Integrand size = 15, antiderivative size = 132 \[ \int \frac {(a+b x) \cosh (c+d x)}{x^4} \, dx=-\frac {a \cosh (c+d x)}{3 x^3}-\frac {b \cosh (c+d x)}{2 x^2}-\frac {a d^2 \cosh (c+d x)}{6 x}+\frac {1}{2} b d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{6} a d^3 \text {Chi}(d x) \sinh (c)-\frac {a d \sinh (c+d x)}{6 x^2}-\frac {b d \sinh (c+d x)}{2 x}+\frac {1}{6} a d^3 \cosh (c) \text {Shi}(d x)+\frac {1}{2} b d^2 \sinh (c) \text {Shi}(d x) \] Output:
-1/3*a*cosh(d*x+c)/x^3-1/2*b*cosh(d*x+c)/x^2-1/6*a*d^2*cosh(d*x+c)/x+1/2*b *d^2*cosh(c)*Chi(d*x)+1/6*a*d^3*Chi(d*x)*sinh(c)-1/6*a*d*sinh(d*x+c)/x^2-1 /2*b*d*sinh(d*x+c)/x+1/6*a*d^3*cosh(c)*Shi(d*x)+1/2*b*d^2*sinh(c)*Shi(d*x)
Time = 0.21 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x) \cosh (c+d x)}{x^4} \, dx=-\frac {2 a \cosh (c+d x)+3 b x \cosh (c+d x)+a d^2 x^2 \cosh (c+d x)-d^2 x^3 \text {Chi}(d x) (3 b \cosh (c)+a d \sinh (c))+a d x \sinh (c+d x)+3 b d x^2 \sinh (c+d x)-d^2 x^3 (a d \cosh (c)+3 b \sinh (c)) \text {Shi}(d x)}{6 x^3} \] Input:
Integrate[((a + b*x)*Cosh[c + d*x])/x^4,x]
Output:
-1/6*(2*a*Cosh[c + d*x] + 3*b*x*Cosh[c + d*x] + a*d^2*x^2*Cosh[c + d*x] - d^2*x^3*CoshIntegral[d*x]*(3*b*Cosh[c] + a*d*Sinh[c]) + a*d*x*Sinh[c + d*x ] + 3*b*d*x^2*Sinh[c + d*x] - d^2*x^3*(a*d*Cosh[c] + 3*b*Sinh[c])*SinhInte gral[d*x])/x^3
Time = 0.56 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, 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.
\(\displaystyle \int \frac {(a+b x) \cosh (c+d x)}{x^4} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {a \cosh (c+d x)}{x^4}+\frac {b \cosh (c+d x)}{x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{6} a d^3 \sinh (c) \text {Chi}(d x)+\frac {1}{6} a d^3 \cosh (c) \text {Shi}(d x)-\frac {a d^2 \cosh (c+d x)}{6 x}-\frac {a \cosh (c+d x)}{3 x^3}-\frac {a d \sinh (c+d x)}{6 x^2}+\frac {1}{2} b d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{2} b d^2 \sinh (c) \text {Shi}(d x)-\frac {b \cosh (c+d x)}{2 x^2}-\frac {b d \sinh (c+d x)}{2 x}\) |
Input:
Int[((a + b*x)*Cosh[c + d*x])/x^4,x]
Output:
-1/3*(a*Cosh[c + d*x])/x^3 - (b*Cosh[c + d*x])/(2*x^2) - (a*d^2*Cosh[c + d *x])/(6*x) + (b*d^2*Cosh[c]*CoshIntegral[d*x])/2 + (a*d^3*CoshIntegral[d*x ]*Sinh[c])/6 - (a*d*Sinh[c + d*x])/(6*x^2) - (b*d*Sinh[c + d*x])/(2*x) + ( a*d^3*Cosh[c]*SinhIntegral[d*x])/6 + (b*d^2*Sinh[c]*SinhIntegral[d*x])/2
Time = 0.58 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.55
method | result | size |
risch | \(-\frac {-{\mathrm e}^{-c} \operatorname {expIntegral}_{1}\left (d x \right ) a \,d^{3} x^{3}+{\mathrm e}^{c} \operatorname {expIntegral}_{1}\left (-d x \right ) a \,d^{3} x^{3}+3 \,{\mathrm e}^{-c} \operatorname {expIntegral}_{1}\left (d x \right ) b \,d^{2} x^{3}+3 \,{\mathrm e}^{c} \operatorname {expIntegral}_{1}\left (-d x \right ) b \,d^{2} x^{3}+a \,d^{2} x^{2} {\mathrm e}^{-d x -c}+a \,d^{2} x^{2} {\mathrm e}^{d x +c}-3 b d \,x^{2} {\mathrm e}^{-d x -c}+3 b d \,x^{2} {\mathrm e}^{d x +c}-a d x \,{\mathrm e}^{-d x -c}+a d x \,{\mathrm e}^{d x +c}+3 \,{\mathrm e}^{-d x -c} b x +3 \,{\mathrm e}^{d x +c} b x +2 \,{\mathrm e}^{-d x -c} a +2 \,{\mathrm e}^{d x +c} a}{12 x^{3}}\) | \(204\) |
meijerg | \(-\frac {d^{2} b \cosh \left (c \right ) \sqrt {\pi }\, \left (-\frac {4 \left (\frac {9 x^{2} d^{2}}{2}+3\right )}{3 \sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \cosh \left (d x \right )}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \sinh \left (d x \right )}{\sqrt {\pi }\, x d}-\frac {4 \left (\operatorname {Chi}\left (d x \right )-\ln \left (d x \right )-\gamma \right )}{\sqrt {\pi }}-\frac {2 \left (2 \gamma -3+2 \ln \left (x \right )+2 \ln \left (i d \right )\right )}{\sqrt {\pi }}+\frac {4}{\sqrt {\pi }\, x^{2} d^{2}}\right )}{8}+\frac {i d^{2} b \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {4 i \cosh \left (d x \right )}{d x \sqrt {\pi }}+\frac {4 i \sinh \left (d x \right )}{x^{2} d^{2} \sqrt {\pi }}-\frac {4 i \operatorname {Shi}\left (d x \right )}{\sqrt {\pi }}\right )}{8}-\frac {i a \cosh \left (c \right ) \sqrt {\pi }\, d^{3} \left (-\frac {8 i \left (x^{2} d^{2}+2\right ) \cosh \left (d x \right )}{3 d^{3} x^{3} \sqrt {\pi }}-\frac {8 i \sinh \left (d x \right )}{3 x^{2} d^{2} \sqrt {\pi }}+\frac {8 i \operatorname {Shi}\left (d x \right )}{3 \sqrt {\pi }}\right )}{16}-\frac {a \sinh \left (c \right ) \sqrt {\pi }\, d^{3} \left (-\frac {8 \left (\frac {55 x^{2} d^{2}}{2}+45\right )}{45 \sqrt {\pi }\, x^{2} d^{2}}+\frac {8 \cosh \left (d x \right )}{3 \sqrt {\pi }\, x^{2} d^{2}}+\frac {16 \left (\frac {5 x^{2} d^{2}}{2}+5\right ) \sinh \left (d x \right )}{15 \sqrt {\pi }\, x^{3} d^{3}}-\frac {8 \left (\operatorname {Chi}\left (d x \right )-\ln \left (d x \right )-\gamma \right )}{3 \sqrt {\pi }}-\frac {4 \left (2 \gamma -\frac {11}{3}+2 \ln \left (x \right )+2 \ln \left (i d \right )\right )}{3 \sqrt {\pi }}+\frac {8}{\sqrt {\pi }\, x^{2} d^{2}}\right )}{16}\) | \(359\) |
Input:
int((b*x+a)*cosh(d*x+c)/x^4,x,method=_RETURNVERBOSE)
Output:
-1/12*(-exp(-c)*Ei(1,d*x)*a*d^3*x^3+exp(c)*Ei(1,-d*x)*a*d^3*x^3+3*exp(-c)* Ei(1,d*x)*b*d^2*x^3+3*exp(c)*Ei(1,-d*x)*b*d^2*x^3+a*d^2*x^2*exp(-d*x-c)+a* d^2*x^2*exp(d*x+c)-3*b*d*x^2*exp(-d*x-c)+3*b*d*x^2*exp(d*x+c)-a*d*x*exp(-d *x-c)+a*d*x*exp(d*x+c)+3*exp(-d*x-c)*b*x+3*exp(d*x+c)*b*x+2*exp(-d*x-c)*a+ 2*exp(d*x+c)*a)/x^3
Time = 0.08 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b x) \cosh (c+d x)}{x^4} \, dx=-\frac {2 \, {\left (a d^{2} x^{2} + 3 \, b x + 2 \, a\right )} \cosh \left (d x + c\right ) - {\left ({\left (a d^{3} + 3 \, b d^{2}\right )} x^{3} {\rm Ei}\left (d x\right ) - {\left (a d^{3} - 3 \, b d^{2}\right )} x^{3} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \, {\left (3 \, b d x^{2} + a d x\right )} \sinh \left (d x + c\right ) - {\left ({\left (a d^{3} + 3 \, b d^{2}\right )} x^{3} {\rm Ei}\left (d x\right ) + {\left (a d^{3} - 3 \, b d^{2}\right )} x^{3} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{12 \, x^{3}} \] Input:
integrate((b*x+a)*cosh(d*x+c)/x^4,x, algorithm="fricas")
Output:
-1/12*(2*(a*d^2*x^2 + 3*b*x + 2*a)*cosh(d*x + c) - ((a*d^3 + 3*b*d^2)*x^3* Ei(d*x) - (a*d^3 - 3*b*d^2)*x^3*Ei(-d*x))*cosh(c) + 2*(3*b*d*x^2 + a*d*x)* sinh(d*x + c) - ((a*d^3 + 3*b*d^2)*x^3*Ei(d*x) + (a*d^3 - 3*b*d^2)*x^3*Ei( -d*x))*sinh(c))/x^3
Timed out. \[ \int \frac {(a+b x) \cosh (c+d x)}{x^4} \, dx=\text {Timed out} \] Input:
integrate((b*x+a)*cosh(d*x+c)/x**4,x)
Output:
Timed out
Time = 0.14 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.59 \[ \int \frac {(a+b x) \cosh (c+d x)}{x^4} \, dx=\frac {1}{12} \, {\left (2 \, a d^{2} e^{\left (-c\right )} \Gamma \left (-2, d x\right ) - 2 \, a d^{2} e^{c} \Gamma \left (-2, -d x\right ) + 3 \, b d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + 3 \, b d e^{c} \Gamma \left (-1, -d x\right )\right )} d - \frac {{\left (3 \, b x + 2 \, a\right )} \cosh \left (d x + c\right )}{6 \, x^{3}} \] Input:
integrate((b*x+a)*cosh(d*x+c)/x^4,x, algorithm="maxima")
Output:
1/12*(2*a*d^2*e^(-c)*gamma(-2, d*x) - 2*a*d^2*e^c*gamma(-2, -d*x) + 3*b*d* e^(-c)*gamma(-1, d*x) + 3*b*d*e^c*gamma(-1, -d*x))*d - 1/6*(3*b*x + 2*a)*c osh(d*x + c)/x^3
Time = 0.14 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.51 \[ \int \frac {(a+b x) \cosh (c+d x)}{x^4} \, dx=-\frac {a d^{3} x^{3} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a d^{3} x^{3} {\rm Ei}\left (d x\right ) e^{c} - 3 \, b d^{2} x^{3} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 3 \, b d^{2} x^{3} {\rm Ei}\left (d x\right ) e^{c} + a d^{2} x^{2} e^{\left (d x + c\right )} + a d^{2} x^{2} e^{\left (-d x - c\right )} + 3 \, b d x^{2} e^{\left (d x + c\right )} - 3 \, b d x^{2} e^{\left (-d x - c\right )} + a d x e^{\left (d x + c\right )} - a d x e^{\left (-d x - c\right )} + 3 \, b x e^{\left (d x + c\right )} + 3 \, b x e^{\left (-d x - c\right )} + 2 \, a e^{\left (d x + c\right )} + 2 \, a e^{\left (-d x - c\right )}}{12 \, x^{3}} \] Input:
integrate((b*x+a)*cosh(d*x+c)/x^4,x, algorithm="giac")
Output:
-1/12*(a*d^3*x^3*Ei(-d*x)*e^(-c) - a*d^3*x^3*Ei(d*x)*e^c - 3*b*d^2*x^3*Ei( -d*x)*e^(-c) - 3*b*d^2*x^3*Ei(d*x)*e^c + a*d^2*x^2*e^(d*x + c) + a*d^2*x^2 *e^(-d*x - c) + 3*b*d*x^2*e^(d*x + c) - 3*b*d*x^2*e^(-d*x - c) + a*d*x*e^( d*x + c) - a*d*x*e^(-d*x - c) + 3*b*x*e^(d*x + c) + 3*b*x*e^(-d*x - c) + 2 *a*e^(d*x + c) + 2*a*e^(-d*x - c))/x^3
Timed out. \[ \int \frac {(a+b x) \cosh (c+d x)}{x^4} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,\left (a+b\,x\right )}{x^4} \,d x \] Input:
int((cosh(c + d*x)*(a + b*x))/x^4,x)
Output:
int((cosh(c + d*x)*(a + b*x))/x^4, x)
Time = 0.15 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.52 \[ \int \frac {(a+b x) \cosh (c+d x)}{x^4} \, dx=\frac {-e^{d x} \mathit {ei} \left (-d x \right ) a \,d^{3} x^{3}+3 e^{d x} \mathit {ei} \left (-d x \right ) b \,d^{2} x^{3}+e^{d x +2 c} \mathit {ei} \left (d x \right ) a \,d^{3} x^{3}+3 e^{d x +2 c} \mathit {ei} \left (d x \right ) b \,d^{2} x^{3}-e^{2 d x +2 c} a \,d^{2} x^{2}-e^{2 d x +2 c} a d x -2 e^{2 d x +2 c} a -3 e^{2 d x +2 c} b d \,x^{2}-3 e^{2 d x +2 c} b x -a \,d^{2} x^{2}+a d x -2 a +3 b d \,x^{2}-3 b x}{12 e^{d x +c} x^{3}} \] Input:
int((b*x+a)*cosh(d*x+c)/x^4,x)
Output:
( - e**(d*x)*ei( - d*x)*a*d**3*x**3 + 3*e**(d*x)*ei( - d*x)*b*d**2*x**3 + e**(2*c + d*x)*ei(d*x)*a*d**3*x**3 + 3*e**(2*c + d*x)*ei(d*x)*b*d**2*x**3 - e**(2*c + 2*d*x)*a*d**2*x**2 - e**(2*c + 2*d*x)*a*d*x - 2*e**(2*c + 2*d* x)*a - 3*e**(2*c + 2*d*x)*b*d*x**2 - 3*e**(2*c + 2*d*x)*b*x - a*d**2*x**2 + a*d*x - 2*a + 3*b*d*x**2 - 3*b*x)/(12*e**(c + d*x)*x**3)