3.1.29 \(\int \frac {x \cosh (c+d x)}{(a+b x)^2} \, dx\) [29]

3.1.29.1 Optimal result

Integrand size = 15, antiderivative size = 125 \[ \int \frac {x \cosh (c+d x)}{(a+b x)^2} \, dx=\frac {a \cosh (c+d x)}{b^2 (a+b x)}+\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^2}-\frac {a d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{b^3}-\frac {a d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^3}+\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^2} \]

output

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

3.1.29.2 Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.78 \[ \int \frac {x \cosh (c+d x)}{(a+b x)^2} \, dx=\frac {\frac {a b \cosh (c+d x)}{a+b x}+\text {Chi}\left (d \left (\frac {a}{b}+x\right )\right ) \left (b \cosh \left (c-\frac {a d}{b}\right )-a d \sinh \left (c-\frac {a d}{b}\right )\right )+\left (-a d \cosh \left (c-\frac {a d}{b}\right )+b \sinh \left (c-\frac {a d}{b}\right )\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )}{b^3} \]

input

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

output

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

3.1.29.3 Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 125, 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.

\(\displaystyle \int \frac {x \cosh (c+d x)}{(a+b x)^2} \, dx\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

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

3.1.29.3.1 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] 
]
 

3.1.29.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(293\) vs. \(2(129)=258\).

Time = 0.20 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.35

method	result	size
risch	\(\frac {d \,{\mathrm e}^{-d x -c} a}{2 b^{2} \left (d x b +d a \right )}-\frac {d \,{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right ) a}{2 b^{3}}-\frac {{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right )}{2 b^{2}}+\frac {{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right ) a b d x +{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right ) a^{2} d -{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right ) b^{2} x -{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right ) a b +{\mathrm e}^{d x +c} a b}{2 b^{3} \left (b x +a \right )}\)	\(294\)

input

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

output

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

3.1.29.5 Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.60 \[ \int \frac {x \cosh (c+d x)}{(a+b x)^2} \, dx=\frac {2 \, a b \cosh \left (d x + c\right ) - {\left ({\left (a^{2} d - a b + {\left (a b d - b^{2}\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\left (a^{2} d + a b + {\left (a b d + b^{2}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) + {\left ({\left (a^{2} d - a b + {\left (a b d - b^{2}\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{2} d + a b + {\left (a b d + b^{2}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{2 \, {\left (b^{4} x + a b^{3}\right )}} \]

input

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

output

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

3.1.29.6 Sympy [F]

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

input

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

output

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

3.1.29.7 Maxima [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.42 \[ \int \frac {x \cosh (c+d x)}{(a+b x)^2} \, dx=-\frac {1}{2} \, {\left (a {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b^{3}} - \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b^{3}}\right )} + \frac {\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b} + \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b}}{b d} + \frac {2 \, \cosh \left (d x + c\right ) \log \left (b x + a\right )}{b^{2} d}\right )} d + {\left (\frac {a}{b^{3} x + a b^{2}} + \frac {\log \left (b x + a\right )}{b^{2}}\right )} \cosh \left (d x + c\right ) \]

input

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

output

-1/2*(a*(e^(-c + a*d/b)*exp_integral_e(1, (b*x + a)*d/b)/b^3 - e^(c - a*d/ 
b)*exp_integral_e(1, -(b*x + a)*d/b)/b^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*d) + 2*cosh(d*x + c)*log(b*x + a)/(b^2*d))*d + (a/(b^3*x + a*b^2) + log 
(b*x + a)/b^2)*cosh(d*x + c)
 

3.1.29.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 994 vs. \(2 (129) = 258\).

Time = 0.31 (sec) , antiderivative size = 994, normalized size of antiderivative = 7.95 \[ \int \frac {x \cosh (c+d x)}{(a+b x)^2} \, dx=-\frac {{\left ({\left (b x + a\right )} a {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d^{2} {\rm Ei}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (\frac {b c - a d}{b}\right )} - a b c d^{2} {\rm Ei}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (\frac {b c - a d}{b}\right )} + a^{2} d^{3} {\rm Ei}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (\frac {b c - a d}{b}\right )} - {\left (b x + a\right )} a {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d^{2} {\rm Ei}\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (-\frac {b c - a d}{b}\right )} + a b c d^{2} {\rm Ei}\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (-\frac {b c - a d}{b}\right )} - a^{2} d^{3} {\rm Ei}\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (-\frac {b c - a d}{b}\right )} - {\left (b x + a\right )} b {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d {\rm Ei}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (\frac {b c - a d}{b}\right )} + b^{2} c d {\rm Ei}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (\frac {b c - a d}{b}\right )} - a b d^{2} {\rm Ei}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (\frac {b c - a d}{b}\right )} - {\left (b x + a\right )} b {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d {\rm Ei}\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (-\frac {b c - a d}{b}\right )} + b^{2} c d {\rm Ei}\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (-\frac {b c - a d}{b}\right )} - a b d^{2} {\rm Ei}\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (-\frac {b c - a d}{b}\right )} - a b d^{2} e^{\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}}{b}\right )} - a b d^{2} e^{\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}}{b}\right )}\right )} b}{2 \, {\left ({\left (b x + a\right )} b^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b^{5} c + a b^{4} d\right )} d} \]

input

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

output

-1/2*((b*x + a)*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^2*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*b* 
c*d^2*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^2*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) - (b*x + a)*a*(b*c/(b*x + a) - a*d/(b* 
x + a) + d)*d^2*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*b*c*d^2*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^2*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) 
- (b*x + a)*b*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d*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^2*c*d*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*b*d^2*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)*b*(b*c/(b*x + a) - a*d/(b*x + a) + 
 d)*d*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^2*c*d*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*b*d^2*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*b*d^2*e^ 
((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) - a*b*d^2*e^(-(b*x + a)* 
(b*c/(b*x + a) - a*d/(b*x + a) + d)/b))*b/(((b*x + a)*b^4*(b*c/(b*x + a...
 

3.1.29.9 Mupad [F(-1)]

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

input

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

output

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