Integrand size = 19, antiderivative size = 500 \[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^2\right )^2} \, dx=-\frac {\cosh (c+d x)}{a^2 x}+\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 \sqrt {b} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{5/2}}+\frac {3 \sqrt {b} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{5/2}}+\frac {d \text {Chi}(d x) \sinh (c)}{a^2}+\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a^2}+\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a^2}+\frac {d \cosh (c) \text {Shi}(d x)}{a^2}-\frac {d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a^2}+\frac {3 \sqrt {b} \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{5/2}}+\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 a^2}+\frac {3 \sqrt {b} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{5/2}} \]
-cosh(d*x+c)/a^2/x+d*cosh(c)*Shi(d*x)/a^2+1/4*d*cosh(c+d*(-a)^(1/2)/b^(1/2 ))*Shi(d*x-d*(-a)^(1/2)/b^(1/2))/a^2+1/4*d*cosh(c-d*(-a)^(1/2)/b^(1/2))*Sh i(d*x+d*(-a)^(1/2)/b^(1/2))/a^2+d*Chi(d*x)*sinh(c)/a^2+1/4*d*Chi(d*x+d*(-a )^(1/2)/b^(1/2))*sinh(c-d*(-a)^(1/2)/b^(1/2))/a^2+1/4*d*Chi(-d*x+d*(-a)^(1 /2)/b^(1/2))*sinh(c+d*(-a)^(1/2)/b^(1/2))/a^2+3/4*Chi(d*x+d*(-a)^(1/2)/b^( 1/2))*cosh(c-d*(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(5/2)-3/4*Chi(-d*x+d*(-a)^ (1/2)/b^(1/2))*cosh(c+d*(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(5/2)+3/4*Shi(d*x +d*(-a)^(1/2)/b^(1/2))*sinh(c-d*(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(5/2)-3/4 *Shi(d*x-d*(-a)^(1/2)/b^(1/2))*sinh(c+d*(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^( 5/2)+1/4*cosh(d*x+c)*b^(1/2)/a^2/((-a)^(1/2)-x*b^(1/2))-1/4*cosh(d*x+c)*b^ (1/2)/a^2/((-a)^(1/2)+x*b^(1/2))
Result contains complex when optimal does not.
Time = 2.15 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.67 \[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^2\right )^2} \, dx=\frac {-\frac {4 \sqrt {a} \left (2 a+3 b x^2\right ) \cosh (c) \cosh (d x)}{x \left (a+b x^2\right )}+e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (3 i \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\left (-3 i \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )-e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (3 i \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\left (-3 i \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )-\frac {4 \sqrt {a} \left (2 a+3 b x^2\right ) \sinh (c) \sinh (d x)}{x \left (a+b x^2\right )}+8 \sqrt {a} d (\text {Chi}(d x) \sinh (c)+\cosh (c) \text {Shi}(d x))}{8 a^{5/2}} \]
((-4*Sqrt[a]*(2*a + 3*b*x^2)*Cosh[c]*Cosh[d*x])/(x*(a + b*x^2)) + E^(c - ( I*Sqrt[a]*d)/Sqrt[b])*(((3*I)*Sqrt[b] + Sqrt[a]*d)*E^(((2*I)*Sqrt[a]*d)/Sq rt[b])*ExpIntegralEi[d*(((-I)*Sqrt[a])/Sqrt[b] + x)] + ((-3*I)*Sqrt[b] + S qrt[a]*d)*ExpIntegralEi[d*((I*Sqrt[a])/Sqrt[b] + x)]) - E^(-c - (I*Sqrt[a] *d)/Sqrt[b])*(((3*I)*Sqrt[b] + Sqrt[a]*d)*E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*Ex pIntegralEi[((-I)*Sqrt[a]*d)/Sqrt[b] - d*x] + ((-3*I)*Sqrt[b] + Sqrt[a]*d) *ExpIntegralEi[(I*Sqrt[a]*d)/Sqrt[b] - d*x]) - (4*Sqrt[a]*(2*a + 3*b*x^2)* Sinh[c]*Sinh[d*x])/(x*(a + b*x^2)) + 8*Sqrt[a]*d*(CoshIntegral[d*x]*Sinh[c ] + Cosh[c]*SinhIntegral[d*x]))/(8*a^(5/2))
Time = 1.60 (sec) , antiderivative size = 500, 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.
| \(\displaystyle \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 5816 |
| \(\displaystyle \int \left (-\frac {b \cosh (c+d x)}{a^2 \left (a+b x^2\right )}+\frac {\cosh (c+d x)}{a^2 x^2}-\frac {b \cosh (c+d x)}{a \left (a+b x^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a^2}+\frac {d \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a^2}-\frac {d \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a^2}+\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a^2}+\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {b} \cosh (c+d x)}{4 a^2 \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d \sinh (c) \text {Chi}(d x)}{a^2}+\frac {d \cosh (c) \text {Shi}(d x)}{a^2}-\frac {\cosh (c+d x)}{a^2 x}-\frac {3 \sqrt {b} \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{5/2}}+\frac {3 \sqrt {b} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{5/2}}+\frac {3 \sqrt {b} \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{5/2}}+\frac {3 \sqrt {b} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{5/2}}\) |
-(Cosh[c + d*x]/(a^2*x)) + (Sqrt[b]*Cosh[c + d*x])/(4*a^2*(Sqrt[-a] - Sqrt [b]*x)) - (Sqrt[b]*Cosh[c + d*x])/(4*a^2*(Sqrt[-a] + Sqrt[b]*x)) - (3*Sqrt [b]*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x ])/(4*(-a)^(5/2)) + (3*Sqrt[b]*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral [(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*(-a)^(5/2)) + (d*CoshIntegral[d*x]*Sinh[c ])/a^2 + (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d) /Sqrt[b]])/(4*a^2) + (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(4*a^2) + (d*Cosh[c]*SinhIntegral[d*x])/a^2 - (d*Co sh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4* a^2) + (3*Sqrt[b]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d) /Sqrt[b] - d*x])/(4*(-a)^(5/2)) + (d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIn tegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*a^2) + (3*Sqrt[b]*Sinh[c - (Sqrt[-a ]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*(-a)^(5/2))
3.1.71.3.1 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 = 0.32 (sec) , antiderivative size = 595, normalized size of antiderivative = 1.19
| method | result | size |
| risch | \(-\frac {3 \,{\mathrm e}^{-d x -c} x \,d^{2} b}{4 a^{2} \left (b \,d^{2} x^{2}+a \,d^{2}\right )}-\frac {{\mathrm e}^{-d x -c} d^{2}}{2 a \left (b \,d^{2} x^{2}+a \,d^{2}\right ) x}+\frac {d \,{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{8 a^{2}}+\frac {d \,{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{8 a^{2}}+\frac {3 \,{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) b}{8 a^{2} \sqrt {-a b}}-\frac {3 \,{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) b}{8 a^{2} \sqrt {-a b}}+\frac {d \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right )}{2 a^{2}}-\frac {3 \,{\mathrm e}^{d x +c} x \,d^{2} b}{4 a^{2} \left (b \,d^{2} x^{2}+a \,d^{2}\right )}-\frac {{\mathrm e}^{d x +c} d^{2}}{2 a \left (b \,d^{2} x^{2}+a \,d^{2}\right ) x}-\frac {d \,{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{8 a^{2}}-\frac {d \,{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{8 a^{2}}+\frac {3 \,{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) b}{8 a^{2} \sqrt {-a b}}-\frac {3 \,{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) b}{8 a^{2} \sqrt {-a b}}-\frac {d \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right )}{2 a^{2}}\) | \(595\) |
-3/4*exp(-d*x-c)/a^2/(b*d^2*x^2+a*d^2)*x*d^2*b-1/2*exp(-d*x-c)/a/(b*d^2*x^ 2+a*d^2)/x*d^2+1/8*d/a^2*exp(-(d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2 )-(d*x+c)*b+c*b)/b)+1/8*d/a^2*exp(-(-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b) ^(1/2)+(d*x+c)*b-c*b)/b)+3/8/a^2/(-a*b)^(1/2)*exp(-(d*(-a*b)^(1/2)+c*b)/b) *Ei(1,-(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*b-3/8/a^2/(-a*b)^(1/2)*exp(-(-d*( -a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*b+1/2*d/a^2*exp (-c)*Ei(1,d*x)-3/4*exp(d*x+c)/a^2/(b*d^2*x^2+a*d^2)*x*d^2*b-1/2*exp(d*x+c) /a/(b*d^2*x^2+a*d^2)/x*d^2-1/8*d/a^2*exp((d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*( -a*b)^(1/2)-(d*x+c)*b+c*b)/b)-1/8*d/a^2*exp((-d*(-a*b)^(1/2)+c*b)/b)*Ei(1, -(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)+3/8/a^2/(-a*b)^(1/2)*exp((d*(-a*b)^(1/2 )+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*b-3/8/a^2/(-a*b)^(1/2)*ex p((-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*b-1/2*d /a^2*exp(c)*Ei(1,-d*x)
Leaf count of result is larger than twice the leaf count of optimal. 1310 vs. \(2 (389) = 778\).
Time = 0.29 (sec) , antiderivative size = 1310, normalized size of antiderivative = 2.62 \[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]
-1/8*(4*(3*a*b*d*x^2 + 2*a^2*d)*cosh(d*x + c) - (((a*b*d^2*x^3 + a^2*d^2*x )*cosh(d*x + c)^2 - (a*b*d^2*x^3 + a^2*d^2*x)*sinh(d*x + c)^2 + 3*((b^2*x^ 3 + a*b*x)*cosh(d*x + c)^2 - (b^2*x^3 + a*b*x)*sinh(d*x + c)^2)*sqrt(-a*d^ 2/b))*Ei(d*x - sqrt(-a*d^2/b)) - ((a*b*d^2*x^3 + a^2*d^2*x)*cosh(d*x + c)^ 2 - (a*b*d^2*x^3 + a^2*d^2*x)*sinh(d*x + c)^2 - 3*((b^2*x^3 + a*b*x)*cosh( d*x + c)^2 - (b^2*x^3 + a*b*x)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(-d*x + sqrt(-a*d^2/b)))*cosh(c + sqrt(-a*d^2/b)) - 4*((a*b*d^2*x^3 + a^2*d^2*x)*E i(d*x) - (a*b*d^2*x^3 + a^2*d^2*x)*Ei(-d*x))*cosh(c) - (((a*b*d^2*x^3 + a^ 2*d^2*x)*cosh(d*x + c)^2 - (a*b*d^2*x^3 + a^2*d^2*x)*sinh(d*x + c)^2 - 3*( (b^2*x^3 + a*b*x)*cosh(d*x + c)^2 - (b^2*x^3 + a*b*x)*sinh(d*x + c)^2)*sqr t(-a*d^2/b))*Ei(d*x + sqrt(-a*d^2/b)) - ((a*b*d^2*x^3 + a^2*d^2*x)*cosh(d* x + c)^2 - (a*b*d^2*x^3 + a^2*d^2*x)*sinh(d*x + c)^2 + 3*((b^2*x^3 + a*b*x )*cosh(d*x + c)^2 - (b^2*x^3 + a*b*x)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei( -d*x - sqrt(-a*d^2/b)))*cosh(-c + sqrt(-a*d^2/b)) - (((a*b*d^2*x^3 + a^2*d ^2*x)*cosh(d*x + c)^2 - (a*b*d^2*x^3 + a^2*d^2*x)*sinh(d*x + c)^2 + 3*((b^ 2*x^3 + a*b*x)*cosh(d*x + c)^2 - (b^2*x^3 + a*b*x)*sinh(d*x + c)^2)*sqrt(- a*d^2/b))*Ei(d*x - sqrt(-a*d^2/b)) + ((a*b*d^2*x^3 + a^2*d^2*x)*cosh(d*x + c)^2 - (a*b*d^2*x^3 + a^2*d^2*x)*sinh(d*x + c)^2 - 3*((b^2*x^3 + a*b*x)*c osh(d*x + c)^2 - (b^2*x^3 + a*b*x)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(-d* x + sqrt(-a*d^2/b)))*sinh(c + sqrt(-a*d^2/b)) - 4*((a*b*d^2*x^3 + a^2*d...
\[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^2\right )^2} \, dx=\int \frac {\cosh {\left (c + d x \right )}}{x^{2} \left (a + b x^{2}\right )^{2}}\, dx \]
\[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^2\right )^2} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{2} x^{2}} \,d x } \]
\[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^2\right )^2} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{2} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^2\right )^2} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )}{x^2\,{\left (b\,x^2+a\right )}^2} \,d x \]