\(\int \frac {x \cosh (c+d x)}{(a+b x)^3} \, dx\) [35]

Optimal result

Integrand size = 15, antiderivative size = 178 \[ \int \frac {x \cosh (c+d x)}{(a+b x)^3} \, dx=\frac {a \cosh (c+d x)}{2 b^2 (a+b x)^2}-\frac {\cosh (c+d x)}{b^2 (a+b x)}-\frac {a d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{2 b^4}+\frac {d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{b^3}+\frac {a d \sinh (c+d x)}{2 b^3 (a+b x)}+\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^3}-\frac {a d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{2 b^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.89 \[ \int \frac {x \cosh (c+d x)}{(a+b x)^3} \, dx=-\frac {b \cosh (d x) (b (a+2 b x) \cosh (c)-a d (a+b x) \sinh (c))-b (a d (a+b x) \cosh (c)-b (a+2 b x) \sinh (c)) \sinh (d x)+d (a+b x)^2 \left (\text {Chi}\left (d \left (\frac {a}{b}+x\right )\right ) \left (a d \cosh \left (c-\frac {a d}{b}\right )-2 b \sinh \left (c-\frac {a d}{b}\right )\right )+\left (-2 b \cosh \left (c-\frac {a d}{b}\right )+a d \sinh \left (c-\frac {a d}{b}\right )\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )\right )}{2 b^4 (a+b x)^2} \] Input:

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

Output:

-1/2*(b*Cosh[d*x]*(b*(a + 2*b*x)*Cosh[c] - a*d*(a + b*x)*Sinh[c]) - b*(a*d 
*(a + b*x)*Cosh[c] - b*(a + 2*b*x)*Sinh[c])*Sinh[d*x] + d*(a + b*x)^2*(Cos 
hIntegral[d*(a/b + x)]*(a*d*Cosh[c - (a*d)/b] - 2*b*Sinh[c - (a*d)/b]) + ( 
-2*b*Cosh[c - (a*d)/b] + a*d*Sinh[c - (a*d)/b])*SinhIntegral[d*(a/b + x)]) 
)/(b^4*(a + b*x)^2)
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {7293, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\cosh (c+d x)}{b (a+b x)^2}-\frac {a \cosh (c+d x)}{b (a+b x)^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{2 b^4}-\frac {a d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{2 b^4}+\frac {d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {a d \sinh (c+d x)}{2 b^3 (a+b x)}-\frac {\cosh (c+d x)}{b^2 (a+b x)}+\frac {a \cosh (c+d x)}{2 b^2 (a+b x)^2}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(434\) vs. \(2(175)=350\).

Time = 0.62 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.44

method	result	size
risch	\(-\frac {d^{3} {\mathrm e}^{-d x -c} a x}{4 b^{2} \left (b^{2} d^{2} x^{2}+2 a b \,d^{2} x +d^{2} a^{2}\right )}-\frac {d^{3} {\mathrm e}^{-d x -c} a^{2}}{4 b^{3} \left (b^{2} d^{2} x^{2}+2 a b \,d^{2} x +d^{2} a^{2}\right )}-\frac {d^{2} {\mathrm e}^{-d x -c} x}{2 b \left (b^{2} d^{2} x^{2}+2 a b \,d^{2} x +d^{2} a^{2}\right )}-\frac {d^{2} {\mathrm e}^{-d x -c} a}{4 b^{2} \left (b^{2} d^{2} x^{2}+2 a b \,d^{2} x +d^{2} a^{2}\right )}+\frac {d^{2} {\mathrm e}^{\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (d x +c +\frac {a d -c b}{b}\right ) a}{4 b^{4}}+\frac {d \,{\mathrm e}^{\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (d x +c +\frac {a d -c b}{b}\right )}{2 b^{3}}+\frac {d^{2} {\mathrm e}^{d x +c} a}{4 b^{4} \left (\frac {d a}{b}+d x \right )^{2}}+\frac {d^{2} {\mathrm e}^{d x +c} a}{4 b^{4} \left (\frac {d a}{b}+d x \right )}+\frac {d^{2} {\mathrm e}^{-\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (-d x -c -\frac {a d -c b}{b}\right ) a}{4 b^{4}}-\frac {d \,{\mathrm e}^{d x +c}}{2 b^{3} \left (\frac {d a}{b}+d x \right )}-\frac {d \,{\mathrm e}^{-\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (-d x -c -\frac {a d -c b}{b}\right )}{2 b^{3}}\)	\(435\)

Input:

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

Output:

-1/4*d^3*exp(-d*x-c)/b^2/(b^2*d^2*x^2+2*a*b*d^2*x+a^2*d^2)*a*x-1/4*d^3*exp 
(-d*x-c)/b^3/(b^2*d^2*x^2+2*a*b*d^2*x+a^2*d^2)*a^2-1/2*d^2*exp(-d*x-c)/b/( 
b^2*d^2*x^2+2*a*b*d^2*x+a^2*d^2)*x-1/4*d^2*exp(-d*x-c)/b^2/(b^2*d^2*x^2+2* 
a*b*d^2*x+a^2*d^2)*a+1/4*d^2/b^4*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)* 
a+1/2*d/b^3*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)+1/4*d^2/b^4*exp(d*x+c 
)/(d/b*a+d*x)^2*a+1/4*d^2/b^4*exp(d*x+c)/(d/b*a+d*x)*a+1/4*d^2/b^4*exp(-(a 
*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a-1/2*d/b^3*exp(d*x+c)/(d/b*a+d*x)-1/2 
*d/b^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. 373 vs. \(2 (175) = 350\).

Time = 0.12 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.10 \[ \int \frac {x \cosh (c+d x)}{(a+b x)^3} \, dx=-\frac {2 \, {\left (2 \, b^{3} x + a b^{2}\right )} \cosh \left (d x + c\right ) + {\left ({\left (a^{3} d^{2} - 2 \, a^{2} b d + {\left (a b^{2} d^{2} - 2 \, b^{3} d\right )} x^{2} + 2 \, {\left (a^{2} b d^{2} - 2 \, a b^{2} d\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{3} d^{2} + 2 \, a^{2} b d + {\left (a b^{2} d^{2} + 2 \, b^{3} d\right )} x^{2} + 2 \, {\left (a^{2} b d^{2} + 2 \, a b^{2} d\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) - 2 \, {\left (a b^{2} d x + a^{2} b d\right )} \sinh \left (d x + c\right ) - {\left ({\left (a^{3} d^{2} - 2 \, a^{2} b d + {\left (a b^{2} d^{2} - 2 \, b^{3} d\right )} x^{2} + 2 \, {\left (a^{2} b d^{2} - 2 \, a b^{2} d\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\left (a^{3} d^{2} + 2 \, a^{2} b d + {\left (a b^{2} d^{2} + 2 \, b^{3} d\right )} x^{2} + 2 \, {\left (a^{2} b d^{2} + 2 \, a b^{2} d\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{4 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 529 vs. \(2 (175) = 350\).

Time = 0.11 (sec) , antiderivative size = 529, normalized size of antiderivative = 2.97 \[ \int \frac {x \cosh (c+d x)}{(a+b x)^3} \, dx=-\frac {a b^{2} d^{2} x^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + a b^{2} d^{2} x^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 2 \, a^{2} b d^{2} x {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} - 2 \, b^{3} d x^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 2 \, a^{2} b d^{2} x {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 2 \, b^{3} d x^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + a^{3} d^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} - 4 \, a b^{2} d x {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + a^{3} d^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 4 \, a b^{2} d x {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} - a b^{2} d x e^{\left (d x + c\right )} + a b^{2} d x e^{\left (-d x - c\right )} - 2 \, a^{2} b d {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 2 \, a^{2} b d {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} - a^{2} b d e^{\left (d x + c\right )} + 2 \, b^{3} x e^{\left (d x + c\right )} + a^{2} b d e^{\left (-d x - c\right )} + 2 \, b^{3} x e^{\left (-d x - c\right )} + a b^{2} e^{\left (d x + c\right )} + a b^{2} e^{\left (-d x - c\right )}}{4 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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