\(\int \frac {x^4 \cosh (c+d x)}{(a+b x)^2} \, dx\) [26]

Optimal result

Integrand size = 17, antiderivative size = 231 \[ \int \frac {x^4 \cosh (c+d x)}{(a+b x)^2} \, dx=\frac {2 a \cosh (c+d x)}{b^3 d^2}-\frac {2 x \cosh (c+d x)}{b^2 d^2}-\frac {a^4 \cosh (c+d x)}{b^5 (a+b x)}-\frac {4 a^3 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^5}+\frac {a^4 d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{b^6}+\frac {2 \sinh (c+d x)}{b^2 d^3}+\frac {3 a^2 \sinh (c+d x)}{b^4 d}-\frac {2 a x \sinh (c+d x)}{b^3 d}+\frac {x^2 \sinh (c+d x)}{b^2 d}+\frac {a^4 d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^6}-\frac {4 a^3 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^5} \] Output:

2*a*cosh(d*x+c)/b^3/d^2-2*x*cosh(d*x+c)/b^2/d^2-a^4*cosh(d*x+c)/b^5/(b*x+a 
)-4*a^3*cosh(-c+a*d/b)*Chi(a*d/b+d*x)/b^5-a^4*d*Chi(a*d/b+d*x)*sinh(-c+a*d 
/b)/b^6+2*sinh(d*x+c)/b^2/d^3+3*a^2*sinh(d*x+c)/b^4/d-2*a*x*sinh(d*x+c)/b^ 
3/d+x^2*sinh(d*x+c)/b^2/d+a^4*d*cosh(-c+a*d/b)*Shi(a*d/b+d*x)/b^6+4*a^3*si 
nh(-c+a*d/b)*Shi(a*d/b+d*x)/b^5
 

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.75 \[ \int \frac {x^4 \cosh (c+d x)}{(a+b x)^2} \, dx=\frac {-\frac {b \left (-2 a^2 b^2+a^4 d^2+2 b^4 x^2\right ) \cosh (c+d x)}{d^2 (a+b x)}+a^3 \text {Chi}\left (d \left (\frac {a}{b}+x\right )\right ) \left (-4 b \cosh \left (c-\frac {a d}{b}\right )+a d \sinh \left (c-\frac {a d}{b}\right )\right )+\frac {b^2 \left (3 a^2 d^2-2 a b d^2 x+b^2 \left (2+d^2 x^2\right )\right ) \sinh (c+d x)}{d^3}+a^3 \left (a d \cosh \left (c-\frac {a d}{b}\right )-4 b \sinh \left (c-\frac {a d}{b}\right )\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )}{b^6} \] Input:

Integrate[(x^4*Cosh[c + d*x])/(a + b*x)^2,x]
 

Output:

(-((b*(-2*a^2*b^2 + a^4*d^2 + 2*b^4*x^2)*Cosh[c + d*x])/(d^2*(a + b*x))) + 
 a^3*CoshIntegral[d*(a/b + x)]*(-4*b*Cosh[c - (a*d)/b] + a*d*Sinh[c - (a*d 
)/b]) + (b^2*(3*a^2*d^2 - 2*a*b*d^2*x + b^2*(2 + d^2*x^2))*Sinh[c + d*x])/ 
d^3 + a^3*(a*d*Cosh[c - (a*d)/b] - 4*b*Sinh[c - (a*d)/b])*SinhIntegral[d*( 
a/b + x)])/b^6
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 231, 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.

\(\displaystyle \int \frac {x^4 \cosh (c+d x)}{(a+b x)^2} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {a^4 \cosh (c+d x)}{b^4 (a+b x)^2}-\frac {4 a^3 \cosh (c+d x)}{b^4 (a+b x)}+\frac {3 a^2 \cosh (c+d x)}{b^4}-\frac {2 a x \cosh (c+d x)}{b^3}+\frac {x^2 \cosh (c+d x)}{b^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^4 d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^6}+\frac {a^4 d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^6}-\frac {a^4 \cosh (c+d x)}{b^5 (a+b x)}-\frac {4 a^3 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^5}-\frac {4 a^3 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^5}+\frac {3 a^2 \sinh (c+d x)}{b^4 d}+\frac {2 a \cosh (c+d x)}{b^3 d^2}-\frac {2 a x \sinh (c+d x)}{b^3 d}+\frac {2 \sinh (c+d x)}{b^2 d^3}-\frac {2 x \cosh (c+d x)}{b^2 d^2}+\frac {x^2 \sinh (c+d x)}{b^2 d}\)

Input:

Int[(x^4*Cosh[c + d*x])/(a + b*x)^2,x]
 

Output:

(2*a*Cosh[c + d*x])/(b^3*d^2) - (2*x*Cosh[c + d*x])/(b^2*d^2) - (a^4*Cosh[ 
c + d*x])/(b^5*(a + b*x)) - (4*a^3*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b 
+ d*x])/b^5 + (a^4*d*CoshIntegral[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/b^6 + 
(2*Sinh[c + d*x])/(b^2*d^3) + (3*a^2*Sinh[c + d*x])/(b^4*d) - (2*a*x*Sinh[ 
c + d*x])/(b^3*d) + (x^2*Sinh[c + d*x])/(b^2*d) + (a^4*d*Cosh[c - (a*d)/b] 
*SinhIntegral[(a*d)/b + d*x])/b^6 - (4*a^3*Sinh[c - (a*d)/b]*SinhIntegral[ 
(a*d)/b + d*x])/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. \(668\) vs. \(2(236)=472\).

Time = 0.68 (sec) , antiderivative size = 669, normalized size of antiderivative = 2.90

method	result	size
risch	\(-\frac {2 \,{\mathrm e}^{-d x -c} b^{5} x +2 \,{\mathrm e}^{-d x -c} a \,b^{4}+3 \,{\mathrm e}^{-d x -c} a^{3} b^{2} d^{2}-2 \,{\mathrm e}^{-d x -c} a^{2} b^{3} d -{\mathrm e}^{\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (d x +c +\frac {a d -c b}{b}\right ) a^{4} b \,d^{4} x -4 \,{\mathrm e}^{\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (d x +c +\frac {a d -c b}{b}\right ) a^{3} b^{2} d^{3} x -{\mathrm e}^{d x +c} b^{5} d^{2} x^{3}+{\mathrm e}^{-\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (-d x -c -\frac {a d -c b}{b}\right ) a^{5} d^{4}+{\mathrm e}^{d x +c} a^{4} b \,d^{3}+2 \,{\mathrm e}^{d x +c} b^{5} d \,x^{2}-3 \,{\mathrm e}^{d x +c} a^{3} b^{2} d^{2}-2 \,{\mathrm e}^{d x +c} a^{2} b^{3} d -{\mathrm e}^{-d x -c} a \,b^{4} d^{2} x^{2}-2 \,{\mathrm e}^{d x +c} b^{5} x -2 \,{\mathrm e}^{d x +c} a \,b^{4}+{\mathrm e}^{-\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (-d x -c -\frac {a d -c b}{b}\right ) a^{4} b \,d^{4} x -4 \,{\mathrm e}^{-\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (-d x -c -\frac {a d -c b}{b}\right ) a^{3} b^{2} d^{3} x +{\mathrm e}^{-d x -c} b^{5} d^{2} x^{3}-{\mathrm e}^{\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (d x +c +\frac {a d -c b}{b}\right ) a^{5} d^{4}+{\mathrm e}^{-d x -c} a^{4} b \,d^{3}+2 \,{\mathrm e}^{-d x -c} b^{5} d \,x^{2}+{\mathrm e}^{d x +c} a \,b^{4} d^{2} x^{2}-4 \,{\mathrm e}^{-\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (-d x -c -\frac {a d -c b}{b}\right ) a^{4} b \,d^{3}-{\mathrm e}^{d x +c} a^{2} b^{3} d^{2} x -4 \,{\mathrm e}^{\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (d x +c +\frac {a d -c b}{b}\right ) a^{4} b \,d^{3}+{\mathrm e}^{-d x -c} a^{2} b^{3} d^{2} x}{2 d^{3} b^{6} \left (b x +a \right )}\)	\(669\)

Input:

int(x^4*cosh(d*x+c)/(b*x+a)^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.61 \[ \int \frac {x^4 \cosh (c+d x)}{(a+b x)^2} \, dx=-\frac {2 \, {\left (a^{4} b d^{3} + 2 \, b^{5} d x^{2} - 2 \, a^{2} b^{3} d\right )} \cosh \left (d x + c\right ) - {\left ({\left (a^{5} d^{4} - 4 \, a^{4} b d^{3} + {\left (a^{4} b d^{4} - 4 \, a^{3} b^{2} d^{3}\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\left (a^{5} d^{4} + 4 \, a^{4} b d^{3} + {\left (a^{4} b d^{4} + 4 \, a^{3} b^{2} d^{3}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) - 2 \, {\left (b^{5} d^{2} x^{3} - a b^{4} d^{2} x^{2} + 3 \, a^{3} b^{2} d^{2} + 2 \, a b^{4} + {\left (a^{2} b^{3} d^{2} + 2 \, b^{5}\right )} x\right )} \sinh \left (d x + c\right ) + {\left ({\left (a^{5} d^{4} - 4 \, a^{4} b d^{3} + {\left (a^{4} b d^{4} - 4 \, a^{3} b^{2} d^{3}\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{5} d^{4} + 4 \, a^{4} b d^{3} + {\left (a^{4} b d^{4} + 4 \, a^{3} b^{2} d^{3}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{2 \, {\left (b^{7} d^{3} x + a b^{6} d^{3}\right )}} \] Input:

integrate(x^4*cosh(d*x+c)/(b*x+a)^2,x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {x^4 \cosh (c+d x)}{(a+b x)^2} \, dx=\int \frac {x^{4} \cosh {\left (c + d x \right )}}{\left (a + b x\right )^{2}}\, dx \] Input:

integrate(x**4*cosh(d*x+c)/(b*x+a)**2,x)
 

Output:

Integral(x**4*cosh(c + d*x)/(a + b*x)**2, x)
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.76 \[ \int \frac {x^4 \cosh (c+d x)}{(a+b x)^2} \, dx=\frac {1}{6} \, {\left (3 \, a^{4} {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b^{6}} - \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b^{6}}\right )} + \frac {12 \, a^{3} {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b} + \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b}\right )}}{b^{4} d} - \frac {9 \, a^{2} {\left (\frac {{\left (d x e^{c} - e^{c}\right )} e^{\left (d x\right )}}{d^{2}} + \frac {{\left (d x + 1\right )} e^{\left (-d x - c\right )}}{d^{2}}\right )}}{b^{4}} + \frac {3 \, a {\left (\frac {{\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} e^{\left (d x\right )}}{d^{3}} + \frac {{\left (d^{2} x^{2} + 2 \, d x + 2\right )} e^{\left (-d x - c\right )}}{d^{3}}\right )}}{b^{3}} - \frac {\frac {{\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} e^{\left (d x\right )}}{d^{4}} + \frac {{\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} e^{\left (-d x - c\right )}}{d^{4}}}{b^{2}} + \frac {24 \, a^{3} \cosh \left (d x + c\right ) \log \left (b x + a\right )}{b^{5} d}\right )} d - \frac {1}{3} \, {\left (\frac {3 \, a^{4}}{b^{6} x + a b^{5}} + \frac {12 \, a^{3} \log \left (b x + a\right )}{b^{5}} - \frac {b^{2} x^{3} - 3 \, a b x^{2} + 9 \, a^{2} x}{b^{4}}\right )} \cosh \left (d x + c\right ) \] Input:

integrate(x^4*cosh(d*x+c)/(b*x+a)^2,x, algorithm="maxima")
 

Output:

1/6*(3*a^4*(e^(-c + a*d/b)*exp_integral_e(1, (b*x + a)*d/b)/b^6 - e^(c - a 
*d/b)*exp_integral_e(1, -(b*x + a)*d/b)/b^6) + 12*a^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^4*d) - 9*a^2*((d*x*e^c - e^c)*e^(d*x)/d^2 + (d*x + 1)*e^(-d*x 
 - c)/d^2)/b^4 + 3*a*((d^2*x^2*e^c - 2*d*x*e^c + 2*e^c)*e^(d*x)/d^3 + (d^2 
*x^2 + 2*d*x + 2)*e^(-d*x - c)/d^3)/b^3 - ((d^3*x^3*e^c - 3*d^2*x^2*e^c + 
6*d*x*e^c - 6*e^c)*e^(d*x)/d^4 + (d^3*x^3 + 3*d^2*x^2 + 6*d*x + 6)*e^(-d*x 
 - c)/d^4)/b^2 + 24*a^3*cosh(d*x + c)*log(b*x + a)/(b^5*d))*d - 1/3*(3*a^4 
/(b^6*x + a*b^5) + 12*a^3*log(b*x + a)/b^5 - (b^2*x^3 - 3*a*b*x^2 + 9*a^2* 
x)/b^4)*cosh(d*x + c)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2979 vs. \(2 (236) = 472\).

Time = 0.17 (sec) , antiderivative size = 2979, normalized size of antiderivative = 12.90 \[ \int \frac {x^4 \cosh (c+d x)}{(a+b x)^2} \, dx=\text {Too large to display} \] Input:

integrate(x^4*cosh(d*x+c)/(b*x+a)^2,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \cosh (c+d x)}{(a+b x)^2} \, dx=\int \frac {x^4\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (a+b\,x\right )}^2} \,d x \] Input:

int((x^4*cosh(c + d*x))/(a + b*x)^2,x)
 

Output:

int((x^4*cosh(c + d*x))/(a + b*x)^2, x)
 

Reduce [F]

\[ \int \frac {x^4 \cosh (c+d x)}{(a+b x)^2} \, dx=\text {too large to display} \] Input:

int(x^4*cosh(d*x+c)/(b*x+a)^2,x)
 

Output:

(e**(2*c + 2*d*x)*a**4*d**4*x - e**(2*c + 2*d*x)*a**4*d**3 - e**(2*c + 2*d 
*x)*a**3*b*d**4*x**2 - e**(2*c + 2*d*x)*a**3*b*d**2 + e**(2*c + 2*d*x)*a** 
2*b**2*d**4*x**3 - 2*e**(2*c + 2*d*x)*a**2*b**2*d**3*x**2 + e**(2*c + 2*d* 
x)*a**2*b**2*d**2*x - 2*e**(2*c + 2*d*x)*a**2*b**2*d + e**(2*c + 2*d*x)*a* 
b**3*d**2*x**2 - 2*e**(2*c + 2*d*x)*a*b**3 - e**(2*c + 2*d*x)*b**4*d**2*x* 
*3 + 2*e**(2*c + 2*d*x)*b**4*d*x**2 - 2*e**(2*c + 2*d*x)*b**4*x - e**(2*c 
+ d*x)*int((e**(d*x)*x)/(a**4*d**2 + 2*a**3*b*d**2*x + a**2*b**2*d**2*x**2 
 - a**2*b**2 - 2*a*b**3*x - b**4*x**2),x)*a**8*d**7 - e**(2*c + d*x)*int(( 
e**(d*x)*x)/(a**4*d**2 + 2*a**3*b*d**2*x + a**2*b**2*d**2*x**2 - a**2*b**2 
 - 2*a*b**3*x - b**4*x**2),x)*a**7*b*d**7*x + 3*e**(2*c + d*x)*int((e**(d* 
x)*x)/(a**4*d**2 + 2*a**3*b*d**2*x + a**2*b**2*d**2*x**2 - a**2*b**2 - 2*a 
*b**3*x - b**4*x**2),x)*a**7*b*d**6 + 3*e**(2*c + d*x)*int((e**(d*x)*x)/(a 
**4*d**2 + 2*a**3*b*d**2*x + a**2*b**2*d**2*x**2 - a**2*b**2 - 2*a*b**3*x 
- b**4*x**2),x)*a**6*b**2*d**6*x + 5*e**(2*c + d*x)*int((e**(d*x)*x)/(a**4 
*d**2 + 2*a**3*b*d**2*x + a**2*b**2*d**2*x**2 - a**2*b**2 - 2*a*b**3*x - b 
**4*x**2),x)*a**6*b**2*d**5 + 5*e**(2*c + d*x)*int((e**(d*x)*x)/(a**4*d**2 
 + 2*a**3*b*d**2*x + a**2*b**2*d**2*x**2 - a**2*b**2 - 2*a*b**3*x - b**4*x 
**2),x)*a**5*b**3*d**5*x - 3*e**(2*c + d*x)*int((e**(d*x)*x)/(a**4*d**2 + 
2*a**3*b*d**2*x + a**2*b**2*d**2*x**2 - a**2*b**2 - 2*a*b**3*x - b**4*x**2 
),x)*a**5*b**3*d**4 - 3*e**(2*c + d*x)*int((e**(d*x)*x)/(a**4*d**2 + 2*...