\(\int \frac {x^2 \cosh (c+d x)}{(a+b x)^3} \, dx\) [34]

Optimal result

Integrand size = 17, antiderivative size = 241 \[ \int \frac {x^2 \cosh (c+d x)}{(a+b x)^3} \, dx=-\frac {a^2 \cosh (c+d x)}{2 b^3 (a+b x)^2}+\frac {2 a \cosh (c+d x)}{b^3 (a+b x)}+\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^3}+\frac {a^2 d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{2 b^5}-\frac {2 a d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{b^4}-\frac {a^2 d \sinh (c+d x)}{2 b^4 (a+b x)}-\frac {2 a d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^4}+\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^3}+\frac {a^2 d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{2 b^5} \] Output:

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

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.63 \[ \int \frac {x^2 \cosh (c+d x)}{(a+b x)^3} \, dx=\frac {\text {Chi}\left (d \left (\frac {a}{b}+x\right )\right ) \left (\left (2 b^2+a^2 d^2\right ) \cosh \left (c-\frac {a d}{b}\right )-4 a b d \sinh \left (c-\frac {a d}{b}\right )\right )-\frac {a b (-b (3 a+4 b x) \cosh (c+d x)+a d (a+b x) \sinh (c+d x))}{(a+b x)^2}+\left (-4 a b d \cosh \left (c-\frac {a d}{b}\right )+\left (2 b^2+a^2 d^2\right ) \sinh \left (c-\frac {a d}{b}\right )\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )}{2 b^5} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^2 d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{2 b^5}+\frac {a^2 d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{2 b^5}-\frac {a^2 d \sinh (c+d x)}{2 b^4 (a+b x)}-\frac {a^2 \cosh (c+d x)}{2 b^3 (a+b x)^2}-\frac {2 a d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^4}-\frac {2 a d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^4}+\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {2 a \cosh (c+d x)}{b^3 (a+b x)}\)

Input:

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

Output:

-1/2*(a^2*Cosh[c + d*x])/(b^3*(a + b*x)^2) + (2*a*Cosh[c + d*x])/(b^3*(a + 
 b*x)) + (Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/b^3 + (a^2*d^2*Co 
sh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/(2*b^5) - (2*a*d*CoshIntegral 
[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/b^4 - (a^2*d*Sinh[c + d*x])/(2*b^4*(a + 
 b*x)) - (2*a*d*Cosh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^4 + (Sinh 
[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^3 + (a^2*d^2*Sinh[c - (a*d)/b 
]*SinhIntegral[(a*d)/b + d*x])/(2*b^5)
 

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. \(916\) vs. \(2(240)=480\).

Time = 0.67 (sec) , antiderivative size = 917, normalized size of antiderivative = 3.80

method	result	size
risch	\(-\frac {{\mathrm e}^{-\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (-d x -c -\frac {a d -c b}{b}\right ) a^{4} d^{2}+2 \,{\mathrm e}^{-\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (-d x -c -\frac {a d -c b}{b}\right ) b^{4} x^{2}+{\mathrm e}^{d x +c} a^{3} b d -4 \,{\mathrm e}^{d x +c} a \,b^{3} x +2 \,{\mathrm e}^{-\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (-d x -c -\frac {a d -c b}{b}\right ) a^{2} b^{2}+{\mathrm e}^{d x +c} a^{2} b^{2} d x -4 \,{\mathrm e}^{-\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (-d x -c -\frac {a d -c b}{b}\right ) a^{3} b d +4 \,{\mathrm e}^{-\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (-d x -c -\frac {a d -c b}{b}\right ) a \,b^{3} x -3 \,{\mathrm e}^{d x +c} a^{2} b^{2}+{\mathrm e}^{\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (d x +c +\frac {a d -c b}{b}\right ) a^{2} b^{2} d^{2} x^{2}+2 \,{\mathrm e}^{\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (d x +c +\frac {a d -c b}{b}\right ) a^{3} b \,d^{2} x +4 \,{\mathrm e}^{\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (d x +c +\frac {a d -c b}{b}\right ) a \,b^{3} d \,x^{2}+8 \,{\mathrm e}^{\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (d x +c +\frac {a d -c b}{b}\right ) a^{2} b^{2} d x +{\mathrm e}^{-\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (-d x -c -\frac {a d -c b}{b}\right ) a^{2} b^{2} d^{2} x^{2}+2 \,{\mathrm e}^{-\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (-d x -c -\frac {a d -c b}{b}\right ) a^{3} b \,d^{2} x -4 \,{\mathrm e}^{-\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (-d x -c -\frac {a d -c b}{b}\right ) a \,b^{3} d \,x^{2}-8 \,{\mathrm e}^{-\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (-d x -c -\frac {a d -c b}{b}\right ) a^{2} b^{2} d x -{\mathrm e}^{-d x -c} a^{2} b^{2} d x +4 \,{\mathrm e}^{\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (d x +c +\frac {a d -c b}{b}\right ) a^{3} b d +4 \,{\mathrm e}^{\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (d x +c +\frac {a d -c b}{b}\right ) a \,b^{3} x -3 \,{\mathrm e}^{-d x -c} a^{2} b^{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} d^{2}+2 \,{\mathrm e}^{\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (d x +c +\frac {a d -c b}{b}\right ) b^{4} x^{2}-{\mathrm e}^{-d x -c} a^{3} b d -4 \,{\mathrm e}^{-d x -c} a \,b^{3} x +2 \,{\mathrm e}^{\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (d x +c +\frac {a d -c b}{b}\right ) a^{2} b^{2}}{4 b^{5} \left (b x +a \right )^{2}}\)	\(917\)

Input:

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

Output:

-1/4*(exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a^4*d^2+2*exp(-(a*d-b*c)/ 
b)*Ei(1,-d*x-c-(a*d-b*c)/b)*b^4*x^2+exp(d*x+c)*a^3*b*d-4*exp(d*x+c)*a*b^3* 
x+2*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a^2*b^2+exp(d*x+c)*a^2*b^2* 
d*x-4*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a^3*b*d+4*exp(-(a*d-b*c)/ 
b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a*b^3*x-3*exp(d*x+c)*a^2*b^2+exp((a*d-b*c)/b)* 
Ei(1,d*x+c+(a*d-b*c)/b)*a^2*b^2*d^2*x^2+2*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d 
-b*c)/b)*a^3*b*d^2*x+4*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*a*b^3*d*x^ 
2+8*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*a^2*b^2*d*x+exp(-(a*d-b*c)/b) 
*Ei(1,-d*x-c-(a*d-b*c)/b)*a^2*b^2*d^2*x^2+2*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c- 
(a*d-b*c)/b)*a^3*b*d^2*x-4*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a*b^ 
3*d*x^2-8*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a^2*b^2*d*x-exp(-d*x- 
c)*a^2*b^2*d*x+4*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*a^3*b*d+4*exp((a 
*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*a*b^3*x-3*exp(-d*x-c)*a^2*b^2+exp((a*d- 
b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*a^4*d^2+2*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d 
-b*c)/b)*b^4*x^2-exp(-d*x-c)*a^3*b*d-4*exp(-d*x-c)*a*b^3*x+2*exp((a*d-b*c) 
/b)*Ei(1,d*x+c+(a*d-b*c)/b)*a^2*b^2)/b^5/(b*x+a)^2
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.97 \[ \int \frac {x^2 \cosh (c+d x)}{(a+b x)^3} \, dx=\frac {2 \, {\left (4 \, a b^{3} x + 3 \, a^{2} b^{2}\right )} \cosh \left (d x + c\right ) + {\left ({\left (a^{4} d^{2} - 4 \, a^{3} b d + 2 \, a^{2} b^{2} + {\left (a^{2} b^{2} d^{2} - 4 \, a b^{3} d + 2 \, b^{4}\right )} x^{2} + 2 \, {\left (a^{3} b d^{2} - 4 \, a^{2} b^{2} d + 2 \, a b^{3}\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{4} d^{2} + 4 \, a^{3} b d + 2 \, a^{2} b^{2} + {\left (a^{2} b^{2} d^{2} + 4 \, a b^{3} d + 2 \, b^{4}\right )} x^{2} + 2 \, {\left (a^{3} b d^{2} + 4 \, a^{2} b^{2} d + 2 \, a b^{3}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) - 2 \, {\left (a^{2} b^{2} d x + a^{3} b d\right )} \sinh \left (d x + c\right ) - {\left ({\left (a^{4} d^{2} - 4 \, a^{3} b d + 2 \, a^{2} b^{2} + {\left (a^{2} b^{2} d^{2} - 4 \, a b^{3} d + 2 \, b^{4}\right )} x^{2} + 2 \, {\left (a^{3} b d^{2} - 4 \, a^{2} b^{2} d + 2 \, a b^{3}\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\left (a^{4} d^{2} + 4 \, a^{3} b d + 2 \, a^{2} b^{2} + {\left (a^{2} b^{2} d^{2} + 4 \, a b^{3} d + 2 \, b^{4}\right )} x^{2} + 2 \, {\left (a^{3} b d^{2} + 4 \, a^{2} b^{2} d + 2 \, a b^{3}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{4 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 741 vs. \(2 (240) = 480\).

Time = 0.12 (sec) , antiderivative size = 741, normalized size of antiderivative = 3.07 \[ \int \frac {x^2 \cosh (c+d x)}{(a+b x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

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*E 
i(-(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) - 4*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) + 4*a*b^3*d*x^2*Ei(-(b*d*x + a*d)/b 
)*e^(-c + a*d/b) + a^4*d^2*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) - 8*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^4*d^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + 8*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^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^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) + 4*a*b^3*x*e^(d*x + c) + 
a^3*b*d*e^(-d*x - c) + 4*a*b^3*x*e^(-d*x - c) + 2*a^2*b^2*Ei((b*d*x + a*d) 
/b)*e^(c - a*d/b) + 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))/(b^7*x^2 + 2*a*b^6*x + a^2*b^5)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2 \cosh (c+d x)}{(a+b x)^3} \, dx=\text {too large to display} \] Input:

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

Output:

(e**(2*c + 2*d*x)*a**2*d**2*x - e**(2*c + 2*d*x)*a**2*d - 2*e**(2*c + 2*d* 
x)*a*b - 4*e**(2*c + 2*d*x)*b**2*x - e**(2*c + d*x)*int((e**(d*x)*x)/(a**5 
*d**2 + 3*a**4*b*d**2*x + 3*a**3*b**2*d**2*x**2 - 4*a**3*b**2 + a**2*b**3* 
d**2*x**3 - 12*a**2*b**3*x - 12*a*b**4*x**2 - 4*b**5*x**3),x)*a**7*d**5 - 
2*e**(2*c + d*x)*int((e**(d*x)*x)/(a**5*d**2 + 3*a**4*b*d**2*x + 3*a**3*b* 
*2*d**2*x**2 - 4*a**3*b**2 + a**2*b**3*d**2*x**3 - 12*a**2*b**3*x - 12*a*b 
**4*x**2 - 4*b**5*x**3),x)*a**6*b*d**5*x + 2*e**(2*c + d*x)*int((e**(d*x)* 
x)/(a**5*d**2 + 3*a**4*b*d**2*x + 3*a**3*b**2*d**2*x**2 - 4*a**3*b**2 + a* 
*2*b**3*d**2*x**3 - 12*a**2*b**3*x - 12*a*b**4*x**2 - 4*b**5*x**3),x)*a**6 
*b*d**4 - e**(2*c + d*x)*int((e**(d*x)*x)/(a**5*d**2 + 3*a**4*b*d**2*x + 3 
*a**3*b**2*d**2*x**2 - 4*a**3*b**2 + a**2*b**3*d**2*x**3 - 12*a**2*b**3*x 
- 12*a*b**4*x**2 - 4*b**5*x**3),x)*a**5*b**2*d**5*x**2 + 4*e**(2*c + d*x)* 
int((e**(d*x)*x)/(a**5*d**2 + 3*a**4*b*d**2*x + 3*a**3*b**2*d**2*x**2 - 4* 
a**3*b**2 + a**2*b**3*d**2*x**3 - 12*a**2*b**3*x - 12*a*b**4*x**2 - 4*b**5 
*x**3),x)*a**5*b**2*d**4*x + 10*e**(2*c + d*x)*int((e**(d*x)*x)/(a**5*d**2 
 + 3*a**4*b*d**2*x + 3*a**3*b**2*d**2*x**2 - 4*a**3*b**2 + a**2*b**3*d**2* 
x**3 - 12*a**2*b**3*x - 12*a*b**4*x**2 - 4*b**5*x**3),x)*a**5*b**2*d**3 + 
2*e**(2*c + d*x)*int((e**(d*x)*x)/(a**5*d**2 + 3*a**4*b*d**2*x + 3*a**3*b* 
*2*d**2*x**2 - 4*a**3*b**2 + a**2*b**3*d**2*x**3 - 12*a**2*b**3*x - 12*a*b 
**4*x**2 - 4*b**5*x**3),x)*a**4*b**3*d**4*x**2 + 20*e**(2*c + d*x)*int(...