Integrand size = 19, antiderivative size = 197 \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )} \, dx=\frac {\cosh (c) \text {Chi}(d x)}{a}-\frac {\cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a}+\frac {\sinh (c) \text {Shi}(d x)}{a}+\frac {\sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a} \]
Chi(d*x)*cosh(c)/a-1/2*Chi(d*x+d*(-a)^(1/2)/b^(1/2))*cosh(c-d*(-a)^(1/2)/b ^(1/2))/a-1/2*Chi(-d*x+d*(-a)^(1/2)/b^(1/2))*cosh(c+d*(-a)^(1/2)/b^(1/2))/ a+Shi(d*x)*sinh(c)/a-1/2*Shi(d*x+d*(-a)^(1/2)/b^(1/2))*sinh(c-d*(-a)^(1/2) /b^(1/2))/a-1/2*Shi(d*x-d*(-a)^(1/2)/b^(1/2))*sinh(c+d*(-a)^(1/2)/b^(1/2)) /a
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.88 \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )} \, dx=-\frac {-4 \cosh (c) \text {Chi}(d x)+e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{2 c+\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+e^{2 c} \operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )-4 \sinh (c) \text {Shi}(d x)}{4 a} \]
-1/4*(-4*Cosh[c]*CoshIntegral[d*x] + E^(-c - (I*Sqrt[a]*d)/Sqrt[b])*(E^(2* c + ((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[d*(((-I)*Sqrt[a])/Sqrt[b] + x )] + E^(2*c)*ExpIntegralEi[d*((I*Sqrt[a])/Sqrt[b] + x)] + E^(((2*I)*Sqrt[a ]*d)/Sqrt[b])*ExpIntegralEi[((-I)*Sqrt[a]*d)/Sqrt[b] - d*x] + ExpIntegralE i[(I*Sqrt[a]*d)/Sqrt[b] - d*x]) - 4*Sinh[c]*SinhIntegral[d*x])/a
Time = 0.62 (sec) , antiderivative size = 197, 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 = {5816, 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.
\(\displaystyle \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )} \, dx\) |
\(\Big \downarrow \) 5816 |
\(\displaystyle \int \left (\frac {\cosh (c+d x)}{a x}-\frac {b x \cosh (c+d x)}{a \left (a+b x^2\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a}+\frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a}+\frac {\cosh (c) \text {Chi}(d x)}{a}+\frac {\sinh (c) \text {Shi}(d x)}{a}\) |
(Cosh[c]*CoshIntegral[d*x])/a - (Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegr al[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a) - (Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*Co shIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a) + (Sinh[c]*SinhIntegral[d*x] )/a + (Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a) - (Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sq rt[b] + d*x])/(2*a)
3.1.62.3.1 Defintions of rubi rules used
Int[Cosh[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Sy mbol] :> Int[ExpandIntegrand[Cosh[c + d*x], x^m*(a + b*x^n)^p, x], x] /; Fr eeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[m] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])
Time = 0.26 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.15
method | result | size |
risch | \(-\frac {{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right )}{2 a}-\frac {{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right )}{2 a}+\frac {{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{4 a}+\frac {{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{4 a}+\frac {{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{4 a}+\frac {{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{4 a}\) | \(227\) |
-1/2/a*exp(-c)*Ei(1,d*x)-1/2/a*exp(c)*Ei(1,-d*x)+1/4/a*exp((d*(-a*b)^(1/2) +c*b)/b)*Ei(1,(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)+1/4/a*exp((-d*(-a*b)^(1/2) +c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)+1/4/a*exp(-(d*(-a*b)^(1/2 )+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)+1/4/a*exp(-(-d*(-a*b)^(1 /2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)
Time = 0.26 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.26 \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )} \, dx=-\frac {{\left ({\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) + {\rm Ei}\left (-d x + \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \cosh \left (c + \sqrt {-\frac {a d^{2}}{b}}\right ) - 2 \, {\left ({\rm Ei}\left (d x\right ) + {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + {\left ({\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \cosh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left ({\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) - {\rm Ei}\left (-d x + \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (c + \sqrt {-\frac {a d^{2}}{b}}\right ) - 2 \, {\left ({\rm Ei}\left (d x\right ) - {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right ) - {\left ({\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right )}{4 \, a} \]
-1/4*((Ei(d*x - sqrt(-a*d^2/b)) + Ei(-d*x + sqrt(-a*d^2/b)))*cosh(c + sqrt (-a*d^2/b)) - 2*(Ei(d*x) + Ei(-d*x))*cosh(c) + (Ei(d*x + sqrt(-a*d^2/b)) + Ei(-d*x - sqrt(-a*d^2/b)))*cosh(-c + sqrt(-a*d^2/b)) + (Ei(d*x - sqrt(-a* d^2/b)) - Ei(-d*x + sqrt(-a*d^2/b)))*sinh(c + sqrt(-a*d^2/b)) - 2*(Ei(d*x) - Ei(-d*x))*sinh(c) - (Ei(d*x + sqrt(-a*d^2/b)) - Ei(-d*x - sqrt(-a*d^2/b )))*sinh(-c + sqrt(-a*d^2/b)))/a
\[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )} \, dx=\int \frac {\cosh {\left (c + d x \right )}}{x \left (a + b x^{2}\right )}\, dx \]
\[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )} x} \,d x } \]
\[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )} x} \,d x } \]
Timed out. \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )}{x\,\left (b\,x^2+a\right )} \,d x \]