\(\int \frac {(a+b x^3)^2 \cosh (c+d x)}{x^5} \, dx\) [93]

Optimal result

Integrand size = 19, antiderivative size = 167 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^5} \, dx=-\frac {b^2 \cosh (c+d x)}{d^2}-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac {2 a b \cosh (c+d x)}{x}+\frac {1}{24} a^2 d^4 \cosh (c) \text {Chi}(d x)+2 a b d \text {Chi}(d x) \sinh (c)-\frac {a^2 d \sinh (c+d x)}{12 x^3}-\frac {a^2 d^3 \sinh (c+d x)}{24 x}+\frac {b^2 x \sinh (c+d x)}{d}+2 a b d \cosh (c) \text {Shi}(d x)+\frac {1}{24} a^2 d^4 \sinh (c) \text {Shi}(d x) \] Output:

-b^2*cosh(d*x+c)/d^2-1/4*a^2*cosh(d*x+c)/x^4-1/24*a^2*d^2*cosh(d*x+c)/x^2- 
2*a*b*cosh(d*x+c)/x+1/24*a^2*d^4*cosh(c)*Chi(d*x)+2*a*b*d*Chi(d*x)*sinh(c) 
-1/12*a^2*d*sinh(d*x+c)/x^3-1/24*a^2*d^3*sinh(d*x+c)/x+b^2*x*sinh(d*x+c)/d 
+2*a*b*d*cosh(c)*Shi(d*x)+1/24*a^2*d^4*sinh(c)*Shi(d*x)
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^5} \, dx=\frac {1}{24} \left (-\frac {24 b^2 \cosh (c+d x)}{d^2}-\frac {6 a^2 \cosh (c+d x)}{x^4}-\frac {a^2 d^2 \cosh (c+d x)}{x^2}-\frac {48 a b \cosh (c+d x)}{x}+a d \text {Chi}(d x) \left (a d^3 \cosh (c)+48 b \sinh (c)\right )-\frac {2 a^2 d \sinh (c+d x)}{x^3}-\frac {a^2 d^3 \sinh (c+d x)}{x}+\frac {24 b^2 x \sinh (c+d x)}{d}+a d \left (48 b \cosh (c)+a d^3 \sinh (c)\right ) \text {Shi}(d x)\right ) \] Input:

Integrate[((a + b*x^3)^2*Cosh[c + d*x])/x^5,x]
 

Output:

((-24*b^2*Cosh[c + d*x])/d^2 - (6*a^2*Cosh[c + d*x])/x^4 - (a^2*d^2*Cosh[c 
 + d*x])/x^2 - (48*a*b*Cosh[c + d*x])/x + a*d*CoshIntegral[d*x]*(a*d^3*Cos 
h[c] + 48*b*Sinh[c]) - (2*a^2*d*Sinh[c + d*x])/x^3 - (a^2*d^3*Sinh[c + d*x 
])/x + (24*b^2*x*Sinh[c + d*x])/d + a*d*(48*b*Cosh[c] + a*d^3*Sinh[c])*Sin 
hIntegral[d*x])/24
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {5810, 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.

\(\displaystyle \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^5} \, dx\)

\(\Big \downarrow \) 5810

\(\displaystyle \int \left (\frac {a^2 \cosh (c+d x)}{x^5}+\frac {2 a b \cosh (c+d x)}{x^2}+b^2 x \cosh (c+d x)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{24} a^2 d^4 \cosh (c) \text {Chi}(d x)+\frac {1}{24} a^2 d^4 \sinh (c) \text {Shi}(d x)-\frac {a^2 d^3 \sinh (c+d x)}{24 x}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a^2 d \sinh (c+d x)}{12 x^3}+2 a b d \sinh (c) \text {Chi}(d x)+2 a b d \cosh (c) \text {Shi}(d x)-\frac {2 a b \cosh (c+d x)}{x}-\frac {b^2 \cosh (c+d x)}{d^2}+\frac {b^2 x \sinh (c+d x)}{d}\)

Input:

Int[((a + b*x^3)^2*Cosh[c + d*x])/x^5,x]
 

Output:

-((b^2*Cosh[c + d*x])/d^2) - (a^2*Cosh[c + d*x])/(4*x^4) - (a^2*d^2*Cosh[c 
 + d*x])/(24*x^2) - (2*a*b*Cosh[c + d*x])/x + (a^2*d^4*Cosh[c]*CoshIntegra 
l[d*x])/24 + 2*a*b*d*CoshIntegral[d*x]*Sinh[c] - (a^2*d*Sinh[c + d*x])/(12 
*x^3) - (a^2*d^3*Sinh[c + d*x])/(24*x) + (b^2*x*Sinh[c + d*x])/d + 2*a*b*d 
*Cosh[c]*SinhIntegral[d*x] + (a^2*d^4*Sinh[c]*SinhIntegral[d*x])/24
 

Defintions of rubi rules used

rule 2009

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

rule 5810

Int[Cosh[(c_.) + (d_.)*(x_)]*((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p 
_.), x_Symbol] :> Int[ExpandIntegrand[Cosh[c + d*x], (e*x)^m*(a + b*x^n)^p, 
 x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(321\) vs. \(2(155)=310\).

Time = 0.87 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.93

method	result	size
risch	\(\frac {-{\mathrm e}^{-c} \operatorname {expIntegral}_{1}\left (d x \right ) a^{2} d^{6} x^{4}-{\mathrm e}^{c} \operatorname {expIntegral}_{1}\left (-d x \right ) a^{2} d^{6} x^{4}+{\mathrm e}^{-d x -c} a^{2} d^{5} x^{3}+48 \,{\mathrm e}^{-c} \operatorname {expIntegral}_{1}\left (d x \right ) a b \,d^{3} x^{4}-48 \,{\mathrm e}^{c} \operatorname {expIntegral}_{1}\left (-d x \right ) a b \,d^{3} x^{4}-{\mathrm e}^{d x +c} a^{2} d^{5} x^{3}-{\mathrm e}^{-d x -c} a^{2} d^{4} x^{2}-24 \,{\mathrm e}^{-d x -c} b^{2} d \,x^{5}-{\mathrm e}^{d x +c} a^{2} d^{4} x^{2}+24 \,{\mathrm e}^{d x +c} b^{2} d \,x^{5}-48 a b \,d^{2} x^{3} {\mathrm e}^{-d x -c}-48 a b \,d^{2} x^{3} {\mathrm e}^{d x +c}+2 \,{\mathrm e}^{-d x -c} a^{2} d^{3} x -24 \,{\mathrm e}^{-d x -c} b^{2} x^{4}-2 \,{\mathrm e}^{d x +c} a^{2} d^{3} x -24 \,{\mathrm e}^{d x +c} b^{2} x^{4}-6 a^{2} d^{2} {\mathrm e}^{-d x -c}-6 a^{2} d^{2} {\mathrm e}^{d x +c}}{48 d^{2} x^{4}}\)	\(322\)
meijerg	\(-\frac {2 b^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {d x \sinh \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{2}}+\frac {b^{2} \sinh \left (c \right ) \left (\cosh \left (d x \right ) x d -\sinh \left (d x \right )\right )}{d^{2}}+\frac {i d a b \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {4 i \cosh \left (d x \right )}{d x \sqrt {\pi }}-\frac {4 i \operatorname {Shi}\left (d x \right )}{\sqrt {\pi }}\right )}{2}+\frac {d b a \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {4}{\sqrt {\pi }}-\frac {4 \sinh \left (d x \right )}{\sqrt {\pi }\, x d}+\frac {4 \,\operatorname {Chi}\left (d x \right )-4 \ln \left (d x \right )-4 \gamma }{\sqrt {\pi }}+\frac {4 \gamma -4+4 \ln \left (x \right )+4 \ln \left (i d \right )}{\sqrt {\pi }}\right )}{2}+\frac {a^{2} \cosh \left (c \right ) \sqrt {\pi }\, d^{4} \left (\frac {\frac {25}{9} d^{4} x^{4}+8 x^{2} d^{2}+8}{\sqrt {\pi }\, x^{4} d^{4}}-\frac {8 \left (\frac {15 x^{2} d^{2}}{2}+45\right ) \cosh \left (d x \right )}{45 \sqrt {\pi }\, x^{4} d^{4}}-\frac {8 \left (\frac {15 x^{2} d^{2}}{2}+15\right ) \sinh \left (d x \right )}{45 \sqrt {\pi }\, x^{3} d^{3}}+\frac {\frac {4 \,\operatorname {Chi}\left (d x \right )}{3}-\frac {4 \ln \left (d x \right )}{3}-\frac {4 \gamma }{3}}{\sqrt {\pi }}+\frac {\frac {4 \gamma }{3}-\frac {25}{9}+\frac {4 \ln \left (x \right )}{3}+\frac {4 \ln \left (i d \right )}{3}}{\sqrt {\pi }}-\frac {8}{\sqrt {\pi }\, x^{4} d^{4}}-\frac {8}{\sqrt {\pi }\, x^{2} d^{2}}\right )}{32}-\frac {i a^{2} \sinh \left (c \right ) \sqrt {\pi }\, d^{4} \left (-\frac {8 i \left (\frac {x^{2} d^{2}}{2}+1\right ) \cosh \left (d x \right )}{3 d^{3} x^{3} \sqrt {\pi }}-\frac {8 i \left (\frac {x^{2} d^{2}}{2}+3\right ) \sinh \left (d x \right )}{3 d^{4} x^{4} \sqrt {\pi }}+\frac {4 i \operatorname {Shi}\left (d x \right )}{3 \sqrt {\pi }}\right )}{32}\)	\(405\)

Input:

int((b*x^3+a)^2*cosh(d*x+c)/x^5,x,method=_RETURNVERBOSE)
 

Output:

1/48/d^2*(-exp(-c)*Ei(1,d*x)*a^2*d^6*x^4-exp(c)*Ei(1,-d*x)*a^2*d^6*x^4+exp 
(-d*x-c)*a^2*d^5*x^3+48*exp(-c)*Ei(1,d*x)*a*b*d^3*x^4-48*exp(c)*Ei(1,-d*x) 
*a*b*d^3*x^4-exp(d*x+c)*a^2*d^5*x^3-exp(-d*x-c)*a^2*d^4*x^2-24*exp(-d*x-c) 
*b^2*d*x^5-exp(d*x+c)*a^2*d^4*x^2+24*exp(d*x+c)*b^2*d*x^5-48*a*b*d^2*x^3*e 
xp(-d*x-c)-48*a*b*d^2*x^3*exp(d*x+c)+2*exp(-d*x-c)*a^2*d^3*x-24*exp(-d*x-c 
)*b^2*x^4-2*exp(d*x+c)*a^2*d^3*x-24*exp(d*x+c)*b^2*x^4-6*a^2*d^2*exp(-d*x- 
c)-6*a^2*d^2*exp(d*x+c))/x^4
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^5} \, dx=-\frac {2 \, {\left (a^{2} d^{4} x^{2} + 48 \, a b d^{2} x^{3} + 24 \, b^{2} x^{4} + 6 \, a^{2} d^{2}\right )} \cosh \left (d x + c\right ) - {\left ({\left (a^{2} d^{6} + 48 \, a b d^{3}\right )} x^{4} {\rm Ei}\left (d x\right ) + {\left (a^{2} d^{6} - 48 \, a b d^{3}\right )} x^{4} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \, {\left (a^{2} d^{5} x^{3} - 24 \, b^{2} d x^{5} + 2 \, a^{2} d^{3} x\right )} \sinh \left (d x + c\right ) - {\left ({\left (a^{2} d^{6} + 48 \, a b d^{3}\right )} x^{4} {\rm Ei}\left (d x\right ) - {\left (a^{2} d^{6} - 48 \, a b d^{3}\right )} x^{4} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{48 \, d^{2} x^{4}} \] Input:

integrate((b*x^3+a)^2*cosh(d*x+c)/x^5,x, algorithm="fricas")
 

Output:

-1/48*(2*(a^2*d^4*x^2 + 48*a*b*d^2*x^3 + 24*b^2*x^4 + 6*a^2*d^2)*cosh(d*x 
+ c) - ((a^2*d^6 + 48*a*b*d^3)*x^4*Ei(d*x) + (a^2*d^6 - 48*a*b*d^3)*x^4*Ei 
(-d*x))*cosh(c) + 2*(a^2*d^5*x^3 - 24*b^2*d*x^5 + 2*a^2*d^3*x)*sinh(d*x + 
c) - ((a^2*d^6 + 48*a*b*d^3)*x^4*Ei(d*x) - (a^2*d^6 - 48*a*b*d^3)*x^4*Ei(- 
d*x))*sinh(c))/(d^2*x^4)
 

Sympy [F]

\[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^5} \, dx=\int \frac {\left (a + b x^{3}\right )^{2} \cosh {\left (c + d x \right )}}{x^{5}}\, dx \] Input:

integrate((b*x**3+a)**2*cosh(d*x+c)/x**5,x)
 

Output:

Integral((a + b*x**3)**2*cosh(c + d*x)/x**5, x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^5} \, dx=\frac {1}{8} \, {\left (a^{2} d^{3} e^{\left (-c\right )} \Gamma \left (-3, d x\right ) + a^{2} d^{3} e^{c} \Gamma \left (-3, -d x\right ) - 8 \, a b {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 8 \, a b {\rm Ei}\left (d x\right ) e^{c} - \frac {2 \, {\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{d^{3}} - \frac {2 \, {\left (d^{2} x^{2} + 2 \, d x + 2\right )} b^{2} e^{\left (-d x - c\right )}}{d^{3}}\right )} d + \frac {1}{4} \, {\left (2 \, b^{2} x^{2} - \frac {8 \, a b x^{3} + a^{2}}{x^{4}}\right )} \cosh \left (d x + c\right ) \] Input:

integrate((b*x^3+a)^2*cosh(d*x+c)/x^5,x, algorithm="maxima")
 

Output:

1/8*(a^2*d^3*e^(-c)*gamma(-3, d*x) + a^2*d^3*e^c*gamma(-3, -d*x) - 8*a*b*E 
i(-d*x)*e^(-c) + 8*a*b*Ei(d*x)*e^c - 2*(d^2*x^2*e^c - 2*d*x*e^c + 2*e^c)*b 
^2*e^(d*x)/d^3 - 2*(d^2*x^2 + 2*d*x + 2)*b^2*e^(-d*x - c)/d^3)*d + 1/4*(2* 
b^2*x^2 - (8*a*b*x^3 + a^2)/x^4)*cosh(d*x + c)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (155) = 310\).

Time = 0.11 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.89 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^5} \, dx=\frac {a^{2} d^{6} x^{4} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{6} x^{4} {\rm Ei}\left (d x\right ) e^{c} - a^{2} d^{5} x^{3} e^{\left (d x + c\right )} + a^{2} d^{5} x^{3} e^{\left (-d x - c\right )} - 48 \, a b d^{3} x^{4} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 48 \, a b d^{3} x^{4} {\rm Ei}\left (d x\right ) e^{c} - a^{2} d^{4} x^{2} e^{\left (d x + c\right )} + 24 \, b^{2} d x^{5} e^{\left (d x + c\right )} - a^{2} d^{4} x^{2} e^{\left (-d x - c\right )} - 24 \, b^{2} d x^{5} e^{\left (-d x - c\right )} - 48 \, a b d^{2} x^{3} e^{\left (d x + c\right )} - 48 \, a b d^{2} x^{3} e^{\left (-d x - c\right )} - 2 \, a^{2} d^{3} x e^{\left (d x + c\right )} - 24 \, b^{2} x^{4} e^{\left (d x + c\right )} + 2 \, a^{2} d^{3} x e^{\left (-d x - c\right )} - 24 \, b^{2} x^{4} e^{\left (-d x - c\right )} - 6 \, a^{2} d^{2} e^{\left (d x + c\right )} - 6 \, a^{2} d^{2} e^{\left (-d x - c\right )}}{48 \, d^{2} x^{4}} \] Input:

integrate((b*x^3+a)^2*cosh(d*x+c)/x^5,x, algorithm="giac")
 

Output:

1/48*(a^2*d^6*x^4*Ei(-d*x)*e^(-c) + a^2*d^6*x^4*Ei(d*x)*e^c - a^2*d^5*x^3* 
e^(d*x + c) + a^2*d^5*x^3*e^(-d*x - c) - 48*a*b*d^3*x^4*Ei(-d*x)*e^(-c) + 
48*a*b*d^3*x^4*Ei(d*x)*e^c - a^2*d^4*x^2*e^(d*x + c) + 24*b^2*d*x^5*e^(d*x 
 + c) - a^2*d^4*x^2*e^(-d*x - c) - 24*b^2*d*x^5*e^(-d*x - c) - 48*a*b*d^2* 
x^3*e^(d*x + c) - 48*a*b*d^2*x^3*e^(-d*x - c) - 2*a^2*d^3*x*e^(d*x + c) - 
24*b^2*x^4*e^(d*x + c) + 2*a^2*d^3*x*e^(-d*x - c) - 24*b^2*x^4*e^(-d*x - c 
) - 6*a^2*d^2*e^(d*x + c) - 6*a^2*d^2*e^(-d*x - c))/(d^2*x^4)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^5} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (b\,x^3+a\right )}^2}{x^5} \,d x \] Input:

int((cosh(c + d*x)*(a + b*x^3)^2)/x^5,x)
 

Output:

int((cosh(c + d*x)*(a + b*x^3)^2)/x^5, x)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.83 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^5} \, dx=\frac {e^{d x} \mathit {ei} \left (-d x \right ) a^{2} d^{6} x^{4}-48 e^{d x} \mathit {ei} \left (-d x \right ) a b \,d^{3} x^{4}+e^{d x +2 c} \mathit {ei} \left (d x \right ) a^{2} d^{6} x^{4}+48 e^{d x +2 c} \mathit {ei} \left (d x \right ) a b \,d^{3} x^{4}-e^{2 d x +2 c} a^{2} d^{5} x^{3}-e^{2 d x +2 c} a^{2} d^{4} x^{2}-2 e^{2 d x +2 c} a^{2} d^{3} x -6 e^{2 d x +2 c} a^{2} d^{2}-48 e^{2 d x +2 c} a b \,d^{2} x^{3}+24 e^{2 d x +2 c} b^{2} d \,x^{5}-24 e^{2 d x +2 c} b^{2} x^{4}+a^{2} d^{5} x^{3}-a^{2} d^{4} x^{2}+2 a^{2} d^{3} x -6 a^{2} d^{2}-48 a b \,d^{2} x^{3}-24 b^{2} d \,x^{5}-24 b^{2} x^{4}}{48 e^{d x +c} d^{2} x^{4}} \] Input:

int((b*x^3+a)^2*cosh(d*x+c)/x^5,x)
 

Output:

(e**(d*x)*ei( - d*x)*a**2*d**6*x**4 - 48*e**(d*x)*ei( - d*x)*a*b*d**3*x**4 
 + e**(2*c + d*x)*ei(d*x)*a**2*d**6*x**4 + 48*e**(2*c + d*x)*ei(d*x)*a*b*d 
**3*x**4 - e**(2*c + 2*d*x)*a**2*d**5*x**3 - e**(2*c + 2*d*x)*a**2*d**4*x* 
*2 - 2*e**(2*c + 2*d*x)*a**2*d**3*x - 6*e**(2*c + 2*d*x)*a**2*d**2 - 48*e* 
*(2*c + 2*d*x)*a*b*d**2*x**3 + 24*e**(2*c + 2*d*x)*b**2*d*x**5 - 24*e**(2* 
c + 2*d*x)*b**2*x**4 + a**2*d**5*x**3 - a**2*d**4*x**2 + 2*a**2*d**3*x - 6 
*a**2*d**2 - 48*a*b*d**2*x**3 - 24*b**2*d*x**5 - 24*b**2*x**4)/(48*e**(c + 
 d*x)*d**2*x**4)