\(\int \frac {(a+b x)^2 \cosh (c+d x)}{x^4} \, dx\) [16]

Optimal result

Integrand size = 17, antiderivative size = 172 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^4} \, dx=-\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {a b \cosh (c+d x)}{x^2}-\frac {b^2 \cosh (c+d x)}{x}-\frac {a^2 d^2 \cosh (c+d x)}{6 x}+a b d^2 \cosh (c) \text {Chi}(d x)+b^2 d \text {Chi}(d x) \sinh (c)+\frac {1}{6} a^2 d^3 \text {Chi}(d x) \sinh (c)-\frac {a^2 d \sinh (c+d x)}{6 x^2}-\frac {a b d \sinh (c+d x)}{x}+b^2 d \cosh (c) \text {Shi}(d x)+\frac {1}{6} a^2 d^3 \cosh (c) \text {Shi}(d x)+a b d^2 \sinh (c) \text {Shi}(d x) \] Output:

-1/3*a^2*cosh(d*x+c)/x^3-a*b*cosh(d*x+c)/x^2-b^2*cosh(d*x+c)/x-1/6*a^2*d^2 
*cosh(d*x+c)/x+a*b*d^2*cosh(c)*Chi(d*x)+b^2*d*Chi(d*x)*sinh(c)+1/6*a^2*d^3 
*Chi(d*x)*sinh(c)-1/6*a^2*d*sinh(d*x+c)/x^2-a*b*d*sinh(d*x+c)/x+b^2*d*cosh 
(c)*Shi(d*x)+1/6*a^2*d^3*cosh(c)*Shi(d*x)+a*b*d^2*sinh(c)*Shi(d*x)
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^4} \, dx=-\frac {2 a^2 \cosh (c+d x)+6 a b x \cosh (c+d x)+6 b^2 x^2 \cosh (c+d x)+a^2 d^2 x^2 \cosh (c+d x)-d x^3 \text {Chi}(d x) \left (6 a b d \cosh (c)+\left (6 b^2+a^2 d^2\right ) \sinh (c)\right )+a^2 d x \sinh (c+d x)+6 a b d x^2 \sinh (c+d x)-d x^3 \left (6 b^2 \cosh (c)+a^2 d^2 \cosh (c)+6 a b d \sinh (c)\right ) \text {Shi}(d x)}{6 x^3} \] Input:

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

Output:

-1/6*(2*a^2*Cosh[c + d*x] + 6*a*b*x*Cosh[c + d*x] + 6*b^2*x^2*Cosh[c + d*x 
] + a^2*d^2*x^2*Cosh[c + d*x] - d*x^3*CoshIntegral[d*x]*(6*a*b*d*Cosh[c] + 
 (6*b^2 + a^2*d^2)*Sinh[c]) + a^2*d*x*Sinh[c + d*x] + 6*a*b*d*x^2*Sinh[c + 
 d*x] - d*x^3*(6*b^2*Cosh[c] + a^2*d^2*Cosh[c] + 6*a*b*d*Sinh[c])*SinhInte 
gral[d*x])/x^3
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 172, 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{6} a^2 d^3 \sinh (c) \text {Chi}(d x)+\frac {1}{6} a^2 d^3 \cosh (c) \text {Shi}(d x)-\frac {a^2 d^2 \cosh (c+d x)}{6 x}-\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {a^2 d \sinh (c+d x)}{6 x^2}+a b d^2 \cosh (c) \text {Chi}(d x)+a b d^2 \sinh (c) \text {Shi}(d x)-\frac {a b \cosh (c+d x)}{x^2}-\frac {a b d \sinh (c+d x)}{x}+b^2 d \sinh (c) \text {Chi}(d x)+b^2 d \cosh (c) \text {Shi}(d x)-\frac {b^2 \cosh (c+d x)}{x}\)

Input:

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

Output:

-1/3*(a^2*Cosh[c + d*x])/x^3 - (a*b*Cosh[c + d*x])/x^2 - (b^2*Cosh[c + d*x 
])/x - (a^2*d^2*Cosh[c + d*x])/(6*x) + a*b*d^2*Cosh[c]*CoshIntegral[d*x] + 
 b^2*d*CoshIntegral[d*x]*Sinh[c] + (a^2*d^3*CoshIntegral[d*x]*Sinh[c])/6 - 
 (a^2*d*Sinh[c + d*x])/(6*x^2) - (a*b*d*Sinh[c + d*x])/x + b^2*d*Cosh[c]*S 
inhIntegral[d*x] + (a^2*d^3*Cosh[c]*SinhIntegral[d*x])/6 + a*b*d^2*Sinh[c] 
*SinhIntegral[d*x]
 

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 [A] (verified)

Time = 0.58 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.70

method	result	size
risch	\(-\frac {{\mathrm e}^{c} \operatorname {expIntegral}_{1}\left (-d x \right ) a^{2} d^{3} x^{3}-{\mathrm e}^{-c} \operatorname {expIntegral}_{1}\left (d x \right ) a^{2} d^{3} x^{3}+6 \,{\mathrm e}^{c} \operatorname {expIntegral}_{1}\left (-d x \right ) a b \,d^{2} x^{3}+6 \,{\mathrm e}^{-c} \operatorname {expIntegral}_{1}\left (d x \right ) a b \,d^{2} x^{3}+6 \,{\mathrm e}^{c} \operatorname {expIntegral}_{1}\left (-d x \right ) b^{2} d \,x^{3}-6 \,{\mathrm e}^{-c} \operatorname {expIntegral}_{1}\left (d x \right ) b^{2} d \,x^{3}+a^{2} d^{2} x^{2} {\mathrm e}^{d x +c}+a^{2} d^{2} x^{2} {\mathrm e}^{-d x -c}+6 a b d \,x^{2} {\mathrm e}^{d x +c}-6 a b d \,x^{2} {\mathrm e}^{-d x -c}+a^{2} d x \,{\mathrm e}^{d x +c}+6 \,{\mathrm e}^{d x +c} b^{2} x^{2}-a^{2} d x \,{\mathrm e}^{-d x -c}+6 \,{\mathrm e}^{-d x -c} b^{2} x^{2}+6 \,{\mathrm e}^{d x +c} a b x +6 \,{\mathrm e}^{-d x -c} a b x +2 \,{\mathrm e}^{d x +c} a^{2}+2 \,{\mathrm e}^{-d x -c} a^{2}}{12 x^{3}}\)	\(292\)
meijerg	\(\frac {i d \,b^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {4 i \cosh \left (d x \right )}{d x \sqrt {\pi }}-\frac {4 i \operatorname {Shi}\left (d x \right )}{\sqrt {\pi }}\right )}{4}+\frac {d \,b^{2} \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {4}{\sqrt {\pi }}-\frac {4 \sinh \left (d x \right )}{\sqrt {\pi }\, x d}+\frac {4 \,\operatorname {Chi}\left (d x \right )-4 \ln \left (d x \right )-4 \gamma }{\sqrt {\pi }}+\frac {4 \gamma -4+4 \ln \left (x \right )+4 \ln \left (i d \right )}{\sqrt {\pi }}\right )}{4}-\frac {d^{2} a b \cosh \left (c \right ) \sqrt {\pi }\, \left (-\frac {4 \left (\frac {9 x^{2} d^{2}}{2}+3\right )}{3 \sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \cosh \left (d x \right )}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \sinh \left (d x \right )}{\sqrt {\pi }\, x d}-\frac {4 \left (\operatorname {Chi}\left (d x \right )-\ln \left (d x \right )-\gamma \right )}{\sqrt {\pi }}-\frac {2 \left (2 \gamma -3+2 \ln \left (x \right )+2 \ln \left (i d \right )\right )}{\sqrt {\pi }}+\frac {4}{\sqrt {\pi }\, x^{2} d^{2}}\right )}{4}+\frac {i d^{2} b a \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {4 i \cosh \left (d x \right )}{d x \sqrt {\pi }}+\frac {4 i \sinh \left (d x \right )}{x^{2} d^{2} \sqrt {\pi }}-\frac {4 i \operatorname {Shi}\left (d x \right )}{\sqrt {\pi }}\right )}{4}-\frac {i a^{2} \cosh \left (c \right ) \sqrt {\pi }\, d^{3} \left (-\frac {8 i \left (x^{2} d^{2}+2\right ) \cosh \left (d x \right )}{3 d^{3} x^{3} \sqrt {\pi }}-\frac {8 i \sinh \left (d x \right )}{3 x^{2} d^{2} \sqrt {\pi }}+\frac {8 i \operatorname {Shi}\left (d x \right )}{3 \sqrt {\pi }}\right )}{16}-\frac {a^{2} \sinh \left (c \right ) \sqrt {\pi }\, d^{3} \left (-\frac {8 \left (\frac {55 x^{2} d^{2}}{2}+45\right )}{45 \sqrt {\pi }\, x^{2} d^{2}}+\frac {8 \cosh \left (d x \right )}{3 \sqrt {\pi }\, x^{2} d^{2}}+\frac {16 \left (\frac {5 x^{2} d^{2}}{2}+5\right ) \sinh \left (d x \right )}{15 \sqrt {\pi }\, x^{3} d^{3}}-\frac {8 \left (\operatorname {Chi}\left (d x \right )-\ln \left (d x \right )-\gamma \right )}{3 \sqrt {\pi }}-\frac {4 \left (2 \gamma -\frac {11}{3}+2 \ln \left (x \right )+2 \ln \left (i d \right )\right )}{3 \sqrt {\pi }}+\frac {8}{\sqrt {\pi }\, x^{2} d^{2}}\right )}{16}\)	\(476\)

Input:

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

Output:

-1/12*(exp(c)*Ei(1,-d*x)*a^2*d^3*x^3-exp(-c)*Ei(1,d*x)*a^2*d^3*x^3+6*exp(c 
)*Ei(1,-d*x)*a*b*d^2*x^3+6*exp(-c)*Ei(1,d*x)*a*b*d^2*x^3+6*exp(c)*Ei(1,-d* 
x)*b^2*d*x^3-6*exp(-c)*Ei(1,d*x)*b^2*d*x^3+a^2*d^2*x^2*exp(d*x+c)+a^2*d^2* 
x^2*exp(-d*x-c)+6*a*b*d*x^2*exp(d*x+c)-6*a*b*d*x^2*exp(-d*x-c)+a^2*d*x*exp 
(d*x+c)+6*exp(d*x+c)*b^2*x^2-a^2*d*x*exp(-d*x-c)+6*exp(-d*x-c)*b^2*x^2+6*e 
xp(d*x+c)*a*b*x+6*exp(-d*x-c)*a*b*x+2*exp(d*x+c)*a^2+2*exp(-d*x-c)*a^2)/x^ 
3
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.13 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^4} \, dx=-\frac {2 \, {\left (6 \, a b x + {\left (a^{2} d^{2} + 6 \, b^{2}\right )} x^{2} + 2 \, a^{2}\right )} \cosh \left (d x + c\right ) - {\left ({\left (a^{2} d^{3} + 6 \, a b d^{2} + 6 \, b^{2} d\right )} x^{3} {\rm Ei}\left (d x\right ) - {\left (a^{2} d^{3} - 6 \, a b d^{2} + 6 \, b^{2} d\right )} x^{3} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \, {\left (6 \, a b d x^{2} + a^{2} d x\right )} \sinh \left (d x + c\right ) - {\left ({\left (a^{2} d^{3} + 6 \, a b d^{2} + 6 \, b^{2} d\right )} x^{3} {\rm Ei}\left (d x\right ) + {\left (a^{2} d^{3} - 6 \, a b d^{2} + 6 \, b^{2} d\right )} x^{3} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{12 \, x^{3}} \] Input:

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

Output:

-1/12*(2*(6*a*b*x + (a^2*d^2 + 6*b^2)*x^2 + 2*a^2)*cosh(d*x + c) - ((a^2*d 
^3 + 6*a*b*d^2 + 6*b^2*d)*x^3*Ei(d*x) - (a^2*d^3 - 6*a*b*d^2 + 6*b^2*d)*x^ 
3*Ei(-d*x))*cosh(c) + 2*(6*a*b*d*x^2 + a^2*d*x)*sinh(d*x + c) - ((a^2*d^3 
+ 6*a*b*d^2 + 6*b^2*d)*x^3*Ei(d*x) + (a^2*d^3 - 6*a*b*d^2 + 6*b^2*d)*x^3*E 
i(-d*x))*sinh(c))/x^3
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.68 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^4} \, dx=\frac {1}{6} \, {\left (a^{2} d^{2} e^{\left (-c\right )} \Gamma \left (-2, d x\right ) - a^{2} d^{2} e^{c} \Gamma \left (-2, -d x\right ) + 3 \, a b d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + 3 \, a b d e^{c} \Gamma \left (-1, -d x\right ) - 3 \, b^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 3 \, b^{2} {\rm Ei}\left (d x\right ) e^{c}\right )} d - \frac {{\left (3 \, b^{2} x^{2} + 3 \, a b x + a^{2}\right )} \cosh \left (d x + c\right )}{3 \, x^{3}} \] Input:

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

Output:

1/6*(a^2*d^2*e^(-c)*gamma(-2, d*x) - a^2*d^2*e^c*gamma(-2, -d*x) + 3*a*b*d 
*e^(-c)*gamma(-1, d*x) + 3*a*b*d*e^c*gamma(-1, -d*x) - 3*b^2*Ei(-d*x)*e^(- 
c) + 3*b^2*Ei(d*x)*e^c)*d - 1/3*(3*b^2*x^2 + 3*a*b*x + a^2)*cosh(d*x + c)/ 
x^3
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.66 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^4} \, dx=-\frac {a^{2} d^{3} x^{3} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a^{2} d^{3} x^{3} {\rm Ei}\left (d x\right ) e^{c} - 6 \, a b d^{2} x^{3} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 6 \, a b d^{2} x^{3} {\rm Ei}\left (d x\right ) e^{c} + 6 \, b^{2} d x^{3} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 6 \, b^{2} d x^{3} {\rm Ei}\left (d x\right ) e^{c} + a^{2} d^{2} x^{2} e^{\left (d x + c\right )} + a^{2} d^{2} x^{2} e^{\left (-d x - c\right )} + 6 \, a b d x^{2} e^{\left (d x + c\right )} - 6 \, a b d x^{2} e^{\left (-d x - c\right )} + a^{2} d x e^{\left (d x + c\right )} + 6 \, b^{2} x^{2} e^{\left (d x + c\right )} - a^{2} d x e^{\left (-d x - c\right )} + 6 \, b^{2} x^{2} e^{\left (-d x - c\right )} + 6 \, a b x e^{\left (d x + c\right )} + 6 \, a b x e^{\left (-d x - c\right )} + 2 \, a^{2} e^{\left (d x + c\right )} + 2 \, a^{2} e^{\left (-d x - c\right )}}{12 \, x^{3}} \] Input:

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

Output:

-1/12*(a^2*d^3*x^3*Ei(-d*x)*e^(-c) - a^2*d^3*x^3*Ei(d*x)*e^c - 6*a*b*d^2*x 
^3*Ei(-d*x)*e^(-c) - 6*a*b*d^2*x^3*Ei(d*x)*e^c + 6*b^2*d*x^3*Ei(-d*x)*e^(- 
c) - 6*b^2*d*x^3*Ei(d*x)*e^c + a^2*d^2*x^2*e^(d*x + c) + a^2*d^2*x^2*e^(-d 
*x - c) + 6*a*b*d*x^2*e^(d*x + c) - 6*a*b*d*x^2*e^(-d*x - c) + a^2*d*x*e^( 
d*x + c) + 6*b^2*x^2*e^(d*x + c) - a^2*d*x*e^(-d*x - c) + 6*b^2*x^2*e^(-d* 
x - c) + 6*a*b*x*e^(d*x + c) + 6*a*b*x*e^(-d*x - c) + 2*a^2*e^(d*x + c) + 
2*a^2*e^(-d*x - c))/x^3
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.69 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^4} \, dx=\frac {-e^{d x} \mathit {ei} \left (-d x \right ) a^{2} d^{3} x^{3}+6 e^{d x} \mathit {ei} \left (-d x \right ) a b \,d^{2} x^{3}-6 e^{d x} \mathit {ei} \left (-d x \right ) b^{2} d \,x^{3}+e^{d x +2 c} \mathit {ei} \left (d x \right ) a^{2} d^{3} x^{3}+6 e^{d x +2 c} \mathit {ei} \left (d x \right ) a b \,d^{2} x^{3}+6 e^{d x +2 c} \mathit {ei} \left (d x \right ) b^{2} d \,x^{3}-e^{2 d x +2 c} a^{2} d^{2} x^{2}-e^{2 d x +2 c} a^{2} d x -2 e^{2 d x +2 c} a^{2}-6 e^{2 d x +2 c} a b d \,x^{2}-6 e^{2 d x +2 c} a b x -6 e^{2 d x +2 c} b^{2} x^{2}-a^{2} d^{2} x^{2}+a^{2} d x -2 a^{2}+6 a b d \,x^{2}-6 a b x -6 b^{2} x^{2}}{12 e^{d x +c} x^{3}} \] Input:

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

Output:

( - e**(d*x)*ei( - d*x)*a**2*d**3*x**3 + 6*e**(d*x)*ei( - d*x)*a*b*d**2*x* 
*3 - 6*e**(d*x)*ei( - d*x)*b**2*d*x**3 + e**(2*c + d*x)*ei(d*x)*a**2*d**3* 
x**3 + 6*e**(2*c + d*x)*ei(d*x)*a*b*d**2*x**3 + 6*e**(2*c + d*x)*ei(d*x)*b 
**2*d*x**3 - e**(2*c + 2*d*x)*a**2*d**2*x**2 - e**(2*c + 2*d*x)*a**2*d*x - 
 2*e**(2*c + 2*d*x)*a**2 - 6*e**(2*c + 2*d*x)*a*b*d*x**2 - 6*e**(2*c + 2*d 
*x)*a*b*x - 6*e**(2*c + 2*d*x)*b**2*x**2 - a**2*d**2*x**2 + a**2*d*x - 2*a 
**2 + 6*a*b*d*x**2 - 6*a*b*x - 6*b**2*x**2)/(12*e**(c + d*x)*x**3)