b^2*Ei(-d*x)*e^(-c) + 2*a^2*b^2*Ei((b*d*x + a*d)/b)*e
^(c - a*d/b) - 2*a^2*b^2*Ei(d*x)*e^c + 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))/(a^3*b^4*x^2 + 2*a^4*b^3*x + a^5*b^2)

Mupad [F(-1)]

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

[In]

int(cosh(c + d*x)/(x*(a + b*x)^3),x)

[Out]

int(cosh(c + d*x)/(x*(a + b*x)^3), x)