Integrand size = 17, antiderivative size = 512 \[ \int \frac {x \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=-\frac {\cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {d^2 \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a b^2}-\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a b^2}+\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d \sinh (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \sinh (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {d^2 \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a b^2}+\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a b^2} \]
-1/4*cosh(d*x+c)/b/(b*x^2+a)^2-1/16*d^2*Chi(d*x+d*(-a)^(1/2)/b^(1/2))*cosh (c-d*(-a)^(1/2)/b^(1/2))/a/b^2-1/16*d^2*Chi(-d*x+d*(-a)^(1/2)/b^(1/2))*cos h(c+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)^(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)^(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)^(3/2)/b^(3/2)-1/16*d^2* Shi(d*x+d*(-a)^(1/2)/b^(1/2))*sinh(c-d*(-a)^(1/2)/b^(1/2))/a/b^2-1/16*d*Ch i(-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*Shi(d*x-d*(-a)^(1/2)/b^(1/2))*sinh(c+d*(-a)^(1/2)/b^(1/2))/a/b ^2-1/16*d*sinh(d*x+c)/a/b^(3/2)/((-a)^(1/2)-x*b^(1/2))+1/16*d*sinh(d*x+c)/ a/b^(3/2)/((-a)^(1/2)+x*b^(1/2))
Result contains complex when optimal does not.
Time = 1.02 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.63 \[ \int \frac {x \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {-d e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (i \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\left (-i \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )-d e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (i \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\left (-i \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )+\frac {4 \sqrt {a} b \cosh (d x) \left (-2 a \cosh (c)+d x \left (a+b x^2\right ) \sinh (c)\right )}{\left (a+b x^2\right )^2}+\frac {4 \sqrt {a} b \left (d x \left (a+b x^2\right ) \cosh (c)-2 a \sinh (c)\right ) \sinh (d x)}{\left (a+b x^2\right )^2}}{32 a^{3/2} b^2} \]
(-(d*E^(c - (I*Sqrt[a]*d)/Sqrt[b])*((I*Sqrt[b] + Sqrt[a]*d)*E^(((2*I)*Sqrt [a]*d)/Sqrt[b])*ExpIntegralEi[d*(((-I)*Sqrt[a])/Sqrt[b] + x)] + ((-I)*Sqrt [b] + Sqrt[a]*d)*ExpIntegralEi[d*((I*Sqrt[a])/Sqrt[b] + x)])) - d*E^(-c - (I*Sqrt[a]*d)/Sqrt[b])*((I*Sqrt[b] + Sqrt[a]*d)*E^(((2*I)*Sqrt[a]*d)/Sqrt[ b])*ExpIntegralEi[((-I)*Sqrt[a]*d)/Sqrt[b] - d*x] + ((-I)*Sqrt[b] + Sqrt[a ]*d)*ExpIntegralEi[(I*Sqrt[a]*d)/Sqrt[b] - d*x]) + (4*Sqrt[a]*b*Cosh[d*x]* (-2*a*Cosh[c] + d*x*(a + b*x^2)*Sinh[c]))/(a + b*x^2)^2 + (4*Sqrt[a]*b*(d* x*(a + b*x^2)*Cosh[c] - 2*a*Sinh[c])*Sinh[d*x])/(a + b*x^2)^2)/(32*a^(3/2) *b^2)
Time = 1.22 (sec) , antiderivative size = 507, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {5812, 5803, 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.
| \(\displaystyle \int \frac {x \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 5812 |
| \(\displaystyle \frac {d \int \frac {\sinh (c+d x)}{\left (b x^2+a\right )^2}dx}{4 b}-\frac {\cosh (c+d x)}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 5803 |
| \(\displaystyle \frac {d \int \left (-\frac {b \sinh (c+d x)}{2 a \left (-b^2 x^2-a b\right )}-\frac {b \sinh (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {b \sinh (c+d x)}{4 a \left (b x+\sqrt {-a} \sqrt {b}\right )^2}\right )dx}{4 b}-\frac {\cosh (c+d x)}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \left (-\frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {d \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a b}-\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a b}+\frac {d \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a b}-\frac {d \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a b}+\frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {\sinh (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sinh (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}\right )}{4 b}-\frac {\cosh (c+d x)}{4 b \left (a+b x^2\right )^2}\) |
-1/4*Cosh[c + d*x]/(b*(a + b*x^2)^2) + (d*(-1/4*(d*Cosh[c + (Sqrt[-a]*d)/S qrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(a*b) - (d*Cosh[c - (Sqr t[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*a*b) + (Cos hIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])/(4*( -a)^(3/2)*Sqrt[b]) - (CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (S qrt[-a]*d)/Sqrt[b]])/(4*(-a)^(3/2)*Sqrt[b]) - Sinh[c + d*x]/(4*a*Sqrt[b]*( Sqrt[-a] - Sqrt[b]*x)) + Sinh[c + d*x]/(4*a*Sqrt[b]*(Sqrt[-a] + Sqrt[b]*x) ) + (Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d* x])/(4*(-a)^(3/2)*Sqrt[b]) + (d*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegra l[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*a*b) + (Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*S inhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*(-a)^(3/2)*Sqrt[b]) - (d*Sinh[ c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*a*b )))/(4*b)
3.1.74.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*Sinh[(c_.) + (d_.)*(x_)], x_Symbol] :> In t[ExpandIntegrand[Sinh[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[Cosh[(c_.) + (d_.)*(x_)]*((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p _), x_Symbol] :> Simp[e^m*(a + b*x^n)^(p + 1)*(Cosh[c + d*x]/(b*n*(p + 1))) , x] - Simp[d*(e^m/(b*n*(p + 1))) Int[(a + b*x^n)^(p + 1)*Sinh[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])
Leaf count of result is larger than twice the leaf count of optimal. \(1502\) vs. \(2(398)=796\).
Time = 0.31 (sec) , antiderivative size = 1503, normalized size of antiderivative = 2.94
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*b^2*d*x^2-2*(-a*b)^(1/2)*exp(-d*x-c)*a*b*d*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^2*d*x^2+exp(-(d*(-a*b )^(1/2)+c*b)/b)*(-a*b)^(1/2)*Ei(1,-(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*b^2*d ^2*x^4+(-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^2*d^2*x^4-4*(-a*b)^(1/2)*exp(-d*x-c)*a*b+(-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^2*d^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^2*d^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+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+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*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)*b^3*d*x^4-2*(-a*b)^(1/2)*exp(-d*x-c)*b^2*d*x^3+2*exp(-(d *(-a*b)^(1/2)+c*b)/b)*(-a*b)^(1/2)*Ei(1,-(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b) *a*b*d^2*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^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)*b^3*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)*b^3*d*x^4+2*(-a*b)^(1/2)*exp(d*x+c)*b^2*d *x^3+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^2*x^2+2*(-a*b)^(1/2)*exp((-d*(-a*b)^(1/2)+c*b)/b)*Ei...
Leaf count of result is larger than twice the leaf count of optimal. 1607 vs. \(2 (399) = 798\).
Time = 0.26 (sec) , antiderivative size = 1607, normalized size of antiderivative = 3.14 \[ \int \frac {x \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\text {Too large to display} \]
-1/32*(8*a^2*b*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 + ((b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*cosh(d*x + c)^2 - (b^3*x^4 + 2*a *b^2*x^2 + a^2*b)*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 - ((b^3*x^4 + 2*a*b^2 *x^2 + a^2*b)*cosh(d*x + c)^2 - (b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*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^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 - ((b^3*x^4 + 2*a*b^2 *x^2 + a^2*b)*cosh(d*x + c)^2 - (b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*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 + ((b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*cosh(d*x + c)^ 2 - (b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*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^2*d*x^3 + a^2*b*d *x)*sinh(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 + (( b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*cosh(d*x + c)^2 - (b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x - sqrt(-a*d^2/b)) - ((a*...
Timed out. \[ \int \frac {x \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {x \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\int { \frac {x \cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3}} \,d x } \]
1/2*(x*e^(d*x + 2*c) - x*e^(-d*x))/(b^3*d*x^6*e^c + 3*a*b^2*d*x^4*e^c + 3* a^2*b*d*x^2*e^c + a^3*d*e^c) + 1/2*integrate((5*b*x^2*e^c - a*e^c)*e^(d*x) /(b^4*d*x^8 + 4*a*b^3*d*x^6 + 6*a^2*b^2*d*x^4 + 4*a^3*b*d*x^2 + a^4*d), x) - 1/2*integrate((5*b*x^2 - a)*e^(-d*x)/(b^4*d*x^8*e^c + 4*a*b^3*d*x^6*e^c + 6*a^2*b^2*d*x^4*e^c + 4*a^3*b*d*x^2*e^c + a^4*d*e^c), x)
\[ \int \frac {x \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\int { \frac {x \cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {x \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\int \frac {x\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (b\,x^2+a\right )}^3} \,d x \]