Integrand size = 17, antiderivative size = 91 \[ \int \frac {x^2 \cosh (c+d x)}{a+b x} \, dx=-\frac {\cosh (c+d x)}{b d^2}+\frac {a^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^3}-\frac {(a-b x) \sinh (c+d x)}{b^2 d}+\frac {a^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^3} \] Output:
-cosh(d*x+c)/b/d^2+a^2*cosh(-c+a*d/b)*Chi(a*d/b+d*x)/b^3-(-b*x+a)*sinh(d*x +c)/b^2/d-a^2*sinh(-c+a*d/b)*Shi(a*d/b+d*x)/b^3
Time = 0.18 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.98 \[ \int \frac {x^2 \cosh (c+d x)}{a+b x} \, dx=\frac {a^2 d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (d \left (\frac {a}{b}+x\right )\right )+b (-b \cosh (c+d x)+d (-a+b x) \sinh (c+d x))+a^2 d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )}{b^3 d^2} \] Input:
Integrate[(x^2*Cosh[c + d*x])/(a + b*x),x]
Output:
(a^2*d^2*Cosh[c - (a*d)/b]*CoshIntegral[d*(a/b + x)] + b*(-(b*Cosh[c + d*x ]) + d*(-a + b*x)*Sinh[c + d*x]) + a^2*d^2*Sinh[c - (a*d)/b]*SinhIntegral[ d*(a/b + x)])/(b^3*d^2)
Time = 0.49 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.10, 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.
\(\displaystyle \int \frac {x^2 \cosh (c+d x)}{a+b x} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {a^2 \cosh (c+d x)}{b^2 (a+b x)}-\frac {a \cosh (c+d x)}{b^2}+\frac {x \cosh (c+d x)}{b}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {a^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^3}-\frac {a \sinh (c+d x)}{b^2 d}-\frac {\cosh (c+d x)}{b d^2}+\frac {x \sinh (c+d x)}{b d}\) |
Input:
Int[(x^2*Cosh[c + d*x])/(a + b*x),x]
Output:
-(Cosh[c + d*x]/(b*d^2)) + (a^2*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d *x])/b^3 - (a*Sinh[c + d*x])/(b^2*d) + (x*Sinh[c + d*x])/(b*d) + (a^2*Sinh [c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^3
Time = 0.46 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.02
method | result | size |
risch | \(-\frac {{\mathrm e}^{-\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (-d x -c -\frac {a d -c b}{b}\right ) a^{2}}{2 b^{3}}-\frac {{\mathrm e}^{\frac {a d -c b}{b}} \operatorname {expIntegral}_{1}\left (d x +c +\frac {a d -c b}{b}\right ) a^{2}}{2 b^{3}}-\frac {{\mathrm e}^{-d x -c} x}{2 d b}+\frac {{\mathrm e}^{d x +c} x}{2 d b}+\frac {{\mathrm e}^{-d x -c} a}{2 d \,b^{2}}-\frac {a \,{\mathrm e}^{d x +c}}{2 d \,b^{2}}-\frac {{\mathrm e}^{-d x -c}}{2 d^{2} b}-\frac {{\mathrm e}^{d x +c}}{2 d^{2} b}\) | \(184\) |
Input:
int(x^2*cosh(d*x+c)/(b*x+a),x,method=_RETURNVERBOSE)
Output:
-1/2/b^3*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a^2-1/2/b^3*exp((a*d-b *c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*a^2-1/2/d/b*exp(-d*x-c)*x+1/2/d/b*exp(d*x+c )*x+1/2/d/b^2*exp(-d*x-c)*a-1/2/d/b^2*a*exp(d*x+c)-1/2/d^2/b*exp(-d*x-c)-1 /2/d^2/b*exp(d*x+c)
Time = 0.09 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.71 \[ \int \frac {x^2 \cosh (c+d x)}{a+b x} \, dx=-\frac {2 \, b^{2} \cosh \left (d x + c\right ) - {\left (a^{2} d^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + a^{2} d^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) - 2 \, {\left (b^{2} d x - a b d\right )} \sinh \left (d x + c\right ) + {\left (a^{2} d^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - a^{2} d^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{2 \, b^{3} d^{2}} \] Input:
integrate(x^2*cosh(d*x+c)/(b*x+a),x, algorithm="fricas")
Output:
-1/2*(2*b^2*cosh(d*x + c) - (a^2*d^2*Ei((b*d*x + a*d)/b) + a^2*d^2*Ei(-(b* d*x + a*d)/b))*cosh(-(b*c - a*d)/b) - 2*(b^2*d*x - a*b*d)*sinh(d*x + c) + (a^2*d^2*Ei((b*d*x + a*d)/b) - a^2*d^2*Ei(-(b*d*x + a*d)/b))*sinh(-(b*c - a*d)/b))/(b^3*d^2)
\[ \int \frac {x^2 \cosh (c+d x)}{a+b x} \, dx=\int \frac {x^{2} \cosh {\left (c + d x \right )}}{a + b x}\, dx \] Input:
integrate(x**2*cosh(d*x+c)/(b*x+a),x)
Output:
Integral(x**2*cosh(c + d*x)/(a + b*x), x)
Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (94) = 188\).
Time = 0.12 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.56 \[ \int \frac {x^2 \cosh (c+d x)}{a+b x} \, dx=-\frac {1}{4} \, d {\left (\frac {2 \, a^{2} {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b} + \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b}\right )}}{b^{2} d} - \frac {2 \, a {\left (\frac {{\left (d x e^{c} - e^{c}\right )} e^{\left (d x\right )}}{d^{2}} + \frac {{\left (d x + 1\right )} e^{\left (-d x - c\right )}}{d^{2}}\right )}}{b^{2}} + \frac {\frac {{\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} e^{\left (d x\right )}}{d^{3}} + \frac {{\left (d^{2} x^{2} + 2 \, d x + 2\right )} e^{\left (-d x - c\right )}}{d^{3}}}{b} + \frac {4 \, a^{2} \cosh \left (d x + c\right ) \log \left (b x + a\right )}{b^{3} d}\right )} + \frac {1}{2} \, {\left (\frac {2 \, a^{2} \log \left (b x + a\right )}{b^{3}} + \frac {b x^{2} - 2 \, a x}{b^{2}}\right )} \cosh \left (d x + c\right ) \] Input:
integrate(x^2*cosh(d*x+c)/(b*x+a),x, algorithm="maxima")
Output:
-1/4*d*(2*a^2*(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)/(b^2*d) - 2*a*((d*x*e^c - e^c) *e^(d*x)/d^2 + (d*x + 1)*e^(-d*x - c)/d^2)/b^2 + ((d^2*x^2*e^c - 2*d*x*e^c + 2*e^c)*e^(d*x)/d^3 + (d^2*x^2 + 2*d*x + 2)*e^(-d*x - c)/d^3)/b + 4*a^2* cosh(d*x + c)*log(b*x + a)/(b^3*d)) + 1/2*(2*a^2*log(b*x + a)/b^3 + (b*x^2 - 2*a*x)/b^2)*cosh(d*x + c)
Time = 0.13 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.63 \[ \int \frac {x^2 \cosh (c+d x)}{a+b x} \, dx=\frac {a^{2} d^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + a^{2} d^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + b^{2} d x e^{\left (d x + c\right )} - b^{2} d x e^{\left (-d x - c\right )} - a b d e^{\left (d x + c\right )} + a b d e^{\left (-d x - c\right )} - b^{2} e^{\left (d x + c\right )} - b^{2} e^{\left (-d x - c\right )}}{2 \, b^{3} d^{2}} \] Input:
integrate(x^2*cosh(d*x+c)/(b*x+a),x, algorithm="giac")
Output:
1/2*(a^2*d^2*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + a^2*d^2*Ei(-(b*d*x + a*d) /b)*e^(-c + a*d/b) + b^2*d*x*e^(d*x + c) - b^2*d*x*e^(-d*x - c) - a*b*d*e^ (d*x + c) + a*b*d*e^(-d*x - c) - b^2*e^(d*x + c) - b^2*e^(-d*x - c))/(b^3* d^2)
Timed out. \[ \int \frac {x^2 \cosh (c+d x)}{a+b x} \, dx=\int \frac {x^2\,\mathrm {cosh}\left (c+d\,x\right )}{a+b\,x} \,d x \] Input:
int((x^2*cosh(c + d*x))/(a + b*x),x)
Output:
int((x^2*cosh(c + d*x))/(a + b*x), x)
\[ \int \frac {x^2 \cosh (c+d x)}{a+b x} \, dx=\frac {-\cosh \left (d x +c \right ) b +\left (\int \frac {\cosh \left (d x +c \right )}{b x +a}d x \right ) a^{2} d^{2}-\sinh \left (d x +c \right ) a d +\sinh \left (d x +c \right ) b d x}{b^{2} d^{2}} \] Input:
int(x^2*cosh(d*x+c)/(b*x+a),x)
Output:
( - cosh(c + d*x)*b + int(cosh(c + d*x)/(a + b*x),x)*a**2*d**2 - sinh(c + d*x)*a*d + sinh(c + d*x)*b*d*x)/(b**2*d**2)