Integrand size = 19, antiderivative size = 746 \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=-\frac {\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {\cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {d^2 \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 \sqrt {-a} b^{5/2}}+\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 \sqrt {-a} b^{5/2}}-\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^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 )}{16 a b^2}-\frac {d \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}+\frac {d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a b^2}+\frac {\sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d^2 \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 \sqrt {-a} b^{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 )}{16 a b^2}+\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 \sqrt {-a} b^{5/2}} \]
-1/4*x*cosh(d*x+c)/b/(b*x^2+a)^2+1/16*Chi(d*x+d*(-a)^(1/2)/b^(1/2))*cosh(c -d*(-a)^(1/2)/b^(1/2))/(-a)^(3/2)/b^(3/2)-1/16*Chi(-d*x+d*(-a)^(1/2)/b^(1/ 2))*cosh(c+d*(-a)^(1/2)/b^(1/2))/(-a)^(3/2)/b^(3/2)-1/16*d*cosh(c+d*(-a)^( 1/2)/b^(1/2))*Shi(d*x-d*(-a)^(1/2)/b^(1/2))/a/b^2-1/16*d*cosh(c-d*(-a)^(1/ 2)/b^(1/2))*Shi(d*x+d*(-a)^(1/2)/b^(1/2))/a/b^2-1/8*d*sinh(d*x+c)/b^2/(b*x ^2+a)-1/16*d*Chi(d*x+d*(-a)^(1/2)/b^(1/2))*sinh(c-d*(-a)^(1/2)/b^(1/2))/a/ b^2+1/16*Shi(d*x+d*(-a)^(1/2)/b^(1/2))*sinh(c-d*(-a)^(1/2)/b^(1/2))/(-a)^( 3/2)/b^(3/2)-1/16*d*Chi(-d*x+d*(-a)^(1/2)/b^(1/2))*sinh(c+d*(-a)^(1/2)/b^( 1/2))/a/b^2-1/16*Shi(d*x-d*(-a)^(1/2)/b^(1/2))*sinh(c+d*(-a)^(1/2)/b^(1/2) )/(-a)^(3/2)/b^(3/2)-1/16*d^2*Chi(d*x+d*(-a)^(1/2)/b^(1/2))*cosh(c-d*(-a)^ (1/2)/b^(1/2))/b^(5/2)/(-a)^(1/2)+1/16*d^2*Chi(-d*x+d*(-a)^(1/2)/b^(1/2))* cosh(c+d*(-a)^(1/2)/b^(1/2))/b^(5/2)/(-a)^(1/2)-1/16*d^2*Shi(d*x+d*(-a)^(1 /2)/b^(1/2))*sinh(c-d*(-a)^(1/2)/b^(1/2))/b^(5/2)/(-a)^(1/2)+1/16*d^2*Shi( d*x-d*(-a)^(1/2)/b^(1/2))*sinh(c+d*(-a)^(1/2)/b^(1/2))/b^(5/2)/(-a)^(1/2)- 1/16*cosh(d*x+c)/a/b^(3/2)/((-a)^(1/2)-x*b^(1/2))+1/16*cosh(d*x+c)/a/b^(3/ 2)/((-a)^(1/2)+x*b^(1/2))
Result contains complex when optimal does not.
Time = 2.73 (sec) , antiderivative size = 380, normalized size of antiderivative = 0.51 \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {-i e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (b-i \sqrt {a} \sqrt {b} d+a d^2\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )-\left (b+i \sqrt {a} \sqrt {b} d+a d^2\right ) \operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )+e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (i b+\sqrt {a} \sqrt {b} d+i a d^2\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )-i \left (b+i \sqrt {a} \sqrt {b} d+a d^2\right ) \operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )-\frac {4 \sqrt {a} \sqrt {b} \cosh (d x) \left (-b x \left (-a+b x^2\right ) \cosh (c)+a d \left (a+b x^2\right ) \sinh (c)\right )}{\left (a+b x^2\right )^2}-\frac {4 \sqrt {a} \sqrt {b} \left (a d \left (a+b x^2\right ) \cosh (c)+b x \left (a-b x^2\right ) \sinh (c)\right ) \sinh (d x)}{\left (a+b x^2\right )^2}}{32 a^{3/2} b^{5/2}} \]
((-I)*E^(c - (I*Sqrt[a]*d)/Sqrt[b])*((b - I*Sqrt[a]*Sqrt[b]*d + a*d^2)*E^( ((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[d*(((-I)*Sqrt[a])/Sqrt[b] + x)] - (b + I*Sqrt[a]*Sqrt[b]*d + a*d^2)*ExpIntegralEi[d*((I*Sqrt[a])/Sqrt[b] + x)]) + E^(-c - (I*Sqrt[a]*d)/Sqrt[b])*((I*b + Sqrt[a]*Sqrt[b]*d + I*a*d^2) *E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[((-I)*Sqrt[a]*d)/Sqrt[b] - d* x] - I*(b + I*Sqrt[a]*Sqrt[b]*d + a*d^2)*ExpIntegralEi[(I*Sqrt[a]*d)/Sqrt[ b] - d*x]) - (4*Sqrt[a]*Sqrt[b]*Cosh[d*x]*(-(b*x*(-a + b*x^2)*Cosh[c]) + a *d*(a + b*x^2)*Sinh[c]))/(a + b*x^2)^2 - (4*Sqrt[a]*Sqrt[b]*(a*d*(a + b*x^ 2)*Cosh[c] + b*x*(a - b*x^2)*Sinh[c])*Sinh[d*x])/(a + b*x^2)^2)/(32*a^(3/2 )*b^(5/2))
Time = 2.15 (sec) , antiderivative size = 759, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5814, 5804, 2009, 5811, 5804, 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.
| \(\displaystyle \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 5814 |
| \(\displaystyle \frac {d \int \frac {x \sinh (c+d x)}{\left (b x^2+a\right )^2}dx}{4 b}+\frac {\int \frac {\cosh (c+d x)}{\left (b x^2+a\right )^2}dx}{4 b}-\frac {x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 5804 |
| \(\displaystyle \frac {\int \left (-\frac {b \cosh (c+d x)}{2 a \left (-b^2 x^2-a b\right )}-\frac {b \cosh (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {b \cosh (c+d x)}{4 a \left (b x+\sqrt {-a} \sqrt {b}\right )^2}\right )dx}{4 b}+\frac {d \int \frac {x \sinh (c+d x)}{\left (b x^2+a\right )^2}dx}{4 b}-\frac {x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \int \frac {x \sinh (c+d x)}{\left (b x^2+a\right )^2}dx}{4 b}+\frac {-\frac {d \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a b}-\frac {d \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a b}-\frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {d \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a b}-\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a b}-\frac {\cosh (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\cosh (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}}{4 b}-\frac {x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 5811 |
| \(\displaystyle \frac {d \left (\frac {d \int \frac {\cosh (c+d x)}{b x^2+a}dx}{2 b}-\frac {\sinh (c+d x)}{2 b \left (a+b x^2\right )}\right )}{4 b}+\frac {-\frac {d \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a b}-\frac {d \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a b}-\frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {d \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a b}-\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a b}-\frac {\cosh (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\cosh (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}}{4 b}-\frac {x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 5804 |
| \(\displaystyle \frac {d \left (\frac {d \int \left (\frac {\sqrt {-a} \cosh (c+d x)}{2 a \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {-a} \cosh (c+d x)}{2 a \left (\sqrt {b} x+\sqrt {-a}\right )}\right )dx}{2 b}-\frac {\sinh (c+d x)}{2 b \left (a+b x^2\right )}\right )}{4 b}+\frac {-\frac {d \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a b}-\frac {d \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a b}-\frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {d \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a b}-\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a b}-\frac {\cosh (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\cosh (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}}{4 b}-\frac {x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}\right )}{2 b}-\frac {\sinh (c+d x)}{2 b \left (a+b x^2\right )}\right )}{4 b}+\frac {-\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 b}-\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 b}-\frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}+\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 b}-\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 b}-\frac {\cosh (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\cosh (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}}{4 b}-\frac {x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}\) |
-1/4*(x*Cosh[c + d*x])/(b*(a + b*x^2)^2) + (-1/4*Cosh[c + d*x]/(a*Sqrt[b]* (Sqrt[-a] - Sqrt[b]*x)) + Cosh[c + d*x]/(4*a*Sqrt[b]*(Sqrt[-a] + Sqrt[b]*x )) - (Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d *x])/(4*(-a)^(3/2)*Sqrt[b]) + (Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral [(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*(-a)^(3/2)*Sqrt[b]) - (d*CoshIntegral[(Sq rt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])/(4*a*b) - (d*Cosh Integral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(4*a* b) + (d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*a*b) + (Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d) /Sqrt[b] - d*x])/(4*(-a)^(3/2)*Sqrt[b]) - (d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b] ]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*a*b) + (Sinh[c - (Sqrt[-a]* d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*(-a)^(3/2)*Sqrt[b ]))/(4*b) + (d*(-1/2*Sinh[c + d*x]/(b*(a + b*x^2)) + (d*((Cosh[c + (Sqrt[- a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*Sqrt[-a]*Sqrt[ b]) - (Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*Sqrt[-a]*Sqrt[b]) - (Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[ (Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*Sqrt[-a]*Sqrt[b]) - (Sinh[c - (Sqrt[-a]*d) /Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*Sqrt[-a]*Sqrt[b]))) /(2*b)))/(4*b)
3.1.73.3.1 Defintions of rubi rules used
Int[Cosh[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> In t[ExpandIntegrand[Cosh[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d }, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sinh[(c_.) + (d_.)*(x_ )], x_Symbol] :> Simp[e^m*(a + b*x^n)^(p + 1)*(Sinh[c + d*x]/(b*n*(p + 1))) , x] - Simp[d*(e^m/(b*n*(p + 1))) Int[(a + b*x^n)^(p + 1)*Cosh[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IntegerQ[p] && EqQ[m - n + 1, 0] && LtQ[p, -1] && (IntegerQ[n] || GtQ[e, 0])
Int[Cosh[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Sy mbol] :> Simp[x^(m - n + 1)*(a + b*x^n)^(p + 1)*(Cosh[c + d*x]/(b*n*(p + 1) )), x] + (-Simp[(m - n + 1)/(b*n*(p + 1)) Int[x^(m - n)*(a + b*x^n)^(p + 1)*Cosh[c + d*x], x], x] - Simp[d/(b*n*(p + 1)) Int[x^(m - n + 1)*(a + b* x^n)^(p + 1)*Sinh[c + d*x], x], x]) /; FreeQ[{a, b, c, d}, x] && ILtQ[p, -1 ] && IGtQ[n, 0] && RationalQ[m] && (GtQ[m - n + 1, 0] || GtQ[n, 2])
Leaf count of result is larger than twice the leaf count of optimal. \(2171\) vs. \(2(574)=1148\).
Time = 0.36 (sec) , antiderivative size = 2172, normalized size of antiderivative = 2.91
1/32/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)*a^2*b*d^2*x^2-2*exp(-(d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)-(d *x+c)*b+c*b)/b)*(-a*b)^(1/2)*a*b*d*x^2-2*(-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)*a*b*d*x^2-2*exp(d*x+c)*a* b*d*x^2*(-a*b)^(1/2)+2*exp(-d*x-c)*a*b*d*x^2*(-a*b)^(1/2)+2*b^2*x^3*(-a*b) ^(1/2)*exp(d*x+c)-exp((d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)-(d*x+c) *b+c*b)/b)*a*b^2*d^2*x^4+exp((-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2 )+(d*x+c)*b-c*b)/b)*a*b^2*d^2*x^4+exp((d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a* b)^(1/2)-(d*x+c)*b+c*b)/b)*(-a*b)^(1/2)*b^2*d*x^4+exp((-d*(-a*b)^(1/2)+c*b )/b)*Ei(1,-(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*(-a*b)^(1/2)*b^2*d*x^4-2*exp( (d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*a^2*b*d^2*x ^2+2*b^2*x^3*(-a*b)^(1/2)*exp(-d*x-c)+2*exp((d*(-a*b)^(1/2)+c*b)/b)*Ei(1,( d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*(-a*b)^(1/2)*a*b*d*x^2+2*exp((-d*(-a*b)^( 1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*(-a*b)^(1/2)*a*b*d*x^ 2-2*exp(-(d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*a *b^2*x^2+2*exp(-(-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)+(d*x+c)*b-c* b)/b)*a*b^2*x^2-exp(-(d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)-(d*x+c) *b+c*b)/b)*(-a*b)^(1/2)*a^2*d-(-a*b)^(1/2)*exp(-(-d*(-a*b)^(1/2)+c*b)/b)*E i(1,(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*a^2*d-2*exp(d*x+c)*a^2*d*(-a*b)^(1/2 )+2*exp(-d*x-c)*a^2*d*(-a*b)^(1/2)-exp((d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*...
Leaf count of result is larger than twice the leaf count of optimal. 2047 vs. \(2 (575) = 1150\).
Time = 0.28 (sec) , antiderivative size = 2047, normalized size of antiderivative = 2.74 \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\text {Too large to display} \]
1/32*(4*(a*b^2*d*x^3 - a^2*b*d*x)*cosh(d*x + c) - (((a*b^2*d^2*x^4 + 2*a^2 *b*d^2*x^2 + a^3*d^2)*cosh(d*x + c)^2 - (a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*sinh(d*x + c)^2 + ((a^3*d^2 + (a*b^2*d^2 + b^3)*x^4 + a^2*b + 2* (a^2*b*d^2 + a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3*d^2 + (a*b^2*d^2 + b^3)*x^ 4 + a^2*b + 2*(a^2*b*d^2 + a*b^2)*x^2)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei (d*x - sqrt(-a*d^2/b)) - ((a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*cosh (d*x + c)^2 - (a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*sinh(d*x + c)^2 - ((a^3*d^2 + (a*b^2*d^2 + b^3)*x^4 + a^2*b + 2*(a^2*b*d^2 + a*b^2)*x^2)*c osh(d*x + c)^2 - (a^3*d^2 + (a*b^2*d^2 + b^3)*x^4 + a^2*b + 2*(a^2*b*d^2 + a*b^2)*x^2)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(-d*x + sqrt(-a*d^2/b)))*c osh(c + sqrt(-a*d^2/b)) - (((a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*co sh(d*x + c)^2 - (a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*sinh(d*x + c)^ 2 - ((a^3*d^2 + (a*b^2*d^2 + b^3)*x^4 + a^2*b + 2*(a^2*b*d^2 + a*b^2)*x^2) *cosh(d*x + c)^2 - (a^3*d^2 + (a*b^2*d^2 + b^3)*x^4 + a^2*b + 2*(a^2*b*d^2 + a*b^2)*x^2)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x + sqrt(-a*d^2/b)) - ((a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*cosh(d*x + c)^2 - (a*b^2*d^2 *x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*sinh(d*x + c)^2 + ((a^3*d^2 + (a*b^2*d^2 + b^3)*x^4 + a^2*b + 2*(a^2*b*d^2 + a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3*d^ 2 + (a*b^2*d^2 + b^3)*x^4 + a^2*b + 2*(a^2*b*d^2 + a*b^2)*x^2)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(-d*x - sqrt(-a*d^2/b)))*cosh(-c + sqrt(-a*d^2/...
Timed out. \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\int { \frac {x^{2} \cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3}} \,d x } \]
1/2*((d*x^2*e^(2*c) + 4*x*e^(2*c))*e^(d*x) - (d*x^2 - 4*x)*e^(-d*x))/(b^3* d^2*x^6*e^c + 3*a*b^2*d^2*x^4*e^c + 3*a^2*b*d^2*x^2*e^c + a^3*d^2*e^c) + 1 /2*integrate(-2*(3*a*d*x*e^c - 10*b*x^2*e^c + 2*a*e^c)*e^(d*x)/(b^4*d^2*x^ 8 + 4*a*b^3*d^2*x^6 + 6*a^2*b^2*d^2*x^4 + 4*a^3*b*d^2*x^2 + a^4*d^2), x) + 1/2*integrate(2*(3*a*d*x + 10*b*x^2 - 2*a)*e^(-d*x)/(b^4*d^2*x^8*e^c + 4* a*b^3*d^2*x^6*e^c + 6*a^2*b^2*d^2*x^4*e^c + 4*a^3*b*d^2*x^2*e^c + a^4*d^2* e^c), x)
\[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\int { \frac {x^{2} \cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\int \frac {x^2\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (b\,x^2+a\right )}^3} \,d x \]