Integrand size = 17, antiderivative size = 190 \[ \int \frac {\cosh (c+d x)}{x^3 (a+b x)} \, dx=-\frac {\cosh (c+d x)}{2 a x^2}+\frac {b \cosh (c+d x)}{a^2 x}+\frac {b^2 \cosh (c) \text {Chi}(d x)}{a^3}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a}-\frac {b^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{a^3}-\frac {b d \text {Chi}(d x) \sinh (c)}{a^2}-\frac {d \sinh (c+d x)}{2 a x}-\frac {b d \cosh (c) \text {Shi}(d x)}{a^2}+\frac {b^2 \sinh (c) \text {Shi}(d x)}{a^3}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a}-\frac {b^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^3} \] Output:
-1/2*cosh(d*x+c)/a/x^2+b*cosh(d*x+c)/a^2/x+b^2*cosh(c)*Chi(d*x)/a^3+1/2*d^ 2*cosh(c)*Chi(d*x)/a-b^2*cosh(-c+a*d/b)*Chi(a*d/b+d*x)/a^3-b*d*Chi(d*x)*si nh(c)/a^2-1/2*d*sinh(d*x+c)/a/x-b*d*cosh(c)*Shi(d*x)/a^2+b^2*sinh(c)*Shi(d *x)/a^3+1/2*d^2*sinh(c)*Shi(d*x)/a+b^2*sinh(-c+a*d/b)*Shi(a*d/b+d*x)/a^3
Time = 0.28 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.94 \[ \int \frac {\cosh (c+d x)}{x^3 (a+b x)} \, dx=\frac {-a^2 \cosh (c+d x)+2 a b x \cosh (c+d x)-2 b^2 x^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (d \left (\frac {a}{b}+x\right )\right )+x^2 \text {Chi}(d x) \left (\left (2 b^2+a^2 d^2\right ) \cosh (c)-2 a b d \sinh (c)\right )-a^2 d x \sinh (c+d x)-2 a b d x^2 \cosh (c) \text {Shi}(d x)+2 b^2 x^2 \sinh (c) \text {Shi}(d x)+a^2 d^2 x^2 \sinh (c) \text {Shi}(d x)-2 b^2 x^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )}{2 a^3 x^2} \] Input:
Integrate[Cosh[c + d*x]/(x^3*(a + b*x)),x]
Output:
(-(a^2*Cosh[c + d*x]) + 2*a*b*x*Cosh[c + d*x] - 2*b^2*x^2*Cosh[c - (a*d)/b ]*CoshIntegral[d*(a/b + x)] + x^2*CoshIntegral[d*x]*((2*b^2 + a^2*d^2)*Cos h[c] - 2*a*b*d*Sinh[c]) - a^2*d*x*Sinh[c + d*x] - 2*a*b*d*x^2*Cosh[c]*Sinh Integral[d*x] + 2*b^2*x^2*Sinh[c]*SinhIntegral[d*x] + a^2*d^2*x^2*Sinh[c]* SinhIntegral[d*x] - 2*b^2*x^2*Sinh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)]) /(2*a^3*x^2)
Time = 0.76 (sec) , antiderivative size = 190, 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.
\(\displaystyle \int \frac {\cosh (c+d x)}{x^3 (a+b x)} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {b^3 \cosh (c+d x)}{a^3 (a+b x)}+\frac {b^2 \cosh (c+d x)}{a^3 x}-\frac {b \cosh (c+d x)}{a^2 x^2}+\frac {\cosh (c+d x)}{a x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b^2 \cosh (c) \text {Chi}(d x)}{a^3}-\frac {b^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{a^3}+\frac {b^2 \sinh (c) \text {Shi}(d x)}{a^3}-\frac {b^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{a^3}-\frac {b d \sinh (c) \text {Chi}(d x)}{a^2}-\frac {b d \cosh (c) \text {Shi}(d x)}{a^2}+\frac {b \cosh (c+d x)}{a^2 x}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a}-\frac {\cosh (c+d x)}{2 a x^2}-\frac {d \sinh (c+d x)}{2 a x}\) |
Input:
Int[Cosh[c + d*x]/(x^3*(a + b*x)),x]
Output:
-1/2*Cosh[c + d*x]/(a*x^2) + (b*Cosh[c + d*x])/(a^2*x) + (b^2*Cosh[c]*Cosh Integral[d*x])/a^3 + (d^2*Cosh[c]*CoshIntegral[d*x])/(2*a) - (b^2*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/a^3 - (b*d*CoshIntegral[d*x]*Sinh[c ])/a^2 - (d*Sinh[c + d*x])/(2*a*x) - (b*d*Cosh[c]*SinhIntegral[d*x])/a^2 + (b^2*Sinh[c]*SinhIntegral[d*x])/a^3 + (d^2*Sinh[c]*SinhIntegral[d*x])/(2* a) - (b^2*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/a^3
Time = 0.52 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.48
method | result | size |
risch | \(\frac {d \,{\mathrm e}^{-d x -c}}{4 a x}+\frac {{\mathrm e}^{-d x -c} b}{2 a^{2} x}-\frac {{\mathrm e}^{-d x -c}}{4 a \,x^{2}}-\frac {d^{2} {\mathrm e}^{-c} \operatorname {expIntegral}_{1}\left (d x \right )}{4 a}-\frac {d \,{\mathrm e}^{-c} \operatorname {expIntegral}_{1}\left (d x \right ) b}{2 a^{2}}-\frac {{\mathrm e}^{-c} \operatorname {expIntegral}_{1}\left (d x \right ) b^{2}}{2 a^{3}}+\frac {b^{2} {\mathrm e}^{\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (d x +c +\frac {a d -c b}{b}\right )}{2 a^{3}}-\frac {b^{2} {\mathrm e}^{c} \operatorname {expIntegral}_{1}\left (-d x \right )}{2 a^{3}}+\frac {b^{2} {\mathrm e}^{-\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (-d x -c -\frac {a d -c b}{b}\right )}{2 a^{3}}-\frac {{\mathrm e}^{d x +c}}{4 a \,x^{2}}-\frac {d \,{\mathrm e}^{d x +c}}{4 a x}-\frac {d^{2} {\mathrm e}^{c} \operatorname {expIntegral}_{1}\left (-d x \right )}{4 a}+\frac {b \,{\mathrm e}^{d x +c}}{2 a^{2} x}+\frac {d b \,{\mathrm e}^{c} \operatorname {expIntegral}_{1}\left (-d x \right )}{2 a^{2}}\) | \(281\) |
Input:
int(cosh(d*x+c)/x^3/(b*x+a),x,method=_RETURNVERBOSE)
Output:
1/4*d*exp(-d*x-c)/a/x+1/2*exp(-d*x-c)/a^2/x*b-1/4*exp(-d*x-c)/a/x^2-1/4*d^ 2/a*exp(-c)*Ei(1,d*x)-1/2*d/a^2*exp(-c)*Ei(1,d*x)*b-1/2/a^3*exp(-c)*Ei(1,d *x)*b^2+1/2*b^2/a^3*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)-1/2*b^2/a^3*e xp(c)*Ei(1,-d*x)+1/2/a^3*b^2*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)-1/ 4/a/x^2*exp(d*x+c)-1/4*d/a/x*exp(d*x+c)-1/4*d^2/a*exp(c)*Ei(1,-d*x)+1/2*b/ a^2/x*exp(d*x+c)+1/2*d*b/a^2*exp(c)*Ei(1,-d*x)
Time = 0.08 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.45 \[ \int \frac {\cosh (c+d x)}{x^3 (a+b x)} \, dx=-\frac {2 \, a^{2} d x \sinh \left (d x + c\right ) - 2 \, {\left (2 \, a b x - a^{2}\right )} \cosh \left (d x + c\right ) - {\left ({\left (a^{2} d^{2} - 2 \, a b d + 2 \, b^{2}\right )} x^{2} {\rm Ei}\left (d x\right ) + {\left (a^{2} d^{2} + 2 \, a b d + 2 \, b^{2}\right )} x^{2} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \, {\left (b^{2} x^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + b^{2} x^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) - {\left ({\left (a^{2} d^{2} - 2 \, a b d + 2 \, b^{2}\right )} x^{2} {\rm Ei}\left (d x\right ) - {\left (a^{2} d^{2} + 2 \, a b d + 2 \, b^{2}\right )} x^{2} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right ) - 2 \, {\left (b^{2} x^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - b^{2} x^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{4 \, a^{3} x^{2}} \] Input:
integrate(cosh(d*x+c)/x^3/(b*x+a),x, algorithm="fricas")
Output:
-1/4*(2*a^2*d*x*sinh(d*x + c) - 2*(2*a*b*x - a^2)*cosh(d*x + c) - ((a^2*d^ 2 - 2*a*b*d + 2*b^2)*x^2*Ei(d*x) + (a^2*d^2 + 2*a*b*d + 2*b^2)*x^2*Ei(-d*x ))*cosh(c) + 2*(b^2*x^2*Ei((b*d*x + a*d)/b) + b^2*x^2*Ei(-(b*d*x + a*d)/b) )*cosh(-(b*c - a*d)/b) - ((a^2*d^2 - 2*a*b*d + 2*b^2)*x^2*Ei(d*x) - (a^2*d ^2 + 2*a*b*d + 2*b^2)*x^2*Ei(-d*x))*sinh(c) - 2*(b^2*x^2*Ei((b*d*x + a*d)/ b) - b^2*x^2*Ei(-(b*d*x + a*d)/b))*sinh(-(b*c - a*d)/b))/(a^3*x^2)
\[ \int \frac {\cosh (c+d x)}{x^3 (a+b x)} \, dx=\int \frac {\cosh {\left (c + d x \right )}}{x^{3} \left (a + b x\right )}\, dx \] Input:
integrate(cosh(d*x+c)/x**3/(b*x+a),x)
Output:
Integral(cosh(c + d*x)/(x**3*(a + b*x)), x)
Time = 0.21 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.27 \[ \int \frac {\cosh (c+d x)}{x^3 (a+b x)} \, dx=\frac {1}{4} \, d {\left (\frac {d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + d e^{c} \Gamma \left (-1, -d x\right )}{a} + \frac {2 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - {\rm Ei}\left (d x\right ) e^{c}\right )} b}{a^{2}} + \frac {2 \, b^{3} {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b} + \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b}\right )}}{a^{3} d} + \frac {4 \, b^{2} \cosh \left (d x + c\right ) \log \left (b x + a\right )}{a^{3} d} - \frac {4 \, b^{2} \cosh \left (d x + c\right ) \log \left (x\right )}{a^{3} d} + \frac {2 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} b^{2}}{a^{3} d}\right )} - \frac {1}{2} \, {\left (\frac {2 \, b^{2} \log \left (b x + a\right )}{a^{3}} - \frac {2 \, b^{2} \log \left (x\right )}{a^{3}} - \frac {2 \, b x - a}{a^{2} x^{2}}\right )} \cosh \left (d x + c\right ) \] Input:
integrate(cosh(d*x+c)/x^3/(b*x+a),x, algorithm="maxima")
Output:
1/4*d*((d*e^(-c)*gamma(-1, d*x) + d*e^c*gamma(-1, -d*x))/a + 2*(Ei(-d*x)*e ^(-c) - Ei(d*x)*e^c)*b/a^2 + 2*b^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)/(a^3*d) + 4*b^2*cosh(d*x + c)*log(b*x + a)/(a^3*d) - 4*b^2*cosh(d*x + c)*log(x)/(a^ 3*d) + 2*(Ei(-d*x)*e^(-c) + Ei(d*x)*e^c)*b^2/(a^3*d)) - 1/2*(2*b^2*log(b*x + a)/a^3 - 2*b^2*log(x)/a^3 - (2*b*x - a)/(a^2*x^2))*cosh(d*x + c)
Time = 0.12 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.31 \[ \int \frac {\cosh (c+d x)}{x^3 (a+b x)} \, dx=\frac {a^{2} d^{2} x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{2} x^{2} {\rm Ei}\left (d x\right ) e^{c} + 2 \, a b d x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 2 \, a b d x^{2} {\rm Ei}\left (d x\right ) e^{c} + 2 \, b^{2} x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 2 \, b^{2} x^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 2 \, b^{2} x^{2} {\rm Ei}\left (d x\right ) e^{c} - 2 \, b^{2} x^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} - a^{2} d x e^{\left (d x + c\right )} + a^{2} d x e^{\left (-d x - c\right )} + 2 \, a b x e^{\left (d x + c\right )} + 2 \, a b x e^{\left (-d x - c\right )} - a^{2} e^{\left (d x + c\right )} - a^{2} e^{\left (-d x - c\right )}}{4 \, a^{3} x^{2}} \] Input:
integrate(cosh(d*x+c)/x^3/(b*x+a),x, algorithm="giac")
Output:
1/4*(a^2*d^2*x^2*Ei(-d*x)*e^(-c) + a^2*d^2*x^2*Ei(d*x)*e^c + 2*a*b*d*x^2*E i(-d*x)*e^(-c) - 2*a*b*d*x^2*Ei(d*x)*e^c + 2*b^2*x^2*Ei(-d*x)*e^(-c) - 2*b ^2*x^2*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 2*b^2*x^2*Ei(d*x)*e^c - 2*b^2*x ^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) - a^2*d*x*e^(d*x + c) + a^2*d*x*e^( -d*x - c) + 2*a*b*x*e^(d*x + c) + 2*a*b*x*e^(-d*x - c) - a^2*e^(d*x + c) - a^2*e^(-d*x - c))/(a^3*x^2)
Timed out. \[ \int \frac {\cosh (c+d x)}{x^3 (a+b x)} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )}{x^3\,\left (a+b\,x\right )} \,d x \] Input:
int(cosh(c + d*x)/(x^3*(a + b*x)),x)
Output:
int(cosh(c + d*x)/(x^3*(a + b*x)), x)
\[ \int \frac {\cosh (c+d x)}{x^3 (a+b x)} \, dx=\int \frac {\cosh \left (d x +c \right )}{b \,x^{4}+a \,x^{3}}d x \] Input:
int(cosh(d*x+c)/x^3/(b*x+a),x)
Output:
int(cosh(c + d*x)/(a*x**3 + b*x**4),x)