Integrand size = 17, antiderivative size = 69 \[ \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x^3} \, dx=-\frac {a \cosh (c+d x)}{2 x^2}+\frac {1}{2} a d^2 \cosh (c) \text {Chi}(d x)+\frac {b \sinh (c+d x)}{d}-\frac {a d \sinh (c+d x)}{2 x}+\frac {1}{2} a d^2 \sinh (c) \text {Shi}(d x) \] Output:
-1/2*a*cosh(d*x+c)/x^2+1/2*a*d^2*cosh(c)*Chi(d*x)+b*sinh(d*x+c)/d-1/2*a*d* sinh(d*x+c)/x+1/2*a*d^2*sinh(c)*Shi(d*x)
Time = 0.07 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x^3} \, dx=\frac {b \cosh (d x) \sinh (c)}{d}-\frac {a \cosh (d x) (\cosh (c)+d x \sinh (c))}{2 x^2}+\frac {b \cosh (c) \sinh (d x)}{d}-\frac {a (d x \cosh (c)+\sinh (c)) \sinh (d x)}{2 x^2}+\frac {1}{2} a d^2 (\cosh (c) \text {Chi}(d x)+\sinh (c) \text {Shi}(d x)) \] Input:
Integrate[((a + b*x^3)*Cosh[c + d*x])/x^3,x]
Output:
(b*Cosh[d*x]*Sinh[c])/d - (a*Cosh[d*x]*(Cosh[c] + d*x*Sinh[c]))/(2*x^2) + (b*Cosh[c]*Sinh[d*x])/d - (a*(d*x*Cosh[c] + Sinh[c])*Sinh[d*x])/(2*x^2) + (a*d^2*(Cosh[c]*CoshIntegral[d*x] + Sinh[c]*SinhIntegral[d*x]))/2
Time = 0.37 (sec) , antiderivative size = 69, 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 = {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.
\(\displaystyle \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x^3} \, dx\) |
\(\Big \downarrow \) 5810 |
\(\displaystyle \int \left (\frac {a \cosh (c+d x)}{x^3}+b \cosh (c+d x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} a d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{2} a d^2 \sinh (c) \text {Shi}(d x)-\frac {a \cosh (c+d x)}{2 x^2}-\frac {a d \sinh (c+d x)}{2 x}+\frac {b \sinh (c+d x)}{d}\) |
Input:
Int[((a + b*x^3)*Cosh[c + d*x])/x^3,x]
Output:
-1/2*(a*Cosh[c + d*x])/x^2 + (a*d^2*Cosh[c]*CoshIntegral[d*x])/2 + (b*Sinh [c + d*x])/d - (a*d*Sinh[c + d*x])/(2*x) + (a*d^2*Sinh[c]*SinhIntegral[d*x ])/2
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]
Time = 0.58 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.72
method | result | size |
risch | \(-\frac {{\mathrm e}^{c} \operatorname {expIntegral}_{1}\left (-d x \right ) a \,d^{3} x^{2}+{\mathrm e}^{-c} \operatorname {expIntegral}_{1}\left (d x \right ) a \,d^{3} x^{2}+{\mathrm e}^{d x +c} a \,d^{2} x -{\mathrm e}^{-d x -c} a \,d^{2} x -2 \,{\mathrm e}^{d x +c} b \,x^{2}+2 \,{\mathrm e}^{-d x -c} b \,x^{2}+d \,{\mathrm e}^{d x +c} a +d \,{\mathrm e}^{-d x -c} a}{4 d \,x^{2}}\) | \(119\) |
meijerg | \(\frac {b \cosh \left (c \right ) \sinh \left (d x \right )}{d}-\frac {b \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (d x \right )}{\sqrt {\pi }}\right )}{d}-\frac {a \cosh \left (c \right ) \sqrt {\pi }\, d^{2} \left (-\frac {4 \left (\frac {9 x^{2} d^{2}}{2}+3\right )}{3 \sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \cosh \left (d x \right )}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \sinh \left (d x \right )}{\sqrt {\pi }\, x d}-\frac {4 \left (\operatorname {Chi}\left (d x \right )-\ln \left (d x \right )-\gamma \right )}{\sqrt {\pi }}-\frac {2 \left (2 \gamma -3+2 \ln \left (x \right )+2 \ln \left (i d \right )\right )}{\sqrt {\pi }}+\frac {4}{\sqrt {\pi }\, x^{2} d^{2}}\right )}{8}+\frac {i a \sinh \left (c \right ) \sqrt {\pi }\, d^{2} \left (\frac {4 i \cosh \left (d x \right )}{d x \sqrt {\pi }}+\frac {4 i \sinh \left (d x \right )}{x^{2} d^{2} \sqrt {\pi }}-\frac {4 i \operatorname {Shi}\left (d x \right )}{\sqrt {\pi }}\right )}{8}\) | \(206\) |
Input:
int((b*x^3+a)*cosh(d*x+c)/x^3,x,method=_RETURNVERBOSE)
Output:
-1/4/d*(exp(c)*Ei(1,-d*x)*a*d^3*x^2+exp(-c)*Ei(1,d*x)*a*d^3*x^2+exp(d*x+c) *a*d^2*x-exp(-d*x-c)*a*d^2*x-2*exp(d*x+c)*b*x^2+2*exp(-d*x-c)*b*x^2+d*exp( d*x+c)*a+d*exp(-d*x-c)*a)/x^2
Time = 0.09 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.46 \[ \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x^3} \, dx=-\frac {2 \, a d \cosh \left (d x + c\right ) - {\left (a d^{3} x^{2} {\rm Ei}\left (d x\right ) + a d^{3} x^{2} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \, {\left (a d^{2} x - 2 \, b x^{2}\right )} \sinh \left (d x + c\right ) - {\left (a d^{3} x^{2} {\rm Ei}\left (d x\right ) - a d^{3} x^{2} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{4 \, d x^{2}} \] Input:
integrate((b*x^3+a)*cosh(d*x+c)/x^3,x, algorithm="fricas")
Output:
-1/4*(2*a*d*cosh(d*x + c) - (a*d^3*x^2*Ei(d*x) + a*d^3*x^2*Ei(-d*x))*cosh( c) + 2*(a*d^2*x - 2*b*x^2)*sinh(d*x + c) - (a*d^3*x^2*Ei(d*x) - a*d^3*x^2* Ei(-d*x))*sinh(c))/(d*x^2)
\[ \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x^3} \, dx=\int \frac {\left (a + b x^{3}\right ) \cosh {\left (c + d x \right )}}{x^{3}}\, dx \] Input:
integrate((b*x**3+a)*cosh(d*x+c)/x**3,x)
Output:
Integral((a + b*x**3)*cosh(c + d*x)/x**3, x)
Time = 0.10 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.26 \[ \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x^3} \, dx=\frac {1}{4} \, {\left (a d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + a d e^{c} \Gamma \left (-1, -d x\right ) - \frac {2 \, {\left (d x e^{c} - e^{c}\right )} b e^{\left (d x\right )}}{d^{2}} - \frac {2 \, {\left (d x + 1\right )} b e^{\left (-d x - c\right )}}{d^{2}}\right )} d + \frac {1}{2} \, {\left (2 \, b x - \frac {a}{x^{2}}\right )} \cosh \left (d x + c\right ) \] Input:
integrate((b*x^3+a)*cosh(d*x+c)/x^3,x, algorithm="maxima")
Output:
1/4*(a*d*e^(-c)*gamma(-1, d*x) + a*d*e^c*gamma(-1, -d*x) - 2*(d*x*e^c - e^ c)*b*e^(d*x)/d^2 - 2*(d*x + 1)*b*e^(-d*x - c)/d^2)*d + 1/2*(2*b*x - a/x^2) *cosh(d*x + c)
Time = 0.14 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.71 \[ \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x^3} \, dx=\frac {a d^{3} x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a d^{3} x^{2} {\rm Ei}\left (d x\right ) e^{c} - a d^{2} x e^{\left (d x + c\right )} + a d^{2} x e^{\left (-d x - c\right )} + 2 \, b x^{2} e^{\left (d x + c\right )} - 2 \, b x^{2} e^{\left (-d x - c\right )} - a d e^{\left (d x + c\right )} - a d e^{\left (-d x - c\right )}}{4 \, d x^{2}} \] Input:
integrate((b*x^3+a)*cosh(d*x+c)/x^3,x, algorithm="giac")
Output:
1/4*(a*d^3*x^2*Ei(-d*x)*e^(-c) + a*d^3*x^2*Ei(d*x)*e^c - a*d^2*x*e^(d*x + c) + a*d^2*x*e^(-d*x - c) + 2*b*x^2*e^(d*x + c) - 2*b*x^2*e^(-d*x - c) - a *d*e^(d*x + c) - a*d*e^(-d*x - c))/(d*x^2)
Timed out. \[ \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x^3} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,\left (b\,x^3+a\right )}{x^3} \,d x \] Input:
int((cosh(c + d*x)*(a + b*x^3))/x^3,x)
Output:
int((cosh(c + d*x)*(a + b*x^3))/x^3, x)
\[ \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x^3} \, dx=\frac {-\cosh \left (d x +c \right ) a d +\left (\int \frac {\cosh \left (d x +c \right )}{x}d x \right ) a \,d^{3} x^{2}-\sinh \left (d x +c \right ) a \,d^{2} x +2 \sinh \left (d x +c \right ) b \,x^{2}}{2 d \,x^{2}} \] Input:
int((b*x^3+a)*cosh(d*x+c)/x^3,x)
Output:
( - cosh(c + d*x)*a*d + int(cosh(c + d*x)/x,x)*a*d**3*x**2 - sinh(c + d*x) *a*d**2*x + 2*sinh(c + d*x)*b*x**2)/(2*d*x**2)